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## [Axiom-developer] 20080427.01.tpd.patch (CATS Schaums-Axiom equivalence

 From: daly Subject: [Axiom-developer] 20080427.01.tpd.patch (CATS Schaums-Axiom equivalence testing (29-34)) Date: Mon, 28 Apr 2008 01:53:34 -0500

  14:668 SCHAUMS AND AXIOM DIFFER (Axiom has closed form)
14:671 SCHAUMS AND AXIOM DIFFER (Axiom has closed form)
In these two problems Axiom constructs a closed form for the integral
whereas Schaums shows a series expansion. Axiom makes the claim that
if there exists a closed form for an integral in elementary terms then
Axiom will return that form. If it returns the original integral then
no closed form exists. Thus, Axiom is a decision procedure. In these
two cases, Axiom has constructed a closed form.

Also, in
14:661 Schaums and Axiom agree
I believe that my copy of Schaums has a typo and that instead of
integral acoth(x/a) = x*acoth(x)+a/2*log(x^2-a^2)
the correct result should be
integral acoth(x/a) = x*acoth(x/a)+a/2*log(x^2-a^2)
This has been fixed proactively but needs to be verified against a
later copy of Schaums. It appears that I've gotten so old that the
equations have changed since my college mathematics education :-)

At this point I have checked Axiom against all of the indefinite
integrals in Schaums. Frankly, I'm quite impressed with the Axiom
team (Trager, Bronstein, et.al.) that created this code (Note that
I did not author any of the integration code). By any measure this
is an impressive showing.

Of the forms that Axiom could not simplify I have two further approaches.
The first approach will be to check the answers using Maxima/MMA/Maple.
Note that I believe the problems are mostly simplification issues.

The second approach is to spend some time thinking about a better,
more general method for automating the simplification, including using
proviso forms to capture assumptions. I have a graph-based procedure
that I'll explain and codify in a different pamphlet.

Another point is that there were a couple simplifications that ended
up with x only in the imaginary portion of the answer. This indicates
to me that there is a sign assumption in the code (see 14.666, around
-S 124 where I expand this more fully in a clearer example). These
assumptions need to be pushed forward in provisos.

Much remains to be done with CATS but this exercise has left me with
the impression that Axiom's integration routines are rock solid and
can be relied on with confidence. The 4Ms could benefit from doing
the same exercise but I don't have the time available to do that yet.
Ideally it could be shown that all of the CASs are "plug-compatible",
giving known-good answers to published problems.

schaum29.input.pamphlet
14:590 Schaums and Axiom agree
14:591 Axiom cannot simplify this expression
14:592 Axiom cannot simplify this expression
14:593 Axiom cannot simplify this expression
14:594 Schaums and Axiom agree
14:595 Schaums and Axiom agree
14:596 Axiom cannot simplify this expression
14:597 Schaums and Axiom agree
14:598 Axiom cannot simplify this expression
14:599 Axiom cannot simplify this expression
14:600 Axiom cannot simplify this expression
14:601 Schaums and Axiom agree
14:602 Schaums and Axiom agree
14:603 Schaums and Axiom agree

schaum30.input.pamphlet
14:604 Schaums and Axiom differ by a constant
14:605 Schaums and Axiom differ by a constant
14:606 Axiom cannot simplify this expression
14:607 Axiom cannot simplify this expression
14:608 Schaums and Axiom agree
14:609 Schaums and Axiom differ by a constant
14:610 Axiom cannot compute this integral
14:611 Schaums and Axiom differ by a constant
14:612 Axiom cannot compute this integral
14:613 Schaums and Axiom differ by a constant
14:614 Axiom cannot compute this integral

schaum31.input.pamphlet
14:615 Schaums and Axiom differ by a constant
14:616 Schaums and Axiom differ by a constant
14:617 Axiom cannot simplify this expression
14:618 Schaums and Axiom agree
14:619 Schaums and Axiom agree
14:620 Schaums and Axiom differ by a constant
14:621 Axiom cannot compute this integral
14:622 Schaums and Axiom differ by a constant
14:623 Axiom cannot compute this integral
14:624 Schaums and Axiom differ by a constant
14:625 Axiom cannot compute this integral

schaum32.input.pamphlet
14:626 Schaums and Axiom agree
14:627 Schaums and Axiom differ by a constant
14:628 Axiom cannot simplify this expression
14:629 Schaums and Axiom agree
14:630 Schaums and Axiom agree
14:631 Axiom cannot compute this integral
14:632 Schaums and Axiom differ by a constant
14:633 Axiom cannot compute this integral
14:634 Schaums and Axiom agree
14:635 Axiom cannot compute this integral

schaum33.input.pamphlet
14:636 Schaums and Axiom agree
14:637 Axiom cannot simplify this expression
14:638 Axiom cannot simplify this expression
14:639 Schaums and Axiom agree
14:640 Schaums and Axiom agree
14:641 Axiom cannot compute this integral
14:642 Axiom cannot simplify this expression
14:643 Axiom cannot compute this integral
14:644 Schaums and Axiom differ by a constant
14:645 Axiom cannot compute this integral

schaum34.input.pamphlet
14:646 Schaums and Axiom agree
14:647 Schaums and Axiom agree
14:648 Schaums and Axiom agree
14:649 Axiom cannot compute this integral
14:650 Schaums and Axiom differ by a constant
14:651 Schaums and Axiom agree
14:652 Schaums and Axiom agree
14:653 Schaums and Axiom agree
14:654 Axiom cannot compute this integral
14:655 Axiom cannot simplify these expressions
14:656 Schaums and Axiom differ by a constant
14:657 Schaums and Axiom agree
14:658 Schaums and Axiom differ by a constant
14:659 Axiom cannot compute this integral
14:660 Schaums and Axiom agree
14:661 Schaums and Axiom agree
14:662 Schaums and Axiom agree
14:663 Schaums and Axiom agree
14:664 Axiom cannot compute this integral
14:665 Schaums and Axiom agree
14:666 Schaums and Axiom agree
14:667 Schaums and Axiom differ by a constant
14:668 SCHAUMS AND AXIOM DIFFER (Axiom has closed form)
14:669 Axiom cannot simplify these expressions
14:670 Axiom cannot simplify these expressions
14:671 SCHAUMS AND AXIOM DIFFER (Axiom has closed form)
14:672 Axiom cannot compute this integral
14:673 Axiom cannot compute this integral
14:674 Axiom cannot compute this integral
14:675 Axiom cannot compute this integral
14:676 Axiom cannot compute this integral
14:677 Axiom cannot compute this integral

===========================================================================
diff --git a/src/input/schaum28.input.pamphlet
b/src/input/schaum28.input.pamphlet
index 5c68b06..3ba0bfc 100644
--- a/src/input/schaum28.input.pamphlet
+++ b/src/input/schaum28.input.pamphlet
@@ -234,6 +234,10 @@ aa:=integrate(x/cosh(a*x),x)
\section{\cite{1}:14.569~~~~~$\displaystyle \int{\cosh^2{ax}}~dx$}
$$\int{\cosh^2{ax}}= +\frac{x}{2}+\frac{\sinh{ax}\cosh{ax}}{2a} +$$
+Note that the Schaums print edition (1968 printing 3) has a typo:
+$$\int{\cosh^2{ax}}= \frac{x}{2}+\frac{\sinh{ax}\cosh{ax}}{2}$$
<<*>>=
@@ -250,40 +254,21 @@ aa:=integrate(cosh(a*x)^2,x)
--E

--S 18
-bb:=x/2+(sinh(a*x)*cosh(a*x))/2
+bb:=x/2+(sinh(a*x)*cosh(a*x))/(2*a)
--R
---R        cosh(a x)sinh(a x) + x
---R   (2)  ----------------------
---R                   2
+--R        cosh(a x)sinh(a x) + a x
+--R   (2)  ------------------------
+--R                   2a
--R                                                     Type: Expression
Integer
--E

---S 19
+--S 19     14:569 Schaums and Axiom agree
cc:=aa-bb
--R
---R        (- a + 1)cosh(a x)sinh(a x)
---R   (3)  ---------------------------
---R                     2a
+--R   (3)  0
--R                                                     Type: Expression
Integer
--E

---S 20
-sinhcoshrule:=rule(sinh(x)*cosh(y) == 1/2*(sinh(x+y)+sinh(x-y)))
---R
---I                             %B sinh(y + x) - %B sinh(y - x)
---I   (4)  %B cosh(y)sinh(x) == -------------------------------
---R                                            2
---R                        Type: RewriteRule(Integer,Integer,Expression
Integer)
---E
-
---S 21     14:569 SCHAUMS AND AXIOM DISAGREE?
-dd:=sinhcoshrule cc
---R
---R        (- a + 1)sinh(2a x)
---R   (5)  -------------------
---R                 4a
---R                                                     Type: Expression
Integer
---E
@

\section{\cite{1}:14.570~~~~~$\displaystyle @@ -294,7 +279,7 @@ $$<<*>>= )clear all ---S 22 +--S 20 aa:=integrate(x*cosh(a*x)^2,x) --R --R @@ -306,7 +291,7 @@ aa:=integrate(x*cosh(a*x)^2,x) --R Type: Union(Expression Integer,...) --E ---S 23 +--S 21 bb:=x^2/4+(x*sinh(2*a*x))/(4*a)-cosh(2*a*x)/(8*a^2) --R --R 2 2 @@ -317,7 +302,7 @@ bb:=x^2/4+(x*sinh(2*a*x))/(4*a)-cosh(2*a*x)/(8*a^2) --R Type: Expression Integer --E ---S 24 +--S 22 cc:=aa-bb --R --R (3) @@ -332,7 +317,7 @@ cc:=aa-bb --R Type: Expression Integer --E ---S 25 +--S 23 sinhsqrrule:=rule(sinh(x)^2 == 1/2*cosh(2*x)-1/2) --R --R 2 cosh(2x) - 1 @@ -341,7 +326,7 @@ sinhsqrrule:=rule(sinh(x)^2 == 1/2*cosh(2*x)-1/2) --R Type: RewriteRule(Integer,Integer,Expression Integer) --E ---S 26 +--S 24 dd:=sinhsqrrule cc --R --R (5) @@ -353,7 +338,7 @@ dd:=sinhsqrrule cc --R Type: Expression Integer --E ---S 27 +--S 25 coshsqrrule:=rule(cosh(x)^2 == 1/2*cosh(2*x)+1/2) --R --R 2 cosh(2x) + 1 @@ -362,7 +347,7 @@ coshsqrrule:=rule(cosh(x)^2 == 1/2*cosh(2*x)+1/2) --R Type: RewriteRule(Integer,Integer,Expression Integer) --E ---S 28 +--S 26 ee:=coshsqrrule dd --R --R - x sinh(2a x) + 2x cosh(a x)sinh(a x) @@ -371,7 +356,7 @@ ee:=coshsqrrule dd --R Type: Expression Integer --E ---S 29 +--S 27 sinhcoshrule:=rule(sinh(x)*cosh(y) == 1/2*(sinh(x+y)+sinh(x-y))) --R --I %S sinh(y + x) - %S sinh(y - x) @@ -380,7 +365,7 @@ sinhcoshrule:=rule(sinh(x)*cosh(y) == 1/2*(sinh(x+y)+sinh(x-y))) --R Type: RewriteRule(Integer,Integer,Expression Integer) --E ---S 30 14:570 Schaums and Axiom agree +--S 28 14:570 Schaums and Axiom agree ff:=sinhcoshrule ee --R --R (9) 0 @@ -396,7 +381,7 @@$$ <<*>>= )clear all ---S 31 +--S 29 aa:=integrate(1/cosh(a*x)^2,x) --R --R @@ -407,7 +392,7 @@ aa:=integrate(1/cosh(a*x)^2,x) --R Type: Union(Expression Integer,...) --E ---S 32 +--S 30 bb:=tanh(a*x)/a --R --R tanh(a x) @@ -416,7 +401,7 @@ bb:=tanh(a*x)/a --R Type: Expression Integer --E ---S 33 +--S 31 cc:=aa-bb --R --R 2 2 @@ -425,155 +410,12 @@ cc:=aa-bb --R 2 2 --R a sinh(a x) + 2a cosh(a x)sinh(a x) + a cosh(a x) + a --R Type: Expression Integer ---E - ---S 34 -tanhrule:=rule(tanh(x) == sinh(x)/cosh(x)) ---R ---R sinh(x) ---R (4) tanh(x) == ------- ---R cosh(x) ---R Type: RewriteRule(Integer,Integer,Expression Integer) ---E - ---S 35 -dd:=tanhrule cc ---R ---R (5) ---R 3 2 2 ---R - sinh(a x) - 2cosh(a x)sinh(a x) + (- cosh(a x) - 1)sinh(a x) ---R + ---R - 2cosh(a x) ---R / ---R 2 2 3 ---R a cosh(a x)sinh(a x) + 2a cosh(a x) sinh(a x) + a cosh(a x) + a cosh(a x) ---R Type: Expression Integer ---E - ---S 36 -sinhsqrrule:=rule(sinh(x)^2 == 1/2*cosh(2*x)-1/2) ---R ---R 2 cosh(2x) - 1 ---R (6) sinh(x) == ------------ ---R 2 ---R Type: RewriteRule(Integer,Integer,Expression Integer) ---E - ---S 37 -ee:=sinhsqrrule dd ---R ---R (7) ---R 3 2 ---R - 2sinh(a x) + (- 2cosh(a x) - 2)sinh(a x) - 2cosh(a x)cosh(2a x) ---R + ---R - 2cosh(a x) ---R / ---R 2 3 ---R 4a cosh(a x) sinh(a x) + a cosh(a x)cosh(2a x) + 2a cosh(a x) ---R + ---R a cosh(a x) ---R Type: Expression Integer ---E - ---S 38 -coshsqrrule:=rule(cosh(x)^2 == 1/2*cosh(2*x)+1/2) ---R ---R 2 cosh(2x) + 1 ---R (8) cosh(x) == ------------ ---R 2 ---R Type: RewriteRule(Integer,Integer,Expression Integer) ---E - ---S 39 -ff:=coshsqrrule ee ---R ---R (9) ---R 3 ---R - 2sinh(a x) + (- cosh(2a x) - 3)sinh(a x) - 2cosh(a x)cosh(2a x) ---R + ---R - 2cosh(a x) ---R / ---R 3 ---R (2a cosh(2a x) + 2a)sinh(a x) + a cosh(a x)cosh(2a x) + 2a cosh(a x) ---R + ---R a cosh(a x) ---R Type: Expression Integer ---E - ---S 40 -sinhcuberule:=rule(sinh(x)^3 == 1/4*sinh(3*x)-3/4*sinh(x)) ---R ---R 3 sinh(3x) - 3sinh(x) ---R (10) sinh(x) == ------------------- ---R 4 ---R Type: RewriteRule(Integer,Integer,Expression Integer) ---E - ---S 41 -gg:=sinhcuberule ff ---R ---R (11) ---R - sinh(3a x) + (- 2cosh(2a x) - 3)sinh(a x) - 4cosh(a x)cosh(2a x) ---R + ---R - 4cosh(a x) ---R / ---R 3 ---R (4a cosh(2a x) + 4a)sinh(a x) + 2a cosh(a x)cosh(2a x) + 4a cosh(a x) ---R + ---R 2a cosh(a x) ---R Type: Expression Integer ---E - ---S 42 -coshcuberule:=rule(cosh(x)^3 == 1/4*cosh(3*x)-3/4*cosh(x)) ---R ---R 3 cosh(3x) - 3cosh(x) ---R (12) cosh(x) == ------------------- ---R 4 ---R Type: RewriteRule(Integer,Integer,Expression Integer) ---E - ---S 43 -hh:=coshcuberule gg ---R ---R (13) ---R - sinh(3a x) + (- 2cosh(2a x) - 3)sinh(a x) - 4cosh(a x)cosh(2a x) ---R + ---R - 4cosh(a x) ---R / ---R (4a cosh(2a x) + 4a)sinh(a x) + a cosh(3a x) + 2a cosh(a x)cosh(2a x) ---R + ---R - a cosh(a x) ---R Type: Expression Integer ---E - ---S 44 -coshcoshrule:=rule(cosh(x)*cosh(y)==1/2*(cosh(x+y)+cosh(x-y))) ---R ---I %U cosh(y + x) + %U cosh(y - x) ---I (14) %U cosh(x)cosh(y) == ------------------------------- ---R 2 ---R Type: RewriteRule(Integer,Integer,Expression Integer) ---E - ---S 45 -ii:=coshcoshrule hh ---R ---R (15) ---R - sinh(3a x) + (- 2cosh(2a x) - 3)sinh(a x) - 2cosh(3a x) - 6cosh(a x) ---R ---------------------------------------------------------------------- ---R (4a cosh(2a x) + 4a)sinh(a x) + 2a cosh(3a x) ---R Type: Expression Integer ---E - ---S 46 14:571 SCHAUMS AND AXIOM DISAGREE? -jj:=complexNormalize ii +--S 32 +dd:=complexNormalize cc --R ---R a x 2 ---R - 2(%e ) - 2 ---R (16) --------------- ---R a x 2 ---R 2a (%e ) - a +--R 1 +--R (4) - - +--R a --R Type: Expression Integer --E @ @@ -586,7 +428,7 @@ $$<<*>>= )clear all ---S 47 +--S 33 aa:=integrate(cosh(a*x)*cosh(p*x),x) --R --R @@ -597,7 +439,7 @@ aa:=integrate(cosh(a*x)*cosh(p*x),x) --R Type: Union(Expression Integer,...) --E ---S 48 +--S 34 bb:=(sinh(a-p)*x)/(2*(a-p))+(sinh(a+p)*x)/(2*(a+p)) --R --R (p - a)x sinh(p + a) + (p + a)x sinh(p - a) @@ -607,7 +449,7 @@ bb:=(sinh(a-p)*x)/(2*(a-p))+(sinh(a+p)*x)/(2*(a+p)) --R Type: Expression Integer --E ---S 49 +--S 35 cc:=aa-bb --R --R (3) @@ -627,7 +469,7 @@ cc:=aa-bb --R Type: Expression Integer --E ---S 50 +--S 36 sinhsqrrule:=rule(sinh(x)^2 == 1/2*cosh(2*x)-1/2) --R --R 2 cosh(2x) - 1 @@ -636,7 +478,7 @@ sinhsqrrule:=rule(sinh(x)^2 == 1/2*cosh(2*x)-1/2) --R Type: RewriteRule(Integer,Integer,Expression Integer) --E ---S 51 +--S 37 dd:=sinhsqrrule cc --R --R (5) @@ -653,7 +495,7 @@ dd:=sinhsqrrule cc --R Type: Expression Integer --E ---S 52 +--S 38 coshsqrrule:=rule(cosh(x)^2 == 1/2*cosh(2*x)+1/2) --R --R 2 cosh(2x) + 1 @@ -662,7 +504,7 @@ coshsqrrule:=rule(cosh(x)^2 == 1/2*cosh(2*x)+1/2) --R Type: RewriteRule(Integer,Integer,Expression Integer) --E ---S 53 +--S 39 ee:=coshsqrrule dd --R --R (7) @@ -675,7 +517,7 @@ ee:=coshsqrrule dd --R Type: Expression Integer --E ---S 54 +--S 40 sinhcoshrule:=rule(sinh(x)*cosh(y) == 1/2*(sinh(x+y)+sinh(x-y))) --R --I %V sinh(y + x) - %V sinh(y - x) @@ -684,7 +526,7 @@ sinhcoshrule:=rule(sinh(x)*cosh(y) == 1/2*(sinh(x+y)+sinh(x-y))) --R Type: RewriteRule(Integer,Integer,Expression Integer) --E ---S 55 14:572 Schaums and Axiom agree +--S 41 14:572 Schaums and Axiom agree ff:=sinhcoshrule ee --R --R (9) @@ -706,7 +548,7 @@$$ <<*>>= )clear all ---S 56 +--S 42 aa:=integrate(cosh(a*x)*sin(p*x),x) --R --R @@ -724,7 +566,7 @@ aa:=integrate(cosh(a*x)*sin(p*x),x) --R Type: Union(Expression Integer,...) --E ---S 57 +--S 43 bb:=(a*sinh(a*x)*sin(p*x)-p*cosh(a*x)*cos(p*x))/(a^2+p^2) --R --R a sin(p x)sinh(a x) - p cos(p x)cosh(a x) @@ -734,7 +576,7 @@ bb:=(a*sinh(a*x)*sin(p*x)-p*cosh(a*x)*cos(p*x))/(a^2+p^2) --R Type: Expression Integer --E ---S 58 +--S 44 cc:=aa-bb --R --R (3) @@ -749,7 +591,7 @@ cc:=aa-bb --R Type: Expression Integer --E ---S 59 +--S 45 coshsqrrule:=rule(cosh(x)^2 == 1/2*cosh(2*x)+1/2) --R --R 2 cosh(2x) + 1 @@ -758,7 +600,7 @@ coshsqrrule:=rule(cosh(x)^2 == 1/2*cosh(2*x)+1/2) --R Type: RewriteRule(Integer,Integer,Expression Integer) --E ---S 60 +--S 46 dd:=coshsqrrule cc --R --R (5) @@ -772,7 +614,7 @@ dd:=coshsqrrule cc --R Type: Expression Integer --E ---S 61 +--S 47 sinhsqrrule:=rule(sinh(x)^2 == 1/2*cosh(2*x)-1/2) --R --R 2 cosh(2x) - 1 @@ -781,7 +623,7 @@ sinhsqrrule:=rule(sinh(x)^2 == 1/2*cosh(2*x)-1/2) --R Type: RewriteRule(Integer,Integer,Expression Integer) --E ---S 62 14:573 Schaums and Axiom agree +--S 48 14:573 Schaums and Axiom agree ee:=sinhsqrrule dd --R --R (7) 0 @@ -797,7 +639,7 @@ $$<<*>>= )clear all ---S 63 +--S 49 aa:=integrate(cosh(a*x)*cos(p*x),x) --R --R @@ -815,7 +657,7 @@ aa:=integrate(cosh(a*x)*cos(p*x),x) --R Type: Union(Expression Integer,...) --E ---S 64 +--S 50 bb:=(a*sinh(a*x)*cos(p*x)+p*cosh(a*x)*sin(p*x))/(a^2+p^2) --R --R a cos(p x)sinh(a x) + p cosh(a x)sin(p x) @@ -825,7 +667,7 @@ bb:=(a*sinh(a*x)*cos(p*x)+p*cosh(a*x)*sin(p*x))/(a^2+p^2) --R Type: Expression Integer --E ---S 65 +--S 51 cc:=aa-bb --R --R (3) @@ -840,7 +682,7 @@ cc:=aa-bb --R Type: Expression Integer --E ---S 66 +--S 52 coshsqrrule:=rule(cosh(x)^2 == 1/2*cosh(2*x)+1/2) --R --R 2 cosh(2x) + 1 @@ -849,7 +691,7 @@ coshsqrrule:=rule(cosh(x)^2 == 1/2*cosh(2*x)+1/2) --R Type: RewriteRule(Integer,Integer,Expression Integer) --E ---S 67 +--S 53 dd:=coshsqrrule cc --R --R (5) @@ -863,7 +705,7 @@ dd:=coshsqrrule cc --R Type: Expression Integer --E ---S 68 +--S 54 sinhsqrrule:=rule(sinh(x)^2 == 1/2*cosh(2*x)-1/2) --R --R 2 cosh(2x) - 1 @@ -872,7 +714,7 @@ sinhsqrrule:=rule(sinh(x)^2 == 1/2*cosh(2*x)-1/2) --R Type: RewriteRule(Integer,Integer,Expression Integer) --E ---S 69 14:574 Schaums and Axiom agree +--S 55 14:574 Schaums and Axiom agree ee:=sinhsqrrule dd --R --R (7) 0 @@ -888,7 +730,7 @@$$ <<*>>= )clear all ---S 70 +--S 56 aa:=integrate(1/(cosh(a*x)+1),x) --R --R @@ -898,7 +740,7 @@ aa:=integrate(1/(cosh(a*x)+1),x) --R Type: Union(Expression Integer,...) --E ---S 71 +--S 57 bb:=1/a*tanh((a*x)/2) --R --R a x @@ -909,7 +751,7 @@ bb:=1/a*tanh((a*x)/2) --R Type: Expression Integer --E ---S 72 +--S 58 cc:=aa-bb --R --R a x @@ -920,7 +762,7 @@ cc:=aa-bb --R Type: Expression Integer --E ---S 73 +--S 59 tanhrule:=rule(tanh(x) == sinh(x)/cosh(x)) --R --R sinh(x) @@ -929,7 +771,7 @@ tanhrule:=rule(tanh(x) == sinh(x)/cosh(x)) --R Type: RewriteRule(Integer,Integer,Expression Integer) --E ---S 74 +--S 60 dd:=tanhrule cc --R --R a x a x a x @@ -942,7 +784,7 @@ dd:=tanhrule cc --R Type: Expression Integer --E ---S 75 +--S 61 sinhcoshrule:=rule(sinh(x)*cosh(y) == 1/2*(sinh(x+y)+sinh(x-y))) --R --I %B sinh(y + x) - %B sinh(y - x) @@ -951,7 +793,7 @@ sinhcoshrule:=rule(sinh(x)*cosh(y) == 1/2*(sinh(x+y)+sinh(x-y))) --R Type: RewriteRule(Integer,Integer,Expression Integer) --E ---S 76 +--S 62 ee:=sinhcoshrule dd --R --R 3a x a x a x a x @@ -964,7 +806,7 @@ ee:=sinhcoshrule dd --R Type: Expression Integer --E ---S 77 +--S 63 sinhsinhrule:=rule(sinh(x)*sinh(y)==1/2*(cosh(x+y)-cosh(x-y))) --R --I %B cosh(y + x) - %B cosh(y - x) @@ -973,7 +815,7 @@ sinhsinhrule:=rule(sinh(x)*sinh(y)==1/2*(cosh(x+y)-cosh(x-y))) --R Type: RewriteRule(Integer,Integer,Expression Integer) --E ---S 78 +--S 64 ff:=sinhsinhrule ee --R --R 3a x a x 3a x a x @@ -986,7 +828,7 @@ ff:=sinhsinhrule ee --R Type: Expression Integer --E ---S 79 +--S 65 coshcoshrule:=rule(cosh(x)*cosh(y)==1/2*(cosh(x+y)+cosh(x-y))) --R --I %BC cosh(y + x) + %BC cosh(y - x) @@ -995,7 +837,7 @@ coshcoshrule:=rule(cosh(x)*cosh(y)==1/2*(cosh(x+y)+cosh(x-y))) --R Type: RewriteRule(Integer,Integer,Expression Integer) --E ---S 80 14:575 Schaums and Axiom differ by a constant +--S 66 14:575 Schaums and Axiom differ by a constant gg:=coshcoshrule ff --R --R 1 @@ -1013,7 +855,7 @@ $$<<*>>= )clear all ---S 81 +--S 67 aa:=integrate(1/(cosh(a*x)-1),x) --R --R @@ -1023,7 +865,7 @@ aa:=integrate(1/(cosh(a*x)-1),x) --R Type: Union(Expression Integer,...) --E ---S 82 +--S 68 bb:=-1/a*coth((a*x)/2) --R --R a x @@ -1034,7 +876,7 @@ bb:=-1/a*coth((a*x)/2) --R Type: Expression Integer --E ---S 83 +--S 69 cc:=aa-bb --R --R a x a x @@ -1045,7 +887,7 @@ cc:=aa-bb --R Type: Expression Integer --E ---S 84 +--S 70 cothrule:=rule(coth(x) == cosh(x)/sinh(x)) --R --R cosh(x) @@ -1054,7 +896,7 @@ cothrule:=rule(coth(x) == cosh(x)/sinh(x)) --R Type: RewriteRule(Integer,Integer,Expression Integer) --E ---S 85 +--S 71 dd:=cothrule cc --R --R a x a x a x a x @@ -1067,7 +909,7 @@ dd:=cothrule cc --R Type: Expression Integer --E ---S 86 +--S 72 sinhcoshrule:=rule(sinh(x)*cosh(y) == 1/2*(sinh(x+y)+sinh(x-y))) --R --I %BD sinh(y + x) - %BD sinh(y - x) @@ -1076,7 +918,7 @@ sinhcoshrule:=rule(sinh(x)*cosh(y) == 1/2*(sinh(x+y)+sinh(x-y))) --R Type: RewriteRule(Integer,Integer,Expression Integer) --E ---S 87 +--S 73 ee:=sinhcoshrule dd --R --R 3a x a x a x a x @@ -1089,7 +931,7 @@ ee:=sinhcoshrule dd --R Type: Expression Integer --E ---S 88 +--S 74 sinhsinhrule:=rule(sinh(x)*sinh(y)==1/2*(cosh(x+y)-cosh(x-y))) --R --I %BE cosh(y + x) - %BE cosh(y - x) @@ -1098,7 +940,7 @@ sinhsinhrule:=rule(sinh(x)*sinh(y)==1/2*(cosh(x+y)-cosh(x-y))) --R Type: RewriteRule(Integer,Integer,Expression Integer) --E ---S 89 +--S 75 ff:=sinhsinhrule ee --R --R 3a x a x a x a x @@ -1111,7 +953,7 @@ ff:=sinhsinhrule ee --R Type: Expression Integer --E ---S 90 +--S 76 coshcoshrule:=rule(cosh(x)*cosh(y)==1/2*(cosh(x+y)+cosh(x-y))) --R --I %BF cosh(y + x) + %BF cosh(y - x) @@ -1120,7 +962,7 @@ coshcoshrule:=rule(cosh(x)*cosh(y)==1/2*(cosh(x+y)+cosh(x-y))) --R Type: RewriteRule(Integer,Integer,Expression Integer) --E ---S 91 14:576 Schaums and Axiom differ by a constant +--S 77 14:576 Schaums and Axiom differ by a constant gg:=coshcoshrule ff --R --R 1 @@ -1138,7 +980,7 @@$$ <<*>>= )clear all ---S 92 +--S 78 aa:=integrate(x/(cosh(a*x)+1),x) --R --R @@ -1152,7 +994,7 @@ aa:=integrate(x/(cosh(a*x)+1),x) --R Type: Union(Expression Integer,...) --E ---S 93 +--S 79 bb:=x/a*tanh((a*x)/2)-2/a^2*log(cosh((a*x)/2)) --R --R a x a x @@ -1164,7 +1006,7 @@ bb:=x/a*tanh((a*x)/2)-2/a^2*log(cosh((a*x)/2)) --R Type: Expression Integer --E ---S 94 +--S 80 cc:=aa-bb --R --R (3) @@ -1185,7 +1027,7 @@ cc:=aa-bb --R Type: Expression Integer --E ---S 95 +--S 81 tanhrule:=rule(tanh(x) == sinh(x)/cosh(x)) --R --R sinh(x) @@ -1194,7 +1036,7 @@ tanhrule:=rule(tanh(x) == sinh(x)/cosh(x)) --R Type: RewriteRule(Integer,Integer,Expression Integer) --E ---S 96 +--S 82 dd:=tanhrule cc --R --R (5) @@ -1222,7 +1064,7 @@ dd:=tanhrule cc --R Type: Expression Integer --E ---S 97 +--S 83 coshcoshrule:=rule(cosh(x)*cosh(y)==1/2*(cosh(x+y)+cosh(x-y))) --R --I %BG cosh(y + x) + %BG cosh(y - x) @@ -1231,7 +1073,7 @@ coshcoshrule:=rule(cosh(x)*cosh(y)==1/2*(cosh(x+y)+cosh(x-y))) --R Type: RewriteRule(Integer,Integer,Expression Integer) --E ---S 98 +--S 84 ee:=coshcoshrule dd --R --R (7) @@ -1259,7 +1101,7 @@ ee:=coshcoshrule dd --R Type: Expression Integer --E ---S 99 +--S 85 sinhcoshrule:=rule(sinh(x)*cosh(y) == 1/2*(sinh(x+y)+sinh(x-y))) --R --I %BH sinh(y + x) - %BH sinh(y - x) @@ -1268,7 +1110,7 @@ sinhcoshrule:=rule(sinh(x)*cosh(y) == 1/2*(sinh(x+y)+sinh(x-y))) --R Type: RewriteRule(Integer,Integer,Expression Integer) --E ---S 100 +--S 86 ff:=sinhcoshrule ee --R --R (9) @@ -1296,7 +1138,7 @@ ff:=sinhcoshrule ee --R Type: Expression Integer --E ---S 101 +--S 87 sinhsinhrule:=rule(sinh(x)*sinh(y)==1/2*(cosh(x+y)-cosh(x-y))) --R --I %BI cosh(y + x) - %BI cosh(y - x) @@ -1305,7 +1147,7 @@ sinhsinhrule:=rule(sinh(x)*sinh(y)==1/2*(cosh(x+y)-cosh(x-y))) --R Type: RewriteRule(Integer,Integer,Expression Integer) --E ---S 102 +--S 88 gg:=sinhsinhrule ff --R --R a x @@ -1317,7 +1159,7 @@ gg:=sinhsinhrule ff --R Type: Expression Integer --E ---S 103 14:577 Schaums and Axiom differ by a constant +--S 89 14:577 Schaums and Axiom differ by a constant complexNormalize gg --R --R 2log(2) @@ -1336,7 +1178,7 @@ $$<<*>>= )clear all ---S 104 +--S 90 aa:=integrate(x/(cosh(a*x)-1),x) --R --R @@ -1350,7 +1192,7 @@ aa:=integrate(x/(cosh(a*x)-1),x) --R Type: Union(Expression Integer,...) --E ---S 105 +--S 91 bb:=-x/a*coth((a*x)/2)+2/a^2*log(sinh((a*x)/2)) --R --R a x a x @@ -1362,7 +1204,7 @@ bb:=-x/a*coth((a*x)/2)+2/a^2*log(sinh((a*x)/2)) --R Type: Expression Integer --E ---S 106 +--S 92 cc:=aa-bb --R --R (3) @@ -1383,7 +1225,7 @@ cc:=aa-bb --R Type: Expression Integer --E ---S 107 +--S 93 cothrule:=rule(coth(x) == cosh(x)/sinh(x)) --R --R cosh(x) @@ -1392,7 +1234,7 @@ cothrule:=rule(coth(x) == cosh(x)/sinh(x)) --R Type: RewriteRule(Integer,Integer,Expression Integer) --E ---S 108 +--S 94 dd:=cothrule cc --R --R (5) @@ -1420,7 +1262,7 @@ dd:=cothrule cc --R Type: Expression Integer --E ---S 109 +--S 95 sinhcoshrule:=rule(sinh(x)*cosh(y) == 1/2*(sinh(x+y)+sinh(x-y))) --R --I %BJ sinh(y + x) - %BJ sinh(y - x) @@ -1429,7 +1271,7 @@ sinhcoshrule:=rule(sinh(x)*cosh(y) == 1/2*(sinh(x+y)+sinh(x-y))) --R Type: RewriteRule(Integer,Integer,Expression Integer) --E ---S 110 +--S 96 ee:=sinhcoshrule dd --R --R (7) @@ -1457,7 +1299,7 @@ ee:=sinhcoshrule dd --R Type: Expression Integer --E ---S 111 +--S 97 sinhsinhrule:=rule(sinh(x)*sinh(y)==1/2*(cosh(x+y)-cosh(x-y))) --R --I %BK cosh(y + x) - %BK cosh(y - x) @@ -1466,7 +1308,7 @@ sinhsinhrule:=rule(sinh(x)*sinh(y)==1/2*(cosh(x+y)-cosh(x-y))) --R Type: RewriteRule(Integer,Integer,Expression Integer) --E ---S 112 +--S 98 ff:=sinhsinhrule ee --R --R (9) @@ -1494,7 +1336,7 @@ ff:=sinhsinhrule ee --R Type: Expression Integer --E ---S 113 +--S 99 coshcoshrule:=rule(cosh(x)*cosh(y)==1/2*(cosh(x+y)+cosh(x-y))) --R --I %BL cosh(y + x) + %BL cosh(y - x) @@ -1503,7 +1345,7 @@ coshcoshrule:=rule(cosh(x)*cosh(y)==1/2*(cosh(x+y)+cosh(x-y))) --R Type: RewriteRule(Integer,Integer,Expression Integer) --E ---S 114 +--S 100 gg:=coshcoshrule ff --R --R a x @@ -1515,7 +1357,7 @@ gg:=coshcoshrule ff --R Type: Expression Integer --E ---S 115 14:578 Schaums and Axiom differ by a constant +--S 101 14:578 Schaums and Axiom differ by a constant hh:=complexNormalize gg --R --R 2log(2) @@ -1534,7 +1376,7 @@$$ <<*>>= )clear all ---S 116 +--S 102 aa:=integrate(1/(cosh(a*x)+1)^2,x) --R --R @@ -1552,7 +1394,7 @@ aa:=integrate(1/(cosh(a*x)+1)^2,x) --R Type: Union(Expression Integer,...) --E ---S 117 +--S 103 bb:=1/(2*a)*tanh((a*x)/2)-1/(6*a)*tanh((a*x)/2)^3 --R --R a x 3 a x @@ -1563,7 +1405,7 @@ bb:=1/(2*a)*tanh((a*x)/2)-1/(6*a)*tanh((a*x)/2)^3 --R Type: Expression Integer --E ---S 118 14:579 Axiom cannot compute this integral +--S 104 14:579 Axiom cannot compute this integral cc:=aa-bb --R --R (3) @@ -1614,7 +1456,7 @@ $$<<*>>= )clear all ---S 119 +--S 105 aa:=integrate(1/(cosh(a*x)-1)^2,x) --R --R @@ -1632,7 +1474,7 @@ aa:=integrate(1/(cosh(a*x)-1)^2,x) --R Type: Union(Expression Integer,...) --E ---S 120 +--S 106 bb:=1/(2*a)*coth((a*x)/2)-1/(6*a)*coth((a*x)/2)^3 --R --R a x 3 a x @@ -1643,7 +1485,7 @@ bb:=1/(2*a)*coth((a*x)/2)-1/(6*a)*coth((a*x)/2)^3 --R Type: Expression Integer --E ---S 121 14:580 Axiom cannot compute this integral +--S 107 14:580 Axiom cannot compute this integral cc:=aa-bb --R --R (3) @@ -1702,7 +1544,7 @@$$ <<*>>= )clear all ---S 122 +--S 108 aa:=integrate(1/(p+q*cosh(a*x)),x) --R --R @@ -1744,7 +1586,7 @@ aa:=integrate(1/(p+q*cosh(a*x)),x) --R Type: Union(List Expression Integer,...) --E ---S 123 +--S 109 bb1:=2/(a*sqrt(q^2-p^2))*atan((q*%e^(a*x)+p)/sqrt(q^2-p^2)) --R --R a x @@ -1760,7 +1602,7 @@ bb1:=2/(a*sqrt(q^2-p^2))*atan((q*%e^(a*x)+p)/sqrt(q^2-p^2)) --R Type: Expression Integer --E ---S 124 +--S 110 bb2:=1/(a*sqrt(p^2-q^2))*log((q*%e^(a*x)+p-sqrt(p^2-q^2))/(q*%e^(a*x)+p+sqrt(p^2-q^2))) --R --R +---------+ @@ -1777,7 +1619,7 @@ bb2:=1/(a*sqrt(p^2-q^2))*log((q*%e^(a*x)+p-sqrt(p^2-q^2))/(q*%e^(a*x)+p+sqrt(p^2 --R Type: Expression Integer --E ---S 125 +--S 111 cc1:=aa.1-bb1 --R --R (4) @@ -1817,7 +1659,7 @@ cc1:=aa.1-bb1 --R Type: Expression Integer --E ---S 126 +--S 112 cc2:=aa.2-bb1 --R --R +-------+ @@ -1834,7 +1676,7 @@ cc2:=aa.2-bb1 --R Type: Expression Integer --E ---S 127 +--S 113 cc3:=aa.1-bb2 --R --R (6) @@ -1871,7 +1713,7 @@ cc3:=aa.1-bb2 --R Type: Expression Integer --E ---S 128 14:581 Axiom cannot simplify this expression +--S 114 14:581 Axiom cannot simplify this expression cc4:=aa.2-bb2 --R --R (7) @@ -1907,7 +1749,7 @@ $$<<*>>= )clear all ---S 129 +--S 115 aa:=integrate(1/(p+q*cosh(a*x))^2,x) --R --R @@ -1989,7 +1831,7 @@ aa:=integrate(1/(p+q*cosh(a*x))^2,x) --R Type: Union(List Expression Integer,...) --E ---S 130 +--S 116 t1:=integrate(1/(p+q*cosh(a*x)),x) --R --R (2) @@ -2030,7 +1872,7 @@ t1:=integrate(1/(p+q*cosh(a*x)),x) --R Type: Union(List Expression Integer,...) --E ---S 131 +--S 117 bb1:=(q*sinh(a*x))/(a*(q^2-p^2)*(p+q*cosh(a*x)))-p/(q^2-p^2)*t1.1 --R --R (3) @@ -2066,7 +1908,7 @@ bb1:=(q*sinh(a*x))/(a*(q^2-p^2)*(p+q*cosh(a*x)))-p/(q^2-p^2)*t1.1 --R Type: Expression Integer --E ---S 132 +--S 118 bb2:=(q*sinh(a*x))/(a*(q^2-p^2)*(p+q*cosh(a*x)))-p/(q^2-p^2)*t1.2 --R --R (4) @@ -2087,7 +1929,7 @@ bb2:=(q*sinh(a*x))/(a*(q^2-p^2)*(p+q*cosh(a*x)))-p/(q^2-p^2)*t1.2 --R Type: Expression Integer --E ---S 133 +--S 119 cc1:=aa.1-bb1 --R --R (5) @@ -2183,7 +2025,7 @@ cc1:=aa.1-bb1 --R Type: Expression Integer --E ---S 134 +--S 120 cc2:=aa.2-bb1 --R --R (6) @@ -2271,7 +2113,7 @@ cc2:=aa.2-bb1 --R Type: Expression Integer --E ---S 135 +--S 121 cc3:=aa.1-bb2 --R --R (7) @@ -2359,7 +2201,7 @@ cc3:=aa.1-bb2 --R Type: Expression Integer --E ---S 136 14:582 Axiom cannot simplify this expression +--S 122 14:582 Axiom cannot simplify this expression cc4:=aa.2-bb2 --R --R (8) @@ -2409,7 +2251,7 @@$$ <<*>>= )clear all ---S 137 +--S 123 aa:=integrate(1/(p^2-q^2*cosh(a*x)^2),x) --R --R @@ -2477,7 +2319,7 @@ aa:=integrate(1/(p^2-q^2*cosh(a*x)^2),x) --R Type: Union(List Expression Integer,...) --E ---S 138 +--S 124 bb1:=1/(2*a*p*sqrt(p^2-q^2))*log((p*tanh(a*x)+sqrt(p^2-q^2))/(p*tanh(a*x)-sqrt(p^2-q^2))) --R --R +---------+ @@ -2494,7 +2336,7 @@ bb1:=1/(2*a*p*sqrt(p^2-q^2))*log((p*tanh(a*x)+sqrt(p^2-q^2))/(p*tanh(a*x)-sqrt(p --R Type: Expression Integer --E ---S 139 +--S 125 bb2:=-1/(a*p*sqrt(q^2-p^2))*atan((p*tanh(a*x))/sqrt(q^2-p^2)) --R --R p tanh(a x) @@ -2509,7 +2351,7 @@ bb2:=-1/(a*p*sqrt(q^2-p^2))*atan((p*tanh(a*x))/sqrt(q^2-p^2)) --R Type: Expression Integer --E ---S 140 +--S 126 cc1:=aa.1-bb1 --R --R (4) @@ -2562,7 +2404,7 @@ cc1:=aa.1-bb1 --R Type: Expression Integer --E ---S 141 +--S 127 cc2:=aa.2-bb1 --R --R (5) @@ -2599,7 +2441,7 @@ cc2:=aa.2-bb1 --R Type: Expression Integer --E ---S 142 +--S 128 cc3:=aa.1-bb2 --R --R (6) @@ -2658,7 +2500,7 @@ cc3:=aa.1-bb2 --R Type: Expression Integer --E ---S 143 14:583 Axiom cannot simplify this expression +--S 129 14:583 Axiom cannot simplify this expression cc4:=aa.2-bb2 --R --R (7) @@ -2708,7 +2550,7 @@ $$<<*>>= )clear all ---S 144 +--S 130 aa:=integrate(1/(p^2+q^2*cosh(a*x)^2),x) --R --R @@ -2754,7 +2596,7 @@ aa:=integrate(1/(p^2+q^2*cosh(a*x)^2),x) --R Type: Union(Expression Integer,...) --E ---S 145 +--S 131 bb1:=1/(2*a*p*sqrt(p^2+q^2))*log((p*tanh(a*x)+sqrt(p^2+q^2))/(p*tanh(a*x)-sqrt(p^2+q^2))) --R --R +-------+ @@ -2771,7 +2613,7 @@ bb1:=1/(2*a*p*sqrt(p^2+q^2))*log((p*tanh(a*x)+sqrt(p^2+q^2))/(p*tanh(a*x)-sqrt(p --R Type: Expression Integer --E ---S 146 +--S 132 bb2:=1/(a*p*sqrt(p^2+q^2))*atan((p*tanh(a*x))/sqrt(p^2+q^2)) --R --R p tanh(a x) @@ -2786,7 +2628,7 @@ bb2:=1/(a*p*sqrt(p^2+q^2))*atan((p*tanh(a*x))/sqrt(p^2+q^2)) --R Type: Expression Integer --E ---S 147 +--S 133 cc1:=aa-bb1 --R --R (4) @@ -2839,7 +2681,7 @@ cc1:=aa-bb1 --R Type: Expression Integer --E ---S 148 14:584 Axiom cannot compute this integral +--S 134 14:584 Axiom cannot compute this integral cc2:=aa-bb2 --R --R (5) @@ -2899,7 +2741,7 @@$$ <<*>>= )clear all ---S 149 14:585 Axiom cannot compute this integral +--S 135 14:585 Axiom cannot compute this integral aa:=integrate(x^m*cosh(a*x),x) --R --R @@ -2919,7 +2761,7 @@ $$<<*>>= )clear all ---S 150 14:586 Axiom cannot compute this integral +--S 136 14:586 Axiom cannot compute this integral aa:=integrate(cosh(a*x)^n,x) --R --R @@ -2940,7 +2782,7 @@$$ <<*>>= )clear all ---S 151 14:587 Axiom cannot compute this integral +--S 137 14:587 Axiom cannot compute this integral aa:=integrate(cosh(a*x)/x^n,x) --R --R @@ -2962,7 +2804,7 @@ $$<<*>>= )clear all ---S 152 14:588 Axiom cannot compute this integral +--S 138 14:588 Axiom cannot compute this integral aa:=integrate(1/cosh(a*x)^n,x) --R --R @@ -2985,7 +2827,7 @@$$ <<*>>= )clear all ---S 153 14:589 Axiom cannot compute this integral +--S 139 14:589 Axiom cannot compute this integral aa:=integrate(1/cosh(a*x)^n,x) --R --R diff --git a/src/input/schaum29.input.pamphlet b/src/input/schaum29.input.pamphlet index 37e2800..1aff49f 100644 --- a/src/input/schaum29.input.pamphlet +++ b/src/input/schaum29.input.pamphlet @@ -18,7 +18,7 @@ $$)set message auto off )clear all ---S 1 of 14 +--S 1 aa:=integrate(sinh(a*x)*cosh(a*x),x) --R --R @@ -28,6 +28,63 @@ aa:=integrate(sinh(a*x)*cosh(a*x),x) --R 4a --R Type: Union(Expression Integer,...) --E + +--S 2 +bb:=sinh(a*x)^2/(2*a) +--R +--R 2 +--R sinh(a x) +--R (2) ---------- +--R 2a +--R Type: Expression Integer +--E + +--S 3 +cc:=aa-bb +--R +--R 2 2 +--R - sinh(a x) + cosh(a x) +--R (3) ------------------------- +--R 4a +--R Type: Expression Integer +--E + +--S 4 +sinhsqrrule:=rule(sinh(x)^2 == 1/2*cosh(2*x)-1/2) +--R +--R 2 cosh(2x) - 1 +--R (4) sinh(x) == ------------ +--R 2 +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 5 +dd:=sinhsqrrule cc +--R +--R 2 +--R - cosh(2a x) + 2cosh(a x) + 1 +--R (5) ------------------------------ +--R 8a +--R Type: Expression Integer +--E + +--S 6 +coshsqrrule:=rule(cosh(x)^2 == 1/2*cosh(2*x)+1/2) +--R +--R 2 cosh(2x) + 1 +--R (6) cosh(x) == ------------ +--R 2 +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 7 14:590 Schaums and Axiom agree +ee:=coshsqrrule dd +--R +--R 1 +--R (7) -- +--R 4a +--R Type: Expression Integer +--E @ \section{\cite{1}:14.591~~~~~\displaystyle @@ -38,7 +95,7 @@$$ <<*>>= )clear all ---S 2 of 14 +--S 8 aa:=integrate(sinh(p*x)*cosh(q*x),x) --R --R @@ -48,6 +105,35 @@ aa:=integrate(sinh(p*x)*cosh(q*x),x) --R (q - p )sinh(p x) + (- q + p )cosh(p x) --R Type: Union(Expression Integer,...) --E + +--S 9 +bb:=(cosh(p+q)*x)/(2*(p+q))+(cosh(p-q)*x)/(2*(p-q)) +--R +--R (q - p)x cosh(q + p) + (- q - p)x cosh(q - p) +--R (2) --------------------------------------------- +--R 2 2 +--R 2q - 2p +--R Type: Expression Integer +--E + +--S 10 14:591 Axiom cannot simplify this expression +cc:=aa-bb +--R +--R (3) +--R - 2q sinh(p x)sinh(q x) +--R + +--R 2 +--R ((- q + p)x cosh(q + p) + (q + p)x cosh(q - p))sinh(p x) +--R + +--R 2p cosh(p x)cosh(q x) +--R + +--R 2 +--R ((q - p)x cosh(q + p) + (- q - p)x cosh(q - p))cosh(p x) +--R / +--R 2 2 2 2 2 2 +--R (2q - 2p )sinh(p x) + (- 2q + 2p )cosh(p x) +--R Type: Expression Integer +--E @ \section{\cite{1}:14.592~~~~~$\displaystyle
@@ -58,7 +144,7 @@ $$<<*>>= )clear all ---S 3 of 14 +--S 11 aa:=integrate(sinh(a*x)^n*cosh(a*x),x) --R --R @@ -68,6 +154,29 @@ aa:=integrate(sinh(a*x)^n*cosh(a*x),x) --R (a n + a)sinh(a x) + (- a n - a)cosh(a x) --R Type: Union(Expression Integer,...) --E + +--S 12 +bb:=sinh(a*x)/((n+1)*a) +--R +--R sinh(a x) +--R (2) --------- +--R a n + a +--R Type: Expression Integer +--E + +--S 13 14:592 Axiom cannot simplify this expression +cc:=aa-bb +--R +--R (3) +--R - sinh(a x)sinh(n log(sinh(a x))) - sinh(a x)cosh(n log(sinh(a x))) +--R + +--R 3 2 +--R - sinh(a x) + cosh(a x) sinh(a x) +--R / +--R 2 2 +--R (a n + a)sinh(a x) + (- a n - a)cosh(a x) +--R Type: Expression Integer +--E @ \section{\cite{1}:14.593~~~~~\displaystyle @@ -78,7 +187,7 @@$$
<<*>>=
)clear all

---S 4 of 14
+--S 14
aa:=integrate(cosh(a*x)^n*sinh(a*x),x)
--R
--R
@@ -88,6 +197,29 @@ aa:=integrate(cosh(a*x)^n*sinh(a*x),x)
--R                    (a n + a)sinh(a x)  + (- a n - a)cosh(a x)
--R                                          Type: Union(Expression
Integer,...)
--E
+
+--S 15
+bb:=cosh(a*x)^(n+1)/((n+1)*a)
+--R
+--R                 n + 1
+--R        cosh(a x)
+--R   (2)  --------------
+--R            a n + a
+--R                                                     Type: Expression
Integer
+--E
+
+--S 16     14:593 Axiom cannot simplify this expression
+--R
+--R   (3)
+--R       - cosh(a x)sinh(n log(cosh(a x))) - cosh(a x)cosh(n log(cosh(a x)))
+--R     +
+--R                   2            2          n + 1
+--R       (- sinh(a x)  + cosh(a x) )cosh(a x)
+--R  /
+--R                       2                       2
+--R     (a n + a)sinh(a x)  + (- a n - a)cosh(a x)
+--R                                                     Type: Expression
Integer
+--E
@

\section{\cite{1}:14.594~~~~~$\displaystyle @@ -98,7 +230,7 @@ $$<<*>>= )clear all ---S 5 of 14 +--S 17 aa:=integrate(sinh(a*x)^2*cosh(a*x)^2,x) --R --R @@ -108,6 +240,22 @@ aa:=integrate(sinh(a*x)^2*cosh(a*x)^2,x) --R 8a --R Type: Union(Expression Integer,...) --E + +--S 18 +bb:=sinh(4*a*x)/(32*a)-x/8 +--R +--R sinh(4a x) - 4a x +--R (2) ----------------- +--R 32a +--R Type: Expression Integer +--E + +--S 19 14:594 Schaums and Axiom agree +cc:=complexNormalize(aa-bb) +--R +--R (3) 0 +--R Type: Expression Integer +--E @ \section{\cite{1}:14.595~~~~~\displaystyle @@ -118,7 +266,7 @@$$ <<*>>= )clear all ---S 6 of 14 +--S 20 aa:=integrate(1/(sinh(a*x)*cosh(a*x)),x) --R --R @@ -129,6 +277,67 @@ aa:=integrate(1/(sinh(a*x)*cosh(a*x)),x) --R a --R Type: Union(Expression Integer,...) --E + +--S 21 +bb:=1/a*log(tanh(a*x)) +--R +--R log(tanh(a x)) +--R (2) -------------- +--R a +--R Type: Expression Integer +--E + +--S 22 +cc:=aa-bb +--R +--R (3) +--R 2cosh(a x) +--R - log(tanh(a x)) - log(- ---------------------) +--R sinh(a x) - cosh(a x) +--R + +--R 2sinh(a x) +--R log(- ---------------------) +--R sinh(a x) - cosh(a x) +--R / +--R a +--R Type: Expression Integer +--E + +--S 23 +dd:=expandLog cc +--R +--R - log(tanh(a x)) + log(sinh(a x)) - log(cosh(a x)) +--R (4) -------------------------------------------------- +--R a +--R Type: Expression Integer +--E + +--S 24 +tanhrule:=rule(tanh(x) == sinh(x)/cosh(x)) +--R +--R sinh(x) +--R (5) tanh(x) == ------- +--R cosh(x) +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 25 +ee:=tanhrule dd +--R +--R sinh(a x) +--R log(sinh(a x)) - log(---------) - log(cosh(a x)) +--R cosh(a x) +--R (6) ------------------------------------------------ +--R a +--R Type: Expression Integer +--E + +--S 26 14:595 Schaums and Axiom agree +ff:=expandLog ee +--R +--R (7) 0 +--R Type: Expression Integer +--E @ \section{\cite{1}:14.596~~~~~$\displaystyle
@@ -139,17 +348,52 @@ $$<<*>>= )clear all ---S 7 of 14 -aa:=integrate(1/(sinh(a*x)^2*cos(a*x)),x) ---R +--S 27 +aa:=integrate(1/(sinh(a*x)^2*cosh(a*x)),x) --R ---R x ---R ++ 1 ---I (1) | -------------------- d%R ---R ++ 2 ---I cos(%R a)sinh(%R a) +--R (1) +--R 2 2 +--R (- 2sinh(a x) - 4cosh(a x)sinh(a x) - 2cosh(a x) + 2) +--R * +--R atan(sinh(a x) + cosh(a x)) +--R + +--R - 2sinh(a x) - 2cosh(a x) +--R / +--R 2 2 +--R a sinh(a x) + 2a cosh(a x)sinh(a x) + a cosh(a x) - a --R Type: Union(Expression Integer,...) --E + +--S 28 +bb:=-1/a*atan(sinh(a*x)-csch(a*x))/a +--R +--R atan(sinh(a x) - csch(a x)) +--R (2) - --------------------------- +--R 2 +--R a +--R Type: Expression Integer +--E + +--S 29 14:596 Axiom cannot simplify this expression +cc:=aa-bb +--R +--R (3) +--R 2 2 +--R (- 2a sinh(a x) - 4a cosh(a x)sinh(a x) - 2a cosh(a x) + 2a) +--R * +--R atan(sinh(a x) + cosh(a x)) +--R + +--R 2 2 +--R (sinh(a x) + 2cosh(a x)sinh(a x) + cosh(a x) - 1) +--R * +--R atan(sinh(a x) - csch(a x)) +--R + +--R - 2a sinh(a x) - 2a cosh(a x) +--R / +--R 2 2 2 2 2 2 +--R a sinh(a x) + 2a cosh(a x)sinh(a x) + a cosh(a x) - a +--R Type: Expression Integer +--E @ \section{\cite{1}:14.597~~~~~\displaystyle @@ -160,7 +404,7 @@$$
<<*>>=
)clear all

---S 8 of 14
+--S 30
aa:=integrate(1/(sinh(a*x)*cosh(a*x)^2),x)
--R
--R
@@ -181,6 +425,272 @@ aa:=integrate(1/(sinh(a*x)*cosh(a*x)^2),x)
--R     a sinh(a x)  + 2a cosh(a x)sinh(a x) + a cosh(a x)  + a
--R                                          Type: Union(Expression
Integer,...)
--E
+
+--S 31
+bb:=sech(a*x)/a+1/a*log(tanh((a*x)/2))
+--R
+--R                 a x
+--R        log(tanh(---)) + sech(a x)
+--R                  2
+--R   (2)  --------------------------
+--R                     a
+--R                                                     Type: Expression
Integer
+--E
+
+--S 32
+cc:=aa-bb
+--R
+--R   (3)
+--R                   2                                  2              a x
+--R       (- sinh(a x)  - 2cosh(a x)sinh(a x) - cosh(a x)  - 1)log(tanh(---))
+--R                                                                      2
+--R     +
+--R                     2                                  2
+--R         (- sinh(a x)  - 2cosh(a x)sinh(a x) - cosh(a x)  - 1)
+--R      *
+--R         log(sinh(a x) + cosh(a x) + 1)
+--R     +
+--R                   2                                  2
+--R         (sinh(a x)  + 2cosh(a x)sinh(a x) + cosh(a x)  + 1)
+--R      *
+--R         log(sinh(a x) + cosh(a x) - 1)
+--R     +
+--R                           2
+--R       - sech(a x)sinh(a x)  + (- 2cosh(a x)sech(a x) + 2)sinh(a x)
+--R     +
+--R                   2
+--R       (- cosh(a x)  - 1)sech(a x) + 2cosh(a x)
+--R  /
+--R                2                                      2
+--R     a sinh(a x)  + 2a cosh(a x)sinh(a x) + a cosh(a x)  + a
+--R                                                     Type: Expression
Integer
+--E
+
+--S 33
+sechrule:=rule(sech(x) == 1/cosh(x))
+--R
+--R                      1
+--R   (4)  sech(x) == -------
+--R                   cosh(x)
+--R                        Type: RewriteRule(Integer,Integer,Expression
Integer)
+--E
+
+--S 34
+dd:=sechrule cc
+--R
+--R   (5)
+--R                              2             2                     3
+--R         (- cosh(a x)sinh(a x)  - 2cosh(a x) sinh(a x) - cosh(a x)  -
cosh(a x))
+--R      *
+--R                  a x
+--R         log(tanh(---))
+--R                   2
+--R     +
+--R                              2             2                     3
+--R         (- cosh(a x)sinh(a x)  - 2cosh(a x) sinh(a x) - cosh(a x)  -
cosh(a x))
+--R      *
+--R         log(sinh(a x) + cosh(a x) + 1)
+--R     +
+--R                            2             2                     3
+--R         (cosh(a x)sinh(a x)  + 2cosh(a x) sinh(a x) + cosh(a x)  + cosh(a
x))
+--R      *
+--R         log(sinh(a x) + cosh(a x) - 1)
+--R     +
+--R                  2            2
+--R       - sinh(a x)  + cosh(a x)  - 1
+--R  /
+--R                         2               2                       3
+--R     a cosh(a x)sinh(a x)  + 2a cosh(a x) sinh(a x) + a cosh(a x)  + a
cosh(a x)
+--R                                                     Type: Expression
Integer
+--E
+
+--S 35
+tanhrule:=rule(tanh(x) == sinh(x)/cosh(x))
+--R
+--R                   sinh(x)
+--R   (6)  tanh(x) == -------
+--R                   cosh(x)
+--R                        Type: RewriteRule(Integer,Integer,Expression
Integer)
+--E
+
+--S 36
+ee:=tanhrule dd
+--R
+--R   (7)
+--R                              2             2                     3
+--R         (- cosh(a x)sinh(a x)  - 2cosh(a x) sinh(a x) - cosh(a x)  -
cosh(a x))
+--R      *
+--R         log(sinh(a x) + cosh(a x) + 1)
+--R     +
+--R                            2             2                     3
+--R         (cosh(a x)sinh(a x)  + 2cosh(a x) sinh(a x) + cosh(a x)  + cosh(a
x))
+--R      *
+--R         log(sinh(a x) + cosh(a x) - 1)
+--R     +
+--R                              2             2                     3
+--R         (- cosh(a x)sinh(a x)  - 2cosh(a x) sinh(a x) - cosh(a x)  -
cosh(a x))
+--R      *
+--R                  a x
+--R             sinh(---)
+--R                   2
+--R         log(---------)
+--R                  a x
+--R             cosh(---)
+--R                   2
+--R     +
+--R                  2            2
+--R       - sinh(a x)  + cosh(a x)  - 1
+--R  /
+--R                         2               2                       3
+--R     a cosh(a x)sinh(a x)  + 2a cosh(a x) sinh(a x) + a cosh(a x)  + a
cosh(a x)
+--R                                                     Type: Expression
Integer
+--E
+
+--S 37
+coshcuberule:=rule(cosh(x)^3 == 1/4*cosh(3*x)-3/4*cosh(x))
+--R
+--R               3    cosh(3x) - 3cosh(x)
+--R   (8)  cosh(x)  == -------------------
+--R                             4
+--R                        Type: RewriteRule(Integer,Integer,Expression
Integer)
+--E
+
+--S 38
+ff:=coshcuberule ee
+--R
+--R   (9)
+--R                                  2             2
+--R             - 4cosh(a x)sinh(a x)  - 8cosh(a x) sinh(a x) - cosh(3a x)
+--R           +
+--R             - cosh(a x)
+--R      *
+--R         log(sinh(a x) + cosh(a x) + 1)
+--R     +
+--R                             2             2
+--R         (4cosh(a x)sinh(a x)  + 8cosh(a x) sinh(a x) + cosh(3a x) + cosh(a
x))
+--R      *
+--R         log(sinh(a x) + cosh(a x) - 1)
+--R     +
+--R                                  2             2
+--R             - 4cosh(a x)sinh(a x)  - 8cosh(a x) sinh(a x) - cosh(3a x)
+--R           +
+--R             - cosh(a x)
+--R      *
+--R                  a x
+--R             sinh(---)
+--R                   2
+--R         log(---------)
+--R                  a x
+--R             cosh(---)
+--R                   2
+--R     +
+--R                   2             2
+--R       - 4sinh(a x)  + 4cosh(a x)  - 4
+--R  /
+--R                            2               2
+--R       4a cosh(a x)sinh(a x)  + 8a cosh(a x) sinh(a x) + a cosh(3a x)
+--R     +
+--R       a cosh(a x)
+--R                                                     Type: Expression
Integer
+--E
+
+--S 39
+coshsqrrule:=rule(cosh(x)^2 == 1/2*cosh(2*x)+1/2)
+--R
+--R                2    cosh(2x) + 1
+--R   (10)  cosh(x)  == ------------
+--R                           2
+--R                        Type: RewriteRule(Integer,Integer,Expression
Integer)
+--E
+
+--S 40
+gg:=coshsqrrule ff
+--R
+--R   (11)
+--R                                2
+--R           - 4cosh(a x)sinh(a x)  + (- 4cosh(2a x) - 4)sinh(a x) - cosh(3a
x)
+--R         +
+--R           - cosh(a x)
+--R      *
+--R         log(sinh(a x) + cosh(a x) + 1)
+--R     +
+--R                              2
+--R           4cosh(a x)sinh(a x)  + (4cosh(2a x) + 4)sinh(a x) + cosh(3a x)
+--R         +
+--R           cosh(a x)
+--R      *
+--R         log(sinh(a x) + cosh(a x) - 1)
+--R     +
+--R                                2
+--R           - 4cosh(a x)sinh(a x)  + (- 4cosh(2a x) - 4)sinh(a x) - cosh(3a
x)
+--R         +
+--R           - cosh(a x)
+--R      *
+--R                  a x
+--R             sinh(---)
+--R                   2
+--R         log(---------)
+--R                  a x
+--R             cosh(---)
+--R                   2
+--R     +
+--R                   2
+--R       - 4sinh(a x)  + 2cosh(2a x) - 2
+--R  /
+--R                            2
+--R       4a cosh(a x)sinh(a x)  + (4a cosh(2a x) + 4a)sinh(a x) + a cosh(3a x)
+--R     +
+--R       a cosh(a x)
+--R                                                     Type: Expression
Integer
+--E
+
+--S 41
+sinhsqrrule:=rule(sinh(x)^2 == 1/2*cosh(2*x)-1/2)
+--R
+--R                2    cosh(2x) - 1
+--R   (12)  sinh(x)  == ------------
+--R                           2
+--R                        Type: RewriteRule(Integer,Integer,Expression
Integer)
+--E
+
+--S 42
+hh:=sinhsqrrule gg
+--R
+--R   (13)
+--R       - log(sinh(a x) + cosh(a x) + 1) + log(sinh(a x) + cosh(a x) - 1)
+--R     +
+--R                  a x
+--R             sinh(---)
+--R                   2
+--R       - log(---------)
+--R                  a x
+--R             cosh(---)
+--R                   2
+--R  /
+--R     a
+--R                                                     Type: Expression
Integer
+--E
+
+--S 43
+ii:=expandLog hh
+--R
+--R   (14)
+--R       - log(sinh(a x) + cosh(a x) + 1) + log(sinh(a x) + cosh(a x) - 1)
+--R     +
+--R                  a x              a x
+--R       - log(sinh(---)) + log(cosh(---))
+--R                   2                2
+--R  /
+--R     a
+--R                                                     Type: Expression
Integer
+--E
+
+--S 44     14:597 Schaums and Axiom agree
+jj:=complexNormalize ii
+--R
+--R   (15)  0
+--R                                                     Type: Expression
Integer
+--E
@

\section{\cite{1}:14.598~~~~~$\displaystyle @@ -191,7 +701,7 @@ $$<<*>>= )clear all ---S 9 of 14 +--S 45 aa:=integrate(1/(sinh(a*x)^2*cosh(a*x)^2),x) --R --R @@ -206,6 +716,35 @@ aa:=integrate(1/(sinh(a*x)^2*cosh(a*x)^2),x) --R 4a cosh(a x) sinh(a x) + a cosh(a x) - a --R Type: Union(Expression Integer,...) --E + +--S 46 +bb:=-(2*coth(2*a*x))/a +--R +--R 2coth(2a x) +--R (2) - ----------- +--R a +--R Type: Expression Integer +--E + +--S 47 14:598 Axiom cannot simplify this expression +--R +--R (3) +--R 4 3 +--R 2coth(2a x)sinh(a x) + 8cosh(a x)coth(2a x)sinh(a x) +--R + +--R 2 2 3 +--R 12cosh(a x) coth(2a x)sinh(a x) + 8cosh(a x) coth(2a x)sinh(a x) +--R + +--R 4 +--R (2cosh(a x) - 2)coth(2a x) - 4 +--R / +--R 4 3 2 2 +--R a sinh(a x) + 4a cosh(a x)sinh(a x) + 6a cosh(a x) sinh(a x) +--R + +--R 3 4 +--R 4a cosh(a x) sinh(a x) + a cosh(a x) - a +--R Type: Expression Integer +--E @ \section{\cite{1}:14.599~~~~~\displaystyle @@ -216,7 +755,7 @@$$ <<*>>= )clear all ---S 10 of 14 +--S 48 aa:=integrate(sinh(a*x)^2/cosh(a*x),x) --R --R @@ -230,6 +769,28 @@ aa:=integrate(sinh(a*x)^2/cosh(a*x),x) --R 2a sinh(a x) + 2a cosh(a x) --R Type: Union(Expression Integer,...) --E + +--S 49 +bb:=sinh(a*x)/a-1/a*atan(sinh(a*x)) +--R +--R - atan(sinh(a x)) + sinh(a x) +--R (2) ----------------------------- +--R a +--R Type: Expression Integer +--E + +--S 50 14:599 Axiom cannot simplify this expression +cc:=aa-bb +--R +--R (3) +--R (- 4sinh(a x) - 4cosh(a x))atan(sinh(a x) + cosh(a x)) +--R + +--R 2 2 +--R (2sinh(a x) + 2cosh(a x))atan(sinh(a x)) - sinh(a x) + cosh(a x) - 1 +--R / +--R 2a sinh(a x) + 2a cosh(a x) +--R Type: Expression Integer +--E @ \section{\cite{1}:14.600~~~~~$\displaystyle
@@ -240,7 +801,7 @@ $$<<*>>= )clear all ---S 11 of 14 +--S 51 aa:=integrate(cosh(a*x)^2/sinh(a*x),x) --R --R @@ -256,6 +817,37 @@ aa:=integrate(cosh(a*x)^2/sinh(a*x),x) --R 2a sinh(a x) + 2a cosh(a x) --R Type: Union(Expression Integer,...) --E + +--S 52 +bb:=cosh(a*x)/a+1/a*log(tanh((a*x)/2)) +--R +--R a x +--R log(tanh(---)) + cosh(a x) +--R 2 +--R (2) -------------------------- +--R a +--R Type: Expression Integer +--E + +--S 53 14:600 Axiom cannot simplify this expression +cc:=aa-bb +--R +--R (3) +--R a x +--R (- 2sinh(a x) - 2cosh(a x))log(tanh(---)) +--R 2 +--R + +--R (- 2sinh(a x) - 2cosh(a x))log(sinh(a x) + cosh(a x) + 1) +--R + +--R 2 +--R (2sinh(a x) + 2cosh(a x))log(sinh(a x) + cosh(a x) - 1) + sinh(a x) +--R + +--R 2 +--R - cosh(a x) + 1 +--R / +--R 2a sinh(a x) + 2a cosh(a x) +--R Type: Expression Integer +--E @ \section{\cite{1}:14.601~~~~~\displaystyle @@ -267,7 +859,7 @@$$
<<*>>=
)clear all

---S 12 of 14
+--S 54
aa:=integrate(1/(cosh(a*x)*(1+sinh(a*x))),x)
--R
--R
@@ -281,6 +873,87 @@ aa:=integrate(1/(cosh(a*x)*(1+sinh(a*x))),x)
--R     2a
--R                                          Type: Union(Expression
Integer,...)
--E
+
+--S 55
+bb:=1/(2*a)*log((1+sinh(a*x))/cosh(a*x))+1/a*atan(%e^(a*x))
+--R
+--R            sinh(a x) + 1            a x
+--R        log(-------------) + 2atan(%e   )
+--R              cosh(a x)
+--R   (2)  ---------------------------------
+--R                        2a
+--R                                                     Type: Expression
Integer
+--E
+
+--S 56
+cc:=aa-bb
+--R
+--R   (3)
+--R             sinh(a x) + 1                2cosh(a x)
+--R       - log(-------------) - log(- ---------------------)
+--R               cosh(a x)            sinh(a x) - cosh(a x)
+--R     +
+--R              - 2sinh(a x) - 2
a x
+--R       log(---------------------) + 2atan(sinh(a x) + cosh(a x)) - 2atan(%e
)
+--R           sinh(a x) - cosh(a x)
+--R  /
+--R     2a
+--R                                                     Type: Expression
Integer
+--E
+
+--S 57
+dd:=expandLog cc
+--R
+--R                                             a x
+--R        atan(sinh(a x) + cosh(a x)) - atan(%e   )
+--R   (4)  -----------------------------------------
+--R                            a
+--R                                                     Type: Expression
Integer
+--E
+
+--S 58
+atanrule:=rule(atan(x) == -%i/2*log((1+%i*x)/(1-%i*x)))
+--R
+--R                            - x + %i
+--R                     %i log(--------)
+--R                             x + %i
+--R   (5)  atan(x) == - ----------------
+--R                             2
+--R        Type: RewriteRule(Integer,Complex Integer,Expression Complex
Integer)
+--E
+
+--S 59
+ee:=atanrule dd
+--R
+--R                   a x
+--R               - %e    + %i           - sinh(a x) - cosh(a x) + %i
+--R        %i log(------------) - %i log(----------------------------)
+--R                  a x                  sinh(a x) + cosh(a x) + %i
+--R                %e    + %i
+--R   (6)  -----------------------------------------------------------
+--R                                     2a
+--R                                             Type: Expression Complex
Integer
+--E
+
+--S 60
+ff:=expandLog ee
+--R
+--R   (7)
+--R       %i log(sinh(a x) + cosh(a x) + %i) - %i log(sinh(a x) + cosh(a x) -
%i)
+--R     +
+--R                  a x                  a x
+--R       - %i log(%e    + %i) + %i log(%e    - %i)
+--R  /
+--R     2a
+--R                                             Type: Expression Complex
Integer
+--E
+
+--S 61     14:601 Schaums and Axiom agree
+gg:=complexNormalize ff
+--R
+--R   (8)  0
+--R                                             Type: Expression Complex
Integer
+--E
@

\section{\cite{1}:14.602~~~~~$\displaystyle @@ -291,7 +964,7 @@ $$<<*>>= )clear all ---S 13 of 14 +--S 62 aa:=integrate(1/(sinh(a*x)*(cosh(a*x)+1)),x) --R --R @@ -316,6 +989,204 @@ aa:=integrate(1/(sinh(a*x)*(cosh(a*x)+1)),x) --R 4a cosh(a x) + 2a --R Type: Union(Expression Integer,...) --E + +--S 63 +bb:=1/(2*a)*log(tanh((a*x)/2))+1/(2*a*(cosh(a*x)+1)) +--R +--R a x +--R (cosh(a x) + 1)log(tanh(---)) + 1 +--R 2 +--R (2) --------------------------------- +--R 2a cosh(a x) + 2a +--R Type: Expression Integer +--E + +--S 64 +cc:=aa-bb +--R +--R (3) +--R 2 +--R (- cosh(a x) - 1)sinh(a x) +--R + +--R 2 3 2 +--R (- 2cosh(a x) - 4cosh(a x) - 2)sinh(a x) - cosh(a x) - 3cosh(a x) +--R + +--R - 3cosh(a x) - 1 +--R * +--R a x +--R log(tanh(---)) +--R 2 +--R + +--R 2 +--R (- cosh(a x) - 1)sinh(a x) +--R + +--R 2 3 2 +--R (- 2cosh(a x) - 4cosh(a x) - 2)sinh(a x) - cosh(a x) - 3cosh(a x) +--R + +--R - 3cosh(a x) - 1 +--R * +--R log(sinh(a x) + cosh(a x) + 1) +--R + +--R 2 2 +--R (cosh(a x) + 1)sinh(a x) + (2cosh(a x) + 4cosh(a x) + 2)sinh(a x) +--R + +--R 3 2 +--R cosh(a x) + 3cosh(a x) + 3cosh(a x) + 1 +--R * +--R log(sinh(a x) + cosh(a x) - 1) +--R + +--R 2 2 +--R - sinh(a x) + cosh(a x) - 1 +--R / +--R 2 +--R (2a cosh(a x) + 2a)sinh(a x) +--R + +--R 2 3 +--R (4a cosh(a x) + 8a cosh(a x) + 4a)sinh(a x) + 2a cosh(a x) +--R + +--R 2 +--R 6a cosh(a x) + 6a cosh(a x) + 2a +--R Type: Expression Integer +--E + +--S 65 +coshcuberule:=rule(cosh(x)^3 == 1/4*cosh(3*x)-3/4*cosh(x)) +--R +--R 3 cosh(3x) - 3cosh(x) +--R (4) cosh(x) == ------------------- +--R 4 +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 66 +dd:=coshcuberule cc +--R +--R (5) +--R 2 +--R (- 4cosh(a x) - 4)sinh(a x) +--R + +--R 2 +--R (- 8cosh(a x) - 16cosh(a x) - 8)sinh(a x) - cosh(3a x) +--R + +--R 2 +--R - 12cosh(a x) - 9cosh(a x) - 4 +--R * +--R a x +--R log(tanh(---)) +--R 2 +--R + +--R 2 +--R (- 4cosh(a x) - 4)sinh(a x) +--R + +--R 2 +--R (- 8cosh(a x) - 16cosh(a x) - 8)sinh(a x) - cosh(3a x) +--R + +--R 2 +--R - 12cosh(a x) - 9cosh(a x) - 4 +--R * +--R log(sinh(a x) + cosh(a x) + 1) +--R + +--R 2 2 +--R (4cosh(a x) + 4)sinh(a x) + (8cosh(a x) + 16cosh(a x) + 8)sinh(a x) +--R + +--R 2 +--R cosh(3a x) + 12cosh(a x) + 9cosh(a x) + 4 +--R * +--R log(sinh(a x) + cosh(a x) - 1) +--R + +--R 2 2 +--R - 4sinh(a x) + 4cosh(a x) - 4 +--R / +--R 2 +--R (8a cosh(a x) + 8a)sinh(a x) +--R + +--R 2 +--R (16a cosh(a x) + 32a cosh(a x) + 16a)sinh(a x) + 2a cosh(3a x) +--R + +--R 2 +--R 24a cosh(a x) + 18a cosh(a x) + 8a +--R Type: Expression Integer +--E + +--S 67 +sinhsqrrule:=rule(sinh(x)^2 == 1/2*cosh(2*x)-1/2) +--R +--R 2 cosh(2x) - 1 +--R (6) sinh(x) == ------------ +--R 2 +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 68 +ee:=sinhsqrrule dd +--R +--R (7) +--R 2 +--R (- 8cosh(a x) - 16cosh(a x) - 8)sinh(a x) - cosh(3a x) +--R + +--R 2 +--R (- 2cosh(a x) - 2)cosh(2a x) - 12cosh(a x) - 7cosh(a x) - 2 +--R * +--R a x +--R log(tanh(---)) +--R 2 +--R + +--R 2 +--R (- 8cosh(a x) - 16cosh(a x) - 8)sinh(a x) - cosh(3a x) +--R + +--R 2 +--R (- 2cosh(a x) - 2)cosh(2a x) - 12cosh(a x) - 7cosh(a x) - 2 +--R * +--R log(sinh(a x) + cosh(a x) + 1) +--R + +--R 2 +--R (8cosh(a x) + 16cosh(a x) + 8)sinh(a x) + cosh(3a x) +--R + +--R 2 +--R (2cosh(a x) + 2)cosh(2a x) + 12cosh(a x) + 7cosh(a x) + 2 +--R * +--R log(sinh(a x) + cosh(a x) - 1) +--R + +--R 2 +--R - 2cosh(2a x) + 4cosh(a x) - 2 +--R / +--R 2 +--R (16a cosh(a x) + 32a cosh(a x) + 16a)sinh(a x) + 2a cosh(3a x) +--R + +--R 2 +--R (4a cosh(a x) + 4a)cosh(2a x) + 24a cosh(a x) + 14a cosh(a x) + 4a +--R Type: Expression Integer +--E + +--S 69 +coshsqrrule:=rule(cosh(x)^2 == 1/2*cosh(2*x)+1/2) +--R +--R 2 cosh(2x) + 1 +--R (8) cosh(x) == ------------ +--R 2 +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 70 +ff:=coshsqrrule ee +--R +--R (9) +--R a x +--R - log(tanh(---)) - log(sinh(a x) + cosh(a x) + 1) +--R 2 +--R + +--R log(sinh(a x) + cosh(a x) - 1) +--R / +--R 2a +--R Type: Expression Integer +--E + +--S 71 14:602 Schaums and Axiom agree +gg:=complexNormalize ff +--R +--R (10) 0 +--R Type: Expression Integer +--E @ \section{\cite{1}:14.603~~~~~\displaystyle @@ -326,7 +1197,7 @@$$ <<*>>= )clear all ---S 14 of 14 +--S 72 aa:=integrate(1/(sinh(a*x)*(cosh(a*x)-1)),x) --R --R @@ -352,6 +1223,199 @@ aa:=integrate(1/(sinh(a*x)*(cosh(a*x)-1)),x) --R Type: Union(Expression Integer,...) --E +--S 73 +bb:=-1/(2*a)*log(tanh((a*x)/2))-1/(2*a*(cosh(a*x)-1)) +--R +--R a x +--R (- cosh(a x) + 1)log(tanh(---)) - 1 +--R 2 +--R (2) ----------------------------------- +--R 2a cosh(a x) - 2a +--R Type: Expression Integer +--E + +--S 74 +cc:=aa-bb +--R +--R (3) +--R 2 2 +--R (cosh(a x) - 1)sinh(a x) + (2cosh(a x) - 4cosh(a x) + 2)sinh(a x) +--R + +--R 3 2 +--R cosh(a x) - 3cosh(a x) + 3cosh(a x) - 1 +--R * +--R a x +--R log(tanh(---)) +--R 2 +--R + +--R 2 2 +--R (cosh(a x) - 1)sinh(a x) + (2cosh(a x) - 4cosh(a x) + 2)sinh(a x) +--R + +--R 3 2 +--R cosh(a x) - 3cosh(a x) + 3cosh(a x) - 1 +--R * +--R log(sinh(a x) + cosh(a x) + 1) +--R + +--R 2 +--R (- cosh(a x) + 1)sinh(a x) +--R + +--R 2 3 2 +--R (- 2cosh(a x) + 4cosh(a x) - 2)sinh(a x) - cosh(a x) + 3cosh(a x) +--R + +--R - 3cosh(a x) + 1 +--R * +--R log(sinh(a x) + cosh(a x) - 1) +--R + +--R 2 2 +--R sinh(a x) - cosh(a x) + 1 +--R / +--R 2 +--R (2a cosh(a x) - 2a)sinh(a x) +--R + +--R 2 3 +--R (4a cosh(a x) - 8a cosh(a x) + 4a)sinh(a x) + 2a cosh(a x) +--R + +--R 2 +--R - 6a cosh(a x) + 6a cosh(a x) - 2a +--R Type: Expression Integer +--E + +--S 75 +coshcuberule:=rule(cosh(x)^3 == 1/4*cosh(3*x)-3/4*cosh(x)) +--R +--R 3 cosh(3x) - 3cosh(x) +--R (4) cosh(x) == ------------------- +--R 4 +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 76 +dd:=coshcuberule cc +--R +--R (5) +--R 2 2 +--R (4cosh(a x) - 4)sinh(a x) + (8cosh(a x) - 16cosh(a x) + 8)sinh(a x) +--R + +--R 2 +--R cosh(3a x) - 12cosh(a x) + 9cosh(a x) - 4 +--R * +--R a x +--R log(tanh(---)) +--R 2 +--R + +--R 2 2 +--R (4cosh(a x) - 4)sinh(a x) + (8cosh(a x) - 16cosh(a x) + 8)sinh(a x) +--R + +--R 2 +--R cosh(3a x) - 12cosh(a x) + 9cosh(a x) - 4 +--R * +--R log(sinh(a x) + cosh(a x) + 1) +--R + +--R 2 +--R (- 4cosh(a x) + 4)sinh(a x) +--R + +--R 2 +--R (- 8cosh(a x) + 16cosh(a x) - 8)sinh(a x) - cosh(3a x) +--R + +--R 2 +--R 12cosh(a x) - 9cosh(a x) + 4 +--R * +--R log(sinh(a x) + cosh(a x) - 1) +--R + +--R 2 2 +--R 4sinh(a x) - 4cosh(a x) + 4 +--R / +--R 2 +--R (8a cosh(a x) - 8a)sinh(a x) +--R + +--R 2 +--R (16a cosh(a x) - 32a cosh(a x) + 16a)sinh(a x) + 2a cosh(3a x) +--R + +--R 2 +--R - 24a cosh(a x) + 18a cosh(a x) - 8a +--R Type: Expression Integer +--E + +--S 77 +sinhsqrrule:=rule(sinh(x)^2 == 1/2*cosh(2*x)-1/2) +--R +--R 2 cosh(2x) - 1 +--R (6) sinh(x) == ------------ +--R 2 +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 78 +ee:=sinhsqrrule dd +--R +--R (7) +--R 2 +--R (8cosh(a x) - 16cosh(a x) + 8)sinh(a x) + cosh(3a x) +--R + +--R 2 +--R (2cosh(a x) - 2)cosh(2a x) - 12cosh(a x) + 7cosh(a x) - 2 +--R * +--R a x +--R log(tanh(---)) +--R 2 +--R + +--R 2 +--R (8cosh(a x) - 16cosh(a x) + 8)sinh(a x) + cosh(3a x) +--R + +--R 2 +--R (2cosh(a x) - 2)cosh(2a x) - 12cosh(a x) + 7cosh(a x) - 2 +--R * +--R log(sinh(a x) + cosh(a x) + 1) +--R + +--R 2 +--R (- 8cosh(a x) + 16cosh(a x) - 8)sinh(a x) - cosh(3a x) +--R + +--R 2 +--R (- 2cosh(a x) + 2)cosh(2a x) + 12cosh(a x) - 7cosh(a x) + 2 +--R * +--R log(sinh(a x) + cosh(a x) - 1) +--R + +--R 2 +--R 2cosh(2a x) - 4cosh(a x) + 2 +--R / +--R 2 +--R (16a cosh(a x) - 32a cosh(a x) + 16a)sinh(a x) + 2a cosh(3a x) +--R + +--R 2 +--R (4a cosh(a x) - 4a)cosh(2a x) - 24a cosh(a x) + 14a cosh(a x) - 4a +--R Type: Expression Integer +--E + +--S 79 +coshsqrrule:=rule(cosh(x)^2 == 1/2*cosh(2*x)+1/2) +--R +--R 2 cosh(2x) + 1 +--R (8) cosh(x) == ------------ +--R 2 +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 80 +ff:=coshsqrrule ee +--R +--R (9) +--R a x +--R log(tanh(---)) + log(sinh(a x) + cosh(a x) + 1) +--R 2 +--R + +--R - log(sinh(a x) + cosh(a x) - 1) +--R / +--R 2a +--R Type: Expression Integer +--E + +--S 81 14:603 Schaums and Axiom agree +gg:=complexNormalize ff +--R +--R (10) 0 +--R Type: Expression Integer +--E + )spool )lisp (bye) @ diff --git a/src/input/schaum30.input.pamphlet b/src/input/schaum30.input.pamphlet index 0e1aaf4..5df48bc 100644 --- a/src/input/schaum30.input.pamphlet +++ b/src/input/schaum30.input.pamphlet @@ -18,7 +18,7 @@ $$)set message auto off )clear all ---S 1 of 11 +--S 1 aa:=integrate(tanh(a*x),x) --R --R @@ -29,6 +29,44 @@ aa:=integrate(tanh(a*x),x) --R a --R Type: Union(Expression Integer,...) --E + +--S 2 +bb:=1/a*log(cosh(a*x)) +--R +--R log(cosh(a x)) +--R (2) -------------- +--R a +--R Type: Expression Integer +--E + +--S 3 +cc:=aa-bb +--R +--R 2cosh(a x) +--R - log(cosh(a x)) + log(- ---------------------) - a x +--R sinh(a x) - cosh(a x) +--R (3) ----------------------------------------------------- +--R a +--R Type: Expression Integer +--E + +--S 4 +dd:=expandLog cc +--R +--R - log(sinh(a x) - cosh(a x)) + log(- 2) - a x +--R (4) --------------------------------------------- +--R a +--R Type: Expression Integer +--E + +--S 5 14:604 Schaums and Axiom differ by a constant +ee:=complexNormalize dd +--R +--R - log(- 1) + log(- 2) +--R (5) --------------------- +--R a +--R Type: Expression Integer +--E @ \section{\cite{1}:14.605~~~~~\displaystyle @@ -39,7 +77,7 @@$$ <<*>>= )clear all ---S 2 of 11 +--S 6 aa:=integrate(tanh(a*x)^2,x) --R --R @@ -48,6 +86,42 @@ aa:=integrate(tanh(a*x)^2,x) --R a cosh(a x) --R Type: Union(Expression Integer,...) --E + +--S 7 +bb:=x-tanh(a*x)/a +--R +--R - tanh(a x) + a x +--R (2) ----------------- +--R a +--R Type: Expression Integer +--E + +--S 8 +cc:=aa-bb +--R +--R cosh(a x)tanh(a x) - sinh(a x) + cosh(a x) +--R (3) ------------------------------------------ +--R a cosh(a x) +--R Type: Expression Integer +--E + +--S 9 +tanhrule:=rule(tanh(x) == sinh(x)/cosh(x)) +--R +--R sinh(x) +--R (4) tanh(x) == ------- +--R cosh(x) +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 10 14:605 Schaums and Axiom differ by a constant +dd:=tanhrule cc +--R +--R 1 +--R (5) - +--R a +--R Type: Expression Integer +--E @ \section{\cite{1}:14.606~~~~~$\displaystyle
@@ -58,7 +132,7 @@ $$<<*>>= )clear all ---S 3 of 11 +--S 11 aa:=integrate(tanh(a*x)^3,x) --R --R @@ -92,6 +166,69 @@ aa:=integrate(tanh(a*x)^3,x) --R (4a cosh(a x) + 4a cosh(a x))sinh(a x) + a cosh(a x) + 2a cosh(a x) + a --R Type: Union(Expression Integer,...) --E + +--S 12 +bb:=1/a*log(cosh(a*x))-tanh(a*x)^2/(2*a) +--R +--R 2 +--R 2log(cosh(a x)) - tanh(a x) +--R (2) ---------------------------- +--R 2a +--R Type: Expression Integer +--E + +--S 13 14:606 Axiom cannot simplify this expression +cc:=aa-bb +--R +--R (3) +--R 4 3 2 2 +--R - 2sinh(a x) - 8cosh(a x)sinh(a x) + (- 12cosh(a x) - 4)sinh(a x) +--R + +--R 3 4 2 +--R (- 8cosh(a x) - 8cosh(a x))sinh(a x) - 2cosh(a x) - 4cosh(a x) - 2 +--R * +--R log(cosh(a x)) +--R + +--R 4 3 2 2 +--R 2sinh(a x) + 8cosh(a x)sinh(a x) + (12cosh(a x) + 4)sinh(a x) +--R + +--R 3 4 2 +--R (8cosh(a x) + 8cosh(a x))sinh(a x) + 2cosh(a x) + 4cosh(a x) + 2 +--R * +--R 2cosh(a x) +--R log(- ---------------------) +--R sinh(a x) - cosh(a x) +--R + +--R 4 3 2 2 +--R sinh(a x) + 4cosh(a x)sinh(a x) + (6cosh(a x) + 2)sinh(a x) +--R + +--R 3 4 2 +--R (4cosh(a x) + 4cosh(a x))sinh(a x) + cosh(a x) + 2cosh(a x) + 1 +--R * +--R 2 +--R tanh(a x) +--R + +--R 4 3 +--R - 2a x sinh(a x) - 8a x cosh(a x)sinh(a x) +--R + +--R 2 2 +--R (- 12a x cosh(a x) - 4a x + 4)sinh(a x) +--R + +--R 3 4 +--R (- 8a x cosh(a x) + (- 8a x + 8)cosh(a x))sinh(a x) - 2a x cosh(a x) +--R + +--R 2 +--R (- 4a x + 4)cosh(a x) - 2a x +--R / +--R 4 3 2 2 +--R 2a sinh(a x) + 8a cosh(a x)sinh(a x) + (12a cosh(a x) + 4a)sinh(a x) +--R + +--R 3 4 2 +--R (8a cosh(a x) + 8a cosh(a x))sinh(a x) + 2a cosh(a x) + 4a cosh(a x) +--R + +--R 2a +--R Type: Expression Integer +--E @ \section{\cite{1}:14.607~~~~~\displaystyle @@ -102,7 +239,7 @@$$
<<*>>=
)clear all

---S 4 of 11
+--S 14
aa:=integrate(tanh(a*x)^n*sech(a*x)^2,x)
--R
--R
@@ -113,6 +250,31 @@ aa:=integrate(tanh(a*x)^n*sech(a*x)^2,x)
--R                                (a n + a)cosh(a x)
--R                                          Type: Union(Expression
Integer,...)
--E
+
+--S 15
+bb:=tanh(a*x)^(n+1)/((n+1)*a)
+--R
+--R                 n + 1
+--R        tanh(a x)
+--R   (2)  --------------
+--R            a n + a
+--R                                                     Type: Expression
Integer
+--E
+
+--S 16     14:607 Axiom cannot simplify this expression
+cc:=aa-bb
+--R
+--R   (3)
+--R                           sinh(a x)                         sinh(a x)
+--R       sinh(a x)sinh(n log(---------)) + sinh(a x)cosh(n log(---------))
+--R                           cosh(a x)                         cosh(a x)
+--R     +
+--R                           n + 1
+--R       - cosh(a x)tanh(a x)
+--R  /
+--R     (a n + a)cosh(a x)
+--R                                                     Type: Expression
Integer
+--E
@

\section{\cite{1}:14.608~~~~~$\displaystyle @@ -123,7 +285,7 @@ $$<<*>>= )clear all ---S 5 of 11 +--S 17 aa:=integrate(sech(a*x)^2/tanh(a*x),x) --R --R @@ -134,6 +296,63 @@ aa:=integrate(sech(a*x)^2/tanh(a*x),x) --R a --R Type: Union(Expression Integer,...) --E + +--S 18 +bb:=1/a*log(tanh(a*x)) +--R +--R log(tanh(a x)) +--R (2) -------------- +--R a +--R Type: Expression Integer +--E + +--S 19 +cc:=aa-bb +--R +--R (3) +--R 2cosh(a x) +--R - log(tanh(a x)) - log(- ---------------------) +--R sinh(a x) - cosh(a x) +--R + +--R 2sinh(a x) +--R log(- ---------------------) +--R sinh(a x) - cosh(a x) +--R / +--R a +--R Type: Expression Integer +--E + +--S 20 +tanhrule:=rule(tanh(x) == sinh(x)/cosh(x)) +--R +--R sinh(x) +--R (4) tanh(x) == ------- +--R cosh(x) +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 21 +dd:=tanhrule cc +--R +--R (5) +--R sinh(a x) 2cosh(a x) +--R - log(---------) - log(- ---------------------) +--R cosh(a x) sinh(a x) - cosh(a x) +--R + +--R 2sinh(a x) +--R log(- ---------------------) +--R sinh(a x) - cosh(a x) +--R / +--R a +--R Type: Expression Integer +--E + +--S 22 14:608 Schaums and Axiom agree +ee:=expandLog dd +--R +--R (6) 0 +--R Type: Expression Integer +--E @ \section{\cite{1}:14.609~~~~~\displaystyle @@ -144,7 +363,7 @@$$ <<*>>= )clear all ---S 6 of 11 +--S 23 aa:=integrate(1/tanh(a*x),x) --R --R @@ -155,6 +374,44 @@ aa:=integrate(1/tanh(a*x),x) --R a --R Type: Union(Expression Integer,...) --E + +--S 24 +bb:=1/a*log(sinh(a*x)) +--R +--R log(sinh(a x)) +--R (2) -------------- +--R a +--R Type: Expression Integer +--E + +--S 25 +cc:=aa-bb +--R +--R 2sinh(a x) +--R - log(sinh(a x)) + log(- ---------------------) - a x +--R sinh(a x) - cosh(a x) +--R (3) ----------------------------------------------------- +--R a +--R Type: Expression Integer +--E + +--S 26 +dd:=expandLog cc +--R +--R - log(sinh(a x) - cosh(a x)) + log(- 2) - a x +--R (4) --------------------------------------------- +--R a +--R Type: Expression Integer +--E + +--S 27 14:609 Schaums and Axiom differ by a constant +ee:=complexNormalize dd +--R +--R - log(- 1) + log(- 2) +--R (5) --------------------- +--R a +--R Type: Expression Integer +--E @ \section{\cite{1}:14.610~~~~~$\displaystyle
@@ -167,7 +424,7 @@ $$<<*>>= )clear all ---S 7 of 11 +--S 28 14:610 Axiom cannot compute this integral aa:=integrate(x*tanh(a*x),x) --R --R @@ -187,7 +444,7 @@$$
<<*>>=
)clear all

---S 8 of 11
+--S 29
aa:=integrate(x*tanh(a*x)^2,x)
--R
--R
@@ -209,6 +466,159 @@ aa:=integrate(x*tanh(a*x)^2,x)
--R     2a sinh(a x)  + 4a cosh(a x)sinh(a x) + 2a cosh(a x)  + 2a
--R                                          Type: Union(Expression
Integer,...)
--E
+
+--S 30
+bb:=x^2/2-(x*tanh(a*x))/a+1/a^2*log(cosh(a*x))
+--R
+--R                                            2 2
+--R        2log(cosh(a x)) - 2a x tanh(a x) + a x
+--R   (2)  ---------------------------------------
+--R                            2
+--R                          2a
+--R                                                     Type: Expression
Integer
+--E
+
+--S 31
+cc:=aa-bb
+--R
+--R   (3)
+--R                   2                                  2
+--R       (- sinh(a x)  - 2cosh(a x)sinh(a x) - cosh(a x)  - 1)log(cosh(a x))
+--R     +
+--R                   2                                  2
+--R         (sinh(a x)  + 2cosh(a x)sinh(a x) + cosh(a x)  + 1)
+--R      *
+--R                     2cosh(a x)
+--R         log(- ---------------------)
+--R               sinh(a x) - cosh(a x)
+--R     +
+--R                       2                                          2
+--R         (a x sinh(a x)  + 2a x cosh(a x)sinh(a x) + a x cosh(a x)  + a x)
+--R      *
+--R         tanh(a x)
+--R     +
+--R                       2                                           2
+--R       - 2a x sinh(a x)  - 4a x cosh(a x)sinh(a x) - 2a x cosh(a x)
+--R  /
+--R      2         2     2                      2         2    2
+--R     a sinh(a x)  + 2a cosh(a x)sinh(a x) + a cosh(a x)  + a
+--R                                                     Type: Expression
Integer
+--E
+
+--S 32
+sinhsqrrule:=rule(sinh(x)^2 == 1/2*cosh(2*x)-1/2)
+--R
+--R               2    cosh(2x) - 1
+--R   (4)  sinh(x)  == ------------
+--R                          2
+--R                        Type: RewriteRule(Integer,Integer,Expression
Integer)
+--E
+
+--S 33
+dd:=sinhsqrrule cc
+--R
+--R   (5)
+--R                                                       2
+--R       (- 4cosh(a x)sinh(a x) - cosh(2a x) - 2cosh(a x)  - 1)log(cosh(a x))
+--R     +
+--R                                                       2
+--R         (4cosh(a x)sinh(a x) + cosh(2a x) + 2cosh(a x)  + 1)
+--R      *
+--R                     2cosh(a x)
+--R         log(- ---------------------)
+--R               sinh(a x) - cosh(a x)
+--R     +
+--R                                                                   2
+--R         (4a x cosh(a x)sinh(a x) + a x cosh(2a x) + 2a x cosh(a x)  + a x)
+--R      *
+--R         tanh(a x)
+--R     +
+--R                                                                   2
+--R       - 8a x cosh(a x)sinh(a x) - 2a x cosh(2a x) - 4a x cosh(a x)  + 2a x
+--R  /
+--R       2                      2               2         2    2
+--R     4a cosh(a x)sinh(a x) + a cosh(2a x) + 2a cosh(a x)  + a
+--R                                                     Type: Expression
Integer
+--E
+
+--S 34
+coshsqrrule:=rule(cosh(x)^2 == 1/2*cosh(2*x)+1/2)
+--R
+--R               2    cosh(2x) + 1
+--R   (6)  cosh(x)  == ------------
+--R                          2
+--R                        Type: RewriteRule(Integer,Integer,Expression
Integer)
+--E
+
+--S 35
+ee:=coshsqrrule dd
+--R
+--R   (7)
+--R       (- 2cosh(a x)sinh(a x) - cosh(2a x) - 1)log(cosh(a x))
+--R     +
+--R                                                         2cosh(a x)
+--R       (2cosh(a x)sinh(a x) + cosh(2a x) + 1)log(- ---------------------)
+--R                                                   sinh(a x) - cosh(a x)
+--R     +
+--R       (2a x cosh(a x)sinh(a x) + a x cosh(2a x) + a x)tanh(a x)
+--R     +
+--R       - 4a x cosh(a x)sinh(a x) - 2a x cosh(2a x)
+--R  /
+--R       2                      2              2
+--R     2a cosh(a x)sinh(a x) + a cosh(2a x) + a
+--R                                                     Type: Expression
Integer
+--E
+
+--S 36
+ff:=expandLog ee
+--R
+--R   (8)
+--R       (- 2cosh(a x)sinh(a x) - cosh(2a x) - 1)log(sinh(a x) - cosh(a x))
+--R     +
+--R       (2a x cosh(a x)sinh(a x) + a x cosh(2a x) + a x)tanh(a x)
+--R     +
+--R       (2log(- 2) - 4a x)cosh(a x)sinh(a x) + (log(- 2) - 2a x)cosh(2a x)
+--R     +
+--R       log(- 2)
+--R  /
+--R       2                      2              2
+--R     2a cosh(a x)sinh(a x) + a cosh(2a x) + a
+--R                                                     Type: Expression
Integer
+--E
+
+--S 37
+sinhcoshrule:=rule(sinh(x)*cosh(y) == 1/2*(sinh(x+y)+sinh(x-y)))
+--R
+--I                             %N sinh(y + x) - %N sinh(y - x)
+--I   (9)  %N cosh(y)sinh(x) == -------------------------------
+--R                                            2
+--R                        Type: RewriteRule(Integer,Integer,Expression
Integer)
+--E
+
+--S 38
+gg:=sinhcoshrule ff
+--R
+--R   (10)
+--R       (- sinh(2a x) - cosh(2a x) - 1)log(sinh(a x) - cosh(a x))
+--R     +
+--R       (a x sinh(2a x) + a x cosh(2a x) + a x)tanh(a x)
+--R     +
+--R       (log(- 2) - 2a x)sinh(2a x) + (log(- 2) - 2a x)cosh(2a x) + log(- 2)
+--R  /
+--R      2              2              2
+--R     a sinh(2a x) + a cosh(2a x) + a
+--R                                                     Type: Expression
Integer
+--E
+
+--S 39     14:611 Schaums and Axiom differ by a constant
+hh:=complexNormalize gg
+--R
+--R         - log(- 1) + log(- 2)
+--R   (11)  ---------------------
+--R                    2
+--R                   a
+--R                                                     Type: Expression
Integer
+--E
@

\section{\cite{1}:14.612~~~~~$\displaystyle @@ -220,7 +630,7 @@ $$<<*>>= )clear all ---S 9 of 11 +--S 40 14:612 Axiom cannot compute this integral aa:=integrate(tanh(a*x)/x,x) --R --R @@ -240,7 +650,7 @@$$ <<*>>= )clear all ---S 10 of 11 +--S 41 aa:=integrate(1/(p+q*tanh(a*x)),x) --R --R @@ -252,6 +662,54 @@ aa:=integrate(1/(p+q*tanh(a*x)),x) --R a q - a p --R Type: Union(Expression Integer,...) --E + +--S 42 +bb:=(p*x)/(p^2-q^2)-q/(a*(p^2-q^2))*log(q*sinh(a*x)+p*cosh(a*x)) +--R +--R q log(q sinh(a x) + p cosh(a x)) - a p x +--R (2) ---------------------------------------- +--R 2 2 +--R a q - a p +--R Type: Expression Integer +--E + +--S 43 +cc:=aa-bb +--R +--R (3) +--R - 2q sinh(a x) - 2p cosh(a x) +--R - q log(q sinh(a x) + p cosh(a x)) + q log(-----------------------------) +--R sinh(a x) - cosh(a x) +--R + +--R - a q x +--R / +--R 2 2 +--R a q - a p +--R Type: Expression Integer +--E + +--S 44 +dd:=expandLog cc +--R +--R (4) +--R - q log(q sinh(a x) + p cosh(a x)) - q log(sinh(a x) - cosh(a x)) +--R + +--R q log(- q sinh(a x) - p cosh(a x)) + q log(2) - a q x +--R / +--R 2 2 +--R a q - a p +--R Type: Expression Integer +--E + +--S 45 14:613 Schaums and Axiom differ by a constant +ee:=complexNormalize dd +--R +--R q log(2) - 2q log(- 1) +--R (5) ---------------------- +--R 2 2 +--R a q - a p +--R Type: Expression Integer +--E @ \section{\cite{1}:14.614~~~~~$\displaystyle
@@ -262,7 +720,7 @@ $$<<*>>= )clear all ---S 11 of 11 +--S 46 14:614 Axiom cannot compute this integral aa:=integrate(tanh(a*x)^n,x) --R --R diff --git a/src/input/schaum31.input.pamphlet b/src/input/schaum31.input.pamphlet index 3e81bf2..343aa2a 100644 --- a/src/input/schaum31.input.pamphlet +++ b/src/input/schaum31.input.pamphlet @@ -18,7 +18,7 @@$$
)set message auto off
)clear all

---S 1 of 11
+--S 1
aa:=integrate(coth(a*x),x)
--R
--R
@@ -29,6 +29,44 @@ aa:=integrate(coth(a*x),x)
--R                         a
--R                                          Type: Union(Expression
Integer,...)
--E
+
+--S 2
+bb:=1/a*log(sinh(a*x))
+--R
+--R        log(sinh(a x))
+--R   (2)  --------------
+--R               a
+--R                                                     Type: Expression
Integer
+--E
+
+--S 3
+cc:=aa-bb
+--R
+--R                                       2sinh(a x)
+--R        - log(sinh(a x)) + log(- ---------------------) - a x
+--R                                 sinh(a x) - cosh(a x)
+--R   (3)  -----------------------------------------------------
+--R                                  a
+--R                                                     Type: Expression
Integer
+--E
+
+--S 4
+dd:=expandLog cc
+--R
+--R        - log(sinh(a x) - cosh(a x)) + log(- 2) - a x
+--R   (4)  ---------------------------------------------
+--R                              a
+--R                                                     Type: Expression
Integer
+--E
+
+--S 5      14:615 Schaums and Axiom differ by a constant
+ee:=complexNormalize dd
+--R
+--R        - log(- 1) + log(- 2)
+--R   (5)  ---------------------
+--R                  a
+--R                                                     Type: Expression
Integer
+--E
@

\section{\cite{1}:14.616~~~~~$\displaystyle @@ -39,7 +77,7 @@ $$<<*>>= )clear all ---S 2 of 11 +--S 6 aa:=integrate(coth(a*x)^2,x) --R --R @@ -48,6 +86,33 @@ aa:=integrate(coth(a*x)^2,x) --R a sinh(a x) --R Type: Union(Expression Integer,...) --E + +--S 7 +bb:=x-coth(a*x)/a +--R +--R - coth(a x) + a x +--R (2) ----------------- +--R a +--R Type: Expression Integer +--E + +--S 8 +cc:=aa-bb +--R +--R (coth(a x) + 1)sinh(a x) - cosh(a x) +--R (3) ------------------------------------ +--R a sinh(a x) +--R Type: Expression Integer +--E + +--S 9 14:616 Schaums and Axiom differ by a constant +dd:=complexNormalize cc +--R +--R 1 +--R (4) - +--R a +--R Type: Expression Integer +--E @ \section{\cite{1}:14.617~~~~~\displaystyle @@ -58,7 +123,7 @@$$ <<*>>= )clear all ---S 3 of 11 +--S 10 aa:=integrate(coth(a*x)^3,x) --R --R @@ -92,6 +157,69 @@ aa:=integrate(coth(a*x)^3,x) --R (4a cosh(a x) - 4a cosh(a x))sinh(a x) + a cosh(a x) - 2a cosh(a x) + a --R Type: Union(Expression Integer,...) --E + +--S 11 +bb:=1/a*log(sinh(a*x)-coth(a*x)^2)/(2*a) +--R +--R 2 +--R log(sinh(a x) - coth(a x) ) +--R (2) --------------------------- +--R 2 +--R 2a +--R Type: Expression Integer +--E + +--S 12 14:617 Axiom cannot simplify this expression +cc:=aa-bb +--R +--R (3) +--R 4 3 2 2 +--R - sinh(a x) - 4cosh(a x)sinh(a x) + (- 6cosh(a x) + 2)sinh(a x) +--R + +--R 3 4 2 +--R (- 4cosh(a x) + 4cosh(a x))sinh(a x) - cosh(a x) + 2cosh(a x) - 1 +--R * +--R 2 +--R log(sinh(a x) - coth(a x) ) +--R + +--R 4 3 +--R 2a sinh(a x) + 8a cosh(a x)sinh(a x) +--R + +--R 2 2 +--R (12a cosh(a x) - 4a)sinh(a x) +--R + +--R 3 4 +--R (8a cosh(a x) - 8a cosh(a x))sinh(a x) + 2a cosh(a x) +--R + +--R 2 +--R - 4a cosh(a x) + 2a +--R * +--R 2sinh(a x) +--R log(- ---------------------) +--R sinh(a x) - cosh(a x) +--R + +--R 2 4 2 3 +--R - 2a x sinh(a x) - 8a x cosh(a x)sinh(a x) +--R + +--R 2 2 2 2 +--R (- 12a x cosh(a x) + 4a x - 4a)sinh(a x) +--R + +--R 2 3 2 2 4 +--R (- 8a x cosh(a x) + (8a x - 8a)cosh(a x))sinh(a x) - 2a x cosh(a x) +--R + +--R 2 2 2 +--R (4a x - 4a)cosh(a x) - 2a x +--R / +--R 2 4 2 3 2 2 2 2 +--R 2a sinh(a x) + 8a cosh(a x)sinh(a x) + (12a cosh(a x) - 4a )sinh(a x) +--R + +--R 2 3 2 2 4 2 2 +--R (8a cosh(a x) - 8a cosh(a x))sinh(a x) + 2a cosh(a x) - 4a cosh(a x) +--R + +--R 2 +--R 2a +--R Type: Expression Integer +--E @ \section{\cite{1}:14.618~~~~~$\displaystyle
@@ -102,7 +230,7 @@ $$<<*>>= )clear all ---S 4 of 11 +--S 13 aa:=integrate(coth(a*x)^n*csch(a*x)^2,x) --R --R @@ -113,6 +241,53 @@ aa:=integrate(coth(a*x)^n*csch(a*x)^2,x) --R (a n + a)sinh(a x) --R Type: Union(Expression Integer,...) --E + +--S 14 +bb:=-coth(a*x)^(n+1)/((n+1)*a) +--R +--R n + 1 +--R coth(a x) +--R (2) - -------------- +--R a n + a +--R Type: Expression Integer +--E + +--S 15 +cc:=aa-bb +--R +--R (3) +--R cosh(a x) cosh(a x) +--R - cosh(a x)sinh(n log(---------)) - cosh(a x)cosh(n log(---------)) +--R sinh(a x) sinh(a x) +--R + +--R n + 1 +--R sinh(a x)coth(a x) +--R / +--R (a n + a)sinh(a x) +--R Type: Expression Integer +--E + +--S 16 +dd:=expandLog cc +--R +--R (4) +--R cosh(a x)sinh(n log(sinh(a x)) - n log(cosh(a x))) +--R + +--R - cosh(a x)cosh(n log(sinh(a x)) - n log(cosh(a x))) +--R + +--R n + 1 +--R sinh(a x)coth(a x) +--R / +--R (a n + a)sinh(a x) +--R Type: Expression Integer +--E + +--S 17 14:618 Schaums and Axiom agree +ee:=complexNormalize dd +--R +--R (5) 0 +--R Type: Expression Integer +--E @ \section{\cite{1}:14.619~~~~~\displaystyle @@ -123,7 +298,7 @@$$
<<*>>=
)clear all

---S 5 of 11
+--S 18
aa:=integrate(csch(a*x)^2/coth(a*x),x)
--R
--R
@@ -134,6 +309,43 @@ aa:=integrate(csch(a*x)^2/coth(a*x),x)
--R                                      a
--R                                          Type: Union(Expression
Integer,...)
--E
+
+--S 19
+bb:=-1/a*log(coth(a*x))
+--R
+--R          log(coth(a x))
+--R   (2)  - --------------
+--R                 a
+--R                                                     Type: Expression
Integer
+--E
+
+--S 20
+cc:=aa-bb
+--R
+--R   (3)
+--R                                2cosh(a x)                     2sinh(a x)
+--R   log(coth(a x)) - log(- ---------------------) + log(-
---------------------)
+--R                          sinh(a x) - cosh(a x)          sinh(a x) - cosh(a
x)
+--R
----------------------------------------------------------------------------
+--R                                         a
+--R                                                     Type: Expression
Integer
+--E
+
+--S 21
+dd:=expandLog cc
+--R
+--R        log(sinh(a x)) + log(coth(a x)) - log(cosh(a x))
+--R   (4)  ------------------------------------------------
+--R                                a
+--R                                                     Type: Expression
Integer
+--E
+
+--S 22     14:619 Schaums and Axiom agree
+ee:=complexNormalize dd
+--R
+--R   (5)  0
+--R                                                     Type: Expression
Integer
+--E
@

\section{\cite{1}:14.620~~~~~$\displaystyle @@ -144,7 +356,7 @@ $$<<*>>= )clear all ---S 6 of 11 +--S 23 aa:=integrate(1/coth(a*x),x) --R --R @@ -155,6 +367,44 @@ aa:=integrate(1/coth(a*x),x) --R a --R Type: Union(Expression Integer,...) --E + +--S 24 +bb:=1/a*log(cosh(a*x)) +--R +--R log(cosh(a x)) +--R (2) -------------- +--R a +--R Type: Expression Integer +--E + +--S 25 +cc:=aa-bb +--R +--R 2cosh(a x) +--R - log(cosh(a x)) + log(- ---------------------) - a x +--R sinh(a x) - cosh(a x) +--R (3) ----------------------------------------------------- +--R a +--R Type: Expression Integer +--E + +--S 26 +dd:=expandLog cc +--R +--R - log(sinh(a x) - cosh(a x)) + log(- 2) - a x +--R (4) --------------------------------------------- +--R a +--R Type: Expression Integer +--E + +--S 27 14:620 Schaums and Axiom differ by a constant +ee:=complexNormalize dd +--R +--R - log(- 1) + log(- 2) +--R (5) --------------------- +--R a +--R Type: Expression Integer +--E @ \section{\cite{1}:14.621~~~~~\displaystyle @@ -167,7 +417,7 @@$$ <<*>>= )clear all ---S 7 of 11 +--S 28 14:621 Axiom cannot compute this integral aa:=integrate(x*coth(a*x),x) --R --R @@ -187,7 +437,7 @@ $$<<*>>= )clear all ---S 8 of 11 +--S 29 aa:=integrate(x*coth(a*x)^2,x) --R --R @@ -209,6 +459,157 @@ aa:=integrate(x*coth(a*x)^2,x) --R 2a sinh(a x) + 4a cosh(a x)sinh(a x) + 2a cosh(a x) - 2a --R Type: Union(Expression Integer,...) --E + +--S 30 +bb:=x^2/2-(x*coth(a*x)/a)+1/a^2*log(sinh(a*x)) +--R +--R 2 2 +--R 2log(sinh(a x)) - 2a x coth(a x) + a x +--R (2) --------------------------------------- +--R 2 +--R 2a +--R Type: Expression Integer +--E + +--S 31 +cc:=aa-bb +--R +--R (3) +--R 2 2 +--R (- sinh(a x) - 2cosh(a x)sinh(a x) - cosh(a x) + 1)log(sinh(a x)) +--R + +--R 2 2 +--R (sinh(a x) + 2cosh(a x)sinh(a x) + cosh(a x) - 1) +--R * +--R 2sinh(a x) +--R log(- ---------------------) +--R sinh(a x) - cosh(a x) +--R + +--R 2 +--R (a x coth(a x) - 2a x)sinh(a x) +--R + +--R (2a x cosh(a x)coth(a x) - 4a x cosh(a x))sinh(a x) +--R + +--R 2 2 +--R (a x cosh(a x) - a x)coth(a x) - 2a x cosh(a x) +--R / +--R 2 2 2 2 2 2 +--R a sinh(a x) + 2a cosh(a x)sinh(a x) + a cosh(a x) - a +--R Type: Expression Integer +--E + +--S 32 +dd:=expandLog cc +--R +--R (4) +--R 2 2 +--R (- sinh(a x) - 2cosh(a x)sinh(a x) - cosh(a x) + 1) +--R * +--R log(sinh(a x) - cosh(a x)) +--R + +--R 2 +--R (a x coth(a x) + log(- 2) - 2a x)sinh(a x) +--R + +--R (2a x cosh(a x)coth(a x) + (2log(- 2) - 4a x)cosh(a x))sinh(a x) +--R + +--R 2 2 +--R (a x cosh(a x) - a x)coth(a x) + (log(- 2) - 2a x)cosh(a x) - log(- 2) +--R / +--R 2 2 2 2 2 2 +--R a sinh(a x) + 2a cosh(a x)sinh(a x) + a cosh(a x) - a +--R Type: Expression Integer +--E + +--S 33 +sinhsqrrule:=rule(sinh(x)^2 == 1/2*cosh(2*x)-1/2) +--R +--R 2 cosh(2x) - 1 +--R (5) sinh(x) == ------------ +--R 2 +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 34 +ee:=sinhsqrrule dd +--R +--R (6) +--R 2 +--R (- 4cosh(a x)sinh(a x) - cosh(2a x) - 2cosh(a x) + 3) +--R * +--R log(sinh(a x) - cosh(a x)) +--R + +--R (4a x cosh(a x)coth(a x) + (4log(- 2) - 8a x)cosh(a x))sinh(a x) +--R + +--R 2 +--R (a x cosh(2a x) + 2a x cosh(a x) - 3a x)coth(a x) +--R + +--R 2 +--R (log(- 2) - 2a x)cosh(2a x) + (2log(- 2) - 4a x)cosh(a x) - 3log(- 2) +--R + +--R 2a x +--R / +--R 2 2 2 2 2 +--R 4a cosh(a x)sinh(a x) + a cosh(2a x) + 2a cosh(a x) - 3a +--R Type: Expression Integer +--E + +--S 35 +coshsqrrule:=rule(cosh(x)^2 == 1/2*cosh(2*x)+1/2) +--R +--R 2 cosh(2x) + 1 +--R (7) cosh(x) == ------------ +--R 2 +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 36 +ff:=coshsqrrule ee +--R +--R (8) +--R (- 2cosh(a x)sinh(a x) - cosh(2a x) + 1)log(sinh(a x) - cosh(a x)) +--R + +--R (2a x cosh(a x)coth(a x) + (2log(- 2) - 4a x)cosh(a x))sinh(a x) +--R + +--R (a x cosh(2a x) - a x)coth(a x) + (log(- 2) - 2a x)cosh(2a x) - log(- 2) +--R / +--R 2 2 2 +--R 2a cosh(a x)sinh(a x) + a cosh(2a x) - a +--R Type: Expression Integer +--E + +--S 37 +sinhcoshrule:=rule(sinh(x)*cosh(y) == 1/2*(sinh(x+y)+sinh(x-y))) +--R +--I %L sinh(y + x) - %L sinh(y - x) +--I (9) %L cosh(y)sinh(x) == ------------------------------- +--R 2 +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 38 +gg:=sinhcoshrule ff +--R +--R (10) +--R (- sinh(2a x) - cosh(2a x) + 1)log(sinh(a x) - cosh(a x)) +--R + +--R (a x coth(a x) + log(- 2) - 2a x)sinh(2a x) +--R + +--R (a x cosh(2a x) - a x)coth(a x) + (log(- 2) - 2a x)cosh(2a x) - log(- 2) +--R / +--R 2 2 2 +--R a sinh(2a x) + a cosh(2a x) - a +--R Type: Expression Integer +--E + +--S 39 14:622 Schaums and Axiom differ by a constant +hh:=complexNormalize gg +--R +--R - log(- 1) + log(- 2) +--R (11) --------------------- +--R 2 +--R a +--R Type: Expression Integer +--E @ \section{\cite{1}:14.623~~~~~\displaystyle @@ -220,7 +621,7 @@$$ <<*>>= )clear all ---S 9 of 11 +--S 40 14:623 Axiom cannot compute this integral aa:=integrate(coth(a*x)/x,x) --R --R @@ -240,7 +641,7 @@ $$<<*>>= )clear all ---S 10 of 11 +--S 41 aa:=integrate(1/(p+q*coth(a*x)),x) --R --R @@ -252,6 +653,54 @@ aa:=integrate(1/(p+q*coth(a*x)),x) --R a q - a p --R Type: Union(Expression Integer,...) --E + +--S 42 +bb:=(p*x)/(p^2-q^2)-q/(a*(p^2-q^2))*log(p*sinh(a*x)+q*cosh(a*x)) +--R +--R q log(p sinh(a x) + q cosh(a x)) - a p x +--R (2) ---------------------------------------- +--R 2 2 +--R a q - a p +--R Type: Expression Integer +--E + +--S 43 +cc:=aa-bb +--R +--R (3) +--R - 2p sinh(a x) - 2q cosh(a x) +--R - q log(p sinh(a x) + q cosh(a x)) + q log(-----------------------------) +--R sinh(a x) - cosh(a x) +--R + +--R - a q x +--R / +--R 2 2 +--R a q - a p +--R Type: Expression Integer +--E + +--S 44 +dd:=expandLog cc +--R +--R (4) +--R - q log(p sinh(a x) + q cosh(a x)) - q log(sinh(a x) - cosh(a x)) +--R + +--R q log(- p sinh(a x) - q cosh(a x)) + q log(2) - a q x +--R / +--R 2 2 +--R a q - a p +--R Type: Expression Integer +--E + +--S 45 14:624 Schaums and Axiom differ by a constant +ee:=complexNormalize dd +--R +--R q log(2) - 2q log(- 1) +--R (5) ---------------------- +--R 2 2 +--R a q - a p +--R Type: Expression Integer +--E @ \section{\cite{1}:14.625~~~~~\displaystyle @@ -262,7 +711,7 @@$$ <<*>>= )clear all ---S 11 of 11 +--S 46 14:625 Axiom cannot compute this integral aa:=integrate(coth(a*x)^n,x) --R --R diff --git a/src/input/schaum32.input.pamphlet b/src/input/schaum32.input.pamphlet index 085ddf0..e5b2409 100644 --- a/src/input/schaum32.input.pamphlet +++ b/src/input/schaum32.input.pamphlet @@ -18,7 +18,7 @@ $$)set message auto off )clear all ---S 1 of 10 +--S 1 aa:=integrate(sech(a*x),x) --R --R @@ -27,6 +27,70 @@ aa:=integrate(sech(a*x),x) --R a --R Type: Union(Expression Integer,...) --E + +--S 2 +bb:=2/a*atan(%e^(a*x)) +--R +--R a x +--R 2atan(%e ) +--R (2) ------------ +--R a +--R Type: Expression Integer +--E + +--S 3 +cc:=aa-bb +--R +--R a x +--R 2atan(sinh(a x) + cosh(a x)) - 2atan(%e ) +--R (3) ------------------------------------------- +--R a +--R Type: Expression Integer +--E + +--S 4 +atanrule:=rule(atan(x) == -%i/2*log((1+%i*x)/(1-%i*x))) +--R +--R - x + %i +--R %i log(--------) +--R x + %i +--R (4) atan(x) == - ---------------- +--R 2 +--R Type: RewriteRule(Integer,Complex Integer,Expression Complex Integer) +--E + +--S 5 +dd:=atanrule cc +--R +--R a x +--R - %e + %i - sinh(a x) - cosh(a x) + %i +--R %i log(------------) - %i log(----------------------------) +--R a x sinh(a x) + cosh(a x) + %i +--R %e + %i +--R (5) ----------------------------------------------------------- +--R a +--R Type: Expression Complex Integer +--E + +--S 6 +ee:=expandLog dd +--R +--R (6) +--R %i log(sinh(a x) + cosh(a x) + %i) - %i log(sinh(a x) + cosh(a x) - %i) +--R + +--R a x a x +--R - %i log(%e + %i) + %i log(%e - %i) +--R / +--R a +--R Type: Expression Complex Integer +--E + +--S 7 14:626 Schaums and Axiom agree +ff:=complexNormalize ee +--R +--R (7) 0 +--R Type: Expression Complex Integer +--E @ \section{\cite{1}:14.627~~~~~\displaystyle @@ -37,7 +101,7 @@$$ <<*>>= )clear all ---S 2 of 10 +--S 8 aa:=integrate(sech(a*x)^2,x) --R --R @@ -47,6 +111,91 @@ aa:=integrate(sech(a*x)^2,x) --R a sinh(a x) + 2a cosh(a x)sinh(a x) + a cosh(a x) + a --R Type: Union(Expression Integer,...) --E + +--S 9 +bb:=tanh(a*x)/a +--R +--R tanh(a x) +--R (2) --------- +--R a +--R Type: Expression Integer +--E + +--S 10 +cc:=aa-bb +--R +--R 2 2 +--R (- sinh(a x) - 2cosh(a x)sinh(a x) - cosh(a x) - 1)tanh(a x) - 2 +--R (3) ------------------------------------------------------------------ +--R 2 2 +--R a sinh(a x) + 2a cosh(a x)sinh(a x) + a cosh(a x) + a +--R Type: Expression Integer +--E + +--S 11 +sinhsqrrule:=rule(sinh(x)^2 == 1/2*cosh(2*x)-1/2) +--R +--R 2 cosh(2x) - 1 +--R (4) sinh(x) == ------------ +--R 2 +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 12 +dd:=sinhsqrrule cc +--R +--R 2 +--R (- 4cosh(a x)sinh(a x) - cosh(2a x) - 2cosh(a x) - 1)tanh(a x) - 4 +--R (5) ------------------------------------------------------------------- +--R 2 +--R 4a cosh(a x)sinh(a x) + a cosh(2a x) + 2a cosh(a x) + a +--R Type: Expression Integer +--E + +--S 13 +coshsqrrule:=rule(cosh(x)^2 == 1/2*cosh(2*x)+1/2) +--R +--R 2 cosh(2x) + 1 +--R (6) cosh(x) == ------------ +--R 2 +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 14 +ee:=coshsqrrule dd +--R +--R (- 2cosh(a x)sinh(a x) - cosh(2a x) - 1)tanh(a x) - 2 +--R (7) ----------------------------------------------------- +--R 2a cosh(a x)sinh(a x) + a cosh(2a x) + a +--R Type: Expression Integer +--E + +--S 15 +sinhcoshrule:=rule(sinh(x)*cosh(y) == 1/2*(sinh(x+y)+sinh(x-y))) +--R +--I %L sinh(y + x) - %L sinh(y - x) +--I (8) %L cosh(y)sinh(x) == ------------------------------- +--R 2 +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 16 +ff:=sinhcoshrule ee +--R +--R (- sinh(2a x) - cosh(2a x) - 1)tanh(a x) - 2 +--R (9) -------------------------------------------- +--R a sinh(2a x) + a cosh(2a x) + a +--R Type: Expression Integer +--E + +--S 17 14:627 Schaums and Axiom differ by a constant +gg:=complexNormalize ff +--R +--R 1 +--R (10) - - +--R a +--R Type: Expression Integer +--E @ \section{\cite{1}:14.628~~~~~$\displaystyle
@@ -57,7 +206,7 @@ $$<<*>>= )clear all ---S 3 of 10 +--S 18 aa:=integrate(sech(a*x)^3,x) --R --R @@ -83,6 +232,65 @@ aa:=integrate(sech(a*x)^3,x) --R (4a cosh(a x) + 4a cosh(a x))sinh(a x) + a cosh(a x) + 2a cosh(a x) + a --R Type: Union(Expression Integer,...) --E + +--S 19 +bb:=(sech(a*x)*tanh(a*x))/(2*a)+1/(2*a)*atan(sinh(a*x)) +--R +--R atan(sinh(a x)) + sech(a x)tanh(a x) +--R (2) ------------------------------------ +--R 2a +--R Type: Expression Integer +--E + +--S 20 14:628 Axiom cannot simplify this expression +cc:=aa-bb +--R +--R (3) +--R 4 3 2 2 +--R 2sinh(a x) + 8cosh(a x)sinh(a x) + (12cosh(a x) + 4)sinh(a x) +--R + +--R 3 4 2 +--R (8cosh(a x) + 8cosh(a x))sinh(a x) + 2cosh(a x) + 4cosh(a x) + 2 +--R * +--R atan(sinh(a x) + cosh(a x)) +--R + +--R 4 3 2 2 +--R - sinh(a x) - 4cosh(a x)sinh(a x) + (- 6cosh(a x) - 2)sinh(a x) +--R + +--R 3 4 2 +--R (- 4cosh(a x) - 4cosh(a x))sinh(a x) - cosh(a x) - 2cosh(a x) - 1 +--R * +--R atan(sinh(a x)) +--R + +--R 4 3 +--R - sech(a x)sinh(a x) - 4cosh(a x)sech(a x)sinh(a x) +--R + +--R 2 2 +--R (- 6cosh(a x) - 2)sech(a x)sinh(a x) +--R + +--R 3 +--R (- 4cosh(a x) - 4cosh(a x))sech(a x)sinh(a x) +--R + +--R 4 2 +--R (- cosh(a x) - 2cosh(a x) - 1)sech(a x) +--R * +--R tanh(a x) +--R + +--R 3 2 2 +--R 2sinh(a x) + 6cosh(a x)sinh(a x) + (6cosh(a x) - 2)sinh(a x) +--R + +--R 3 +--R 2cosh(a x) - 2cosh(a x) +--R / +--R 4 3 2 2 +--R 2a sinh(a x) + 8a cosh(a x)sinh(a x) + (12a cosh(a x) + 4a)sinh(a x) +--R + +--R 3 4 2 +--R (8a cosh(a x) + 8a cosh(a x))sinh(a x) + 2a cosh(a x) + 4a cosh(a x) +--R + +--R 2a +--R Type: Expression Integer +--E @ \section{\cite{1}:14.629~~~~~\displaystyle @@ -93,7 +301,7 @@$$
<<*>>=
)clear all

---S 4 of 10
+--S 21
aa:=integrate(sech(a*x)^n*tanh(a*x),x)
--R
--R
@@ -111,6 +319,100 @@ aa:=integrate(sech(a*x)^n*tanh(a*x),x)
--R     a n
--R                                          Type: Union(Expression
Integer,...)
--E
+
+--S 22
+bb:=-sech(a*x)^n/(n*a)
+--R
+--R                   n
+--R          sech(a x)
+--R   (2)  - ----------
+--R              a n
+--R                                                     Type: Expression
Integer
+--E
+
+--S 23
+cc:=aa-bb
+--R
+--R   (3)
+--R                                 2sinh(a x) + 2cosh(a x)
+--R       - sinh(n log(-------------------------------------------------))
+--R                             2                                  2
+--R                    sinh(a x)  + 2cosh(a x)sinh(a x) + cosh(a x)  + 1
+--R     +
+--R                                 2sinh(a x) + 2cosh(a x)
+--R       - cosh(n log(-------------------------------------------------))
+--R                             2                                  2
+--R                    sinh(a x)  + 2cosh(a x)sinh(a x) + cosh(a x)  + 1
+--R     +
+--R                n
+--R       sech(a x)
+--R  /
+--R     a n
+--R                                                     Type: Expression
Integer
+--E
+
+--S 24
+sechrule:=rule(sech(x) == 1/cosh(x))
+--R
+--R                      1
+--R   (4)  sech(x) == -------
+--R                   cosh(x)
+--R                        Type: RewriteRule(Integer,Integer,Expression
Integer)
+--E
+
+--S 25
+dd:=sechrule cc
+--R
+--R   (5)
+--R                                 2sinh(a x) + 2cosh(a x)
+--R       - sinh(n log(-------------------------------------------------))
+--R                             2                                  2
+--R                    sinh(a x)  + 2cosh(a x)sinh(a x) + cosh(a x)  + 1
+--R     +
+--R                                 2sinh(a x) + 2cosh(a x)
+--R       - cosh(n log(-------------------------------------------------))
+--R                             2                                  2
+--R                    sinh(a x)  + 2cosh(a x)sinh(a x) + cosh(a x)  + 1
+--R     +
+--R            1     n
+--R       (---------)
+--R        cosh(a x)
+--R  /
+--R     a n
+--R                                                     Type: Expression
Integer
+--E
+
+--S 26
+ee:=expandLog dd
+--R
+--R   (6)
+--R       sinh
+--R                           2                                  2
+--R            n log(sinh(a x)  + 2cosh(a x)sinh(a x) + cosh(a x)  + 1)
+--R          +
+--R            - n log(sinh(a x) + cosh(a x)) - n log(2)
+--R     +
+--R       -
+--R          cosh
+--R                              2                                  2
+--R               n log(sinh(a x)  + 2cosh(a x)sinh(a x) + cosh(a x)  + 1)
+--R             +
+--R               - n log(sinh(a x) + cosh(a x)) - n log(2)
+--R     +
+--R            1     n
+--R       (---------)
+--R        cosh(a x)
+--R  /
+--R     a n
+--R                                                     Type: Expression
Integer
+--E
+
+--S 27     14:629 Schaums and Axiom agree
+ff:=complexNormalize ee
+--R
+--R   (7)  0
+--R                                                     Type: Expression
Integer
+--E
@

\section{\cite{1}:14.630~~~~~$\displaystyle @@ -121,7 +423,7 @@ $$<<*>>= )clear all ---S 5 of 10 +--S 28 aa:=integrate(1/sech(a*x),x) --R --R @@ -130,6 +432,22 @@ aa:=integrate(1/sech(a*x),x) --R a --R Type: Union(Expression Integer,...) --E + +--S 29 +bb:=sinh(a*x)/a +--R +--R sinh(a x) +--R (2) --------- +--R a +--R Type: Expression Integer +--E + +--S 30 14:630 Schaums and Axiom agree +cc:=aa-bb +--R +--R (3) 0 +--R Type: Expression Integer +--E @ \section{\cite{1}:14.631~~~~~\displaystyle @@ -142,7 +460,7 @@$$ <<*>>= )clear all ---S 6 of 10 +--S 31 14:631 Axiom cannot compute this integral aa:=integrate(x*sech(a*x),x) --R --R @@ -162,7 +480,7 @@ $$<<*>>= )clear all ---S 7 of 10 +--S 32 aa:=integrate(x*sech(a*x)^2,x) --R --R @@ -181,6 +499,161 @@ aa:=integrate(x*sech(a*x)^2,x) --R a sinh(a x) + 2a cosh(a x)sinh(a x) + a cosh(a x) + a --R Type: Union(Expression Integer,...) --E + +--S 33 +bb:=(x*tanh(a*x))/a-1/a^2*log(cosh(a*x)) +--R +--R - log(cosh(a x)) + a x tanh(a x) +--R (2) -------------------------------- +--R 2 +--R a +--R Type: Expression Integer +--E + +--S 34 +cc:=aa-bb +--R +--R (3) +--R 2 2 +--R (sinh(a x) + 2cosh(a x)sinh(a x) + cosh(a x) + 1)log(cosh(a x)) +--R + +--R 2 2 +--R (- sinh(a x) - 2cosh(a x)sinh(a x) - cosh(a x) - 1) +--R * +--R 2cosh(a x) +--R log(- ---------------------) +--R sinh(a x) - cosh(a x) +--R + +--R 2 2 +--R (- a x sinh(a x) - 2a x cosh(a x)sinh(a x) - a x cosh(a x) - a x) +--R * +--R tanh(a x) +--R + +--R 2 2 +--R 2a x sinh(a x) + 4a x cosh(a x)sinh(a x) + 2a x cosh(a x) +--R / +--R 2 2 2 2 2 2 +--R a sinh(a x) + 2a cosh(a x)sinh(a x) + a cosh(a x) + a +--R Type: Expression Integer +--E + +--S 35 +dd:=expandLog cc +--R +--R (4) +--R 2 2 +--R (sinh(a x) + 2cosh(a x)sinh(a x) + cosh(a x) + 1) +--R * +--R log(sinh(a x) - cosh(a x)) +--R + +--R 2 2 +--R (- a x sinh(a x) - 2a x cosh(a x)sinh(a x) - a x cosh(a x) - a x) +--R * +--R tanh(a x) +--R + +--R 2 +--R (- log(- 2) + 2a x)sinh(a x) + (- 2log(- 2) + 4a x)cosh(a x)sinh(a x) +--R + +--R 2 +--R (- log(- 2) + 2a x)cosh(a x) - log(- 2) +--R / +--R 2 2 2 2 2 2 +--R a sinh(a x) + 2a cosh(a x)sinh(a x) + a cosh(a x) + a +--R Type: Expression Integer +--E + +--S 36 +sinhsqrrule:=rule(sinh(x)^2 == 1/2*cosh(2*x)-1/2) +--R +--R 2 cosh(2x) - 1 +--R (5) sinh(x) == ------------ +--R 2 +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 37 +ee:=sinhsqrrule dd +--R +--R (6) +--R 2 +--R (4cosh(a x)sinh(a x) + cosh(2a x) + 2cosh(a x) + 1) +--R * +--R log(sinh(a x) - cosh(a x)) +--R + +--R 2 +--R (- 4a x cosh(a x)sinh(a x) - a x cosh(2a x) - 2a x cosh(a x) - a x) +--R * +--R tanh(a x) +--R + +--R (- 4log(- 2) + 8a x)cosh(a x)sinh(a x) + (- log(- 2) + 2a x)cosh(2a x) +--R + +--R 2 +--R (- 2log(- 2) + 4a x)cosh(a x) - log(- 2) - 2a x +--R / +--R 2 2 2 2 2 +--R 4a cosh(a x)sinh(a x) + a cosh(2a x) + 2a cosh(a x) + a +--R Type: Expression Integer +--E + +--S 38 +coshsqrrule:=rule(cosh(x)^2 == 1/2*cosh(2*x)+1/2) +--R +--R 2 cosh(2x) + 1 +--R (7) cosh(x) == ------------ +--R 2 +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 39 +ff:=coshsqrrule ee +--R +--R (8) +--R (2cosh(a x)sinh(a x) + cosh(2a x) + 1)log(sinh(a x) - cosh(a x)) +--R + +--R (- 2a x cosh(a x)sinh(a x) - a x cosh(2a x) - a x)tanh(a x) +--R + +--R (- 2log(- 2) + 4a x)cosh(a x)sinh(a x) + (- log(- 2) + 2a x)cosh(2a x) +--R + +--R - log(- 2) +--R / +--R 2 2 2 +--R 2a cosh(a x)sinh(a x) + a cosh(2a x) + a +--R Type: Expression Integer +--E + +--S 40 +sinhcoshrule:=rule(sinh(x)*cosh(y) == 1/2*(sinh(x+y)+sinh(x-y))) +--R +--I %P sinh(y + x) - %P sinh(y - x) +--I (9) %P cosh(y)sinh(x) == ------------------------------- +--R 2 +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 41 +gg:=sinhcoshrule ff +--R +--R (10) +--R (sinh(2a x) + cosh(2a x) + 1)log(sinh(a x) - cosh(a x)) +--R + +--R (- a x sinh(2a x) - a x cosh(2a x) - a x)tanh(a x) +--R + +--R (- log(- 2) + 2a x)sinh(2a x) + (- log(- 2) + 2a x)cosh(2a x) - log(- 2) +--R / +--R 2 2 2 +--R a sinh(2a x) + a cosh(2a x) + a +--R Type: Expression Integer +--E + +--S 42 14:632 Schaums and Axiom differ by a constant +hh:=complexNormalize gg +--R +--R log(- 1) - log(- 2) +--R (11) ------------------- +--R 2 +--R a +--R Type: Expression Integer +--E @ \section{\cite{1}:14.633~~~~~\displaystyle @@ -192,7 +665,7 @@$$ <<*>>= )clear all ---S 8 of 10 +--S 43 14:633 Axiom cannot compute this integral aa:=integrate(sech(a*x)/x,x) --R --R @@ -212,7 +685,7 @@ $$<<*>>= )clear all ---S 9 of 10 +--S 44 aa:=integrate(1/(q+p*sech(a*x)),x) --R --R @@ -259,6 +732,238 @@ aa:=integrate(1/(q+p*sech(a*x)),x) --R a q\|q - p --R Type: Union(List Expression Integer,...) --E + +--S 45 +t1:=integrate(1/(p+q*cosh(a*x)),x) +--R +--R (2) +--R [ +--R log +--R 2 2 2 2 2 +--R q sinh(a x) + (2q cosh(a x) + 2p q)sinh(a x) + q cosh(a x) +--R + +--R 2 2 +--R 2p q cosh(a x) - q + 2p +--R * +--R +---------+ +--R | 2 2 +--R \|- q + p +--R + +--R 3 2 3 2 2 3 +--R (2q - 2p q)sinh(a x) + (2q - 2p q)cosh(a x) + 2p q - 2p +--R / +--R 2 2 +--R q sinh(a x) + (2q cosh(a x) + 2p)sinh(a x) + q cosh(a x) +--R + +--R 2p cosh(a x) + q +--R / +--R +---------+ +--R | 2 2 +--R a\|- q + p +--R , +--R +-------+ +--R | 2 2 +--R (q sinh(a x) + q cosh(a x) + p)\|q - p +--R 2atan(-----------------------------------------) +--R 2 2 +--R q - p +--R ------------------------------------------------] +--R +-------+ +--R | 2 2 +--R a\|q - p +--R Type: Union(List Expression Integer,...) +--E + +--S 46 +bb1:=x/q-p/q*t1.1 +--R +--R (3) +--R - +--R p +--R * +--R log +--R 2 2 2 +--R q sinh(a x) + (2q cosh(a x) + 2p q)sinh(a x) +--R + +--R 2 2 2 2 +--R q cosh(a x) + 2p q cosh(a x) - q + 2p +--R * +--R +---------+ +--R | 2 2 +--R \|- q + p +--R + +--R 3 2 3 2 2 3 +--R (2q - 2p q)sinh(a x) + (2q - 2p q)cosh(a x) + 2p q - 2p +--R / +--R 2 2 +--R q sinh(a x) + (2q cosh(a x) + 2p)sinh(a x) + q cosh(a x) +--R + +--R 2p cosh(a x) + q +--R + +--R +---------+ +--R | 2 2 +--R a x\|- q + p +--R / +--R +---------+ +--R | 2 2 +--R a q\|- q + p +--R Type: Expression Integer +--E + +--S 47 +bb2:=x/q-p/q*t1.2 +--R +--R +-------+ +--R | 2 2 +-------+ +--R (q sinh(a x) + q cosh(a x) + p)\|q - p | 2 2 +--R - 2p atan(-----------------------------------------) + a x\|q - p +--R 2 2 +--R q - p +--R (4) -------------------------------------------------------------------- +--R +-------+ +--R | 2 2 +--R a q\|q - p +--R Type: Expression Integer +--E + +--S 48 +cc1:=aa.1-bb1 +--R +--R (5) +--R p +--R * +--R log +--R 2 2 2 2 2 +--R q sinh(a x) + (2q cosh(a x) + 2p q)sinh(a x) + q cosh(a x) +--R + +--R 2 2 +--R 2p q cosh(a x) - q + 2p +--R * +--R +---------+ +--R | 2 2 +--R \|- q + p +--R + +--R 3 2 3 2 2 3 +--R (2q - 2p q)sinh(a x) + (2q - 2p q)cosh(a x) + 2p q - 2p +--R / +--R 2 2 +--R q sinh(a x) + (2q cosh(a x) + 2p)sinh(a x) + q cosh(a x) +--R + +--R 2p cosh(a x) + q +--R + +--R p +--R * +--R log +--R 2 2 2 2 2 +--R q sinh(a x) + (2q cosh(a x) + 2p q)sinh(a x) + q cosh(a x) +--R + +--R 2 2 +--R 2p q cosh(a x) - q + 2p +--R * +--R +---------+ +--R | 2 2 +--R \|- q + p +--R + +--R 3 2 3 2 2 3 +--R (- 2q + 2p q)sinh(a x) + (- 2q + 2p q)cosh(a x) - 2p q + 2p +--R / +--R 2 2 +--R q sinh(a x) + (2q cosh(a x) + 2p)sinh(a x) + q cosh(a x) +--R + +--R 2p cosh(a x) + q +--R / +--R +---------+ +--R | 2 2 +--R a q\|- q + p +--R Type: Expression Integer +--E + +--S 49 +cc2:=aa.2-bb1 +--R +--R (6) +--R +-------+ +--R | 2 2 +--R p\|q - p +--R * +--R log +--R 2 2 2 2 2 +--R q sinh(a x) + (2q cosh(a x) + 2p q)sinh(a x) + q cosh(a x) +--R + +--R 2 2 +--R 2p q cosh(a x) - q + 2p +--R * +--R +---------+ +--R | 2 2 +--R \|- q + p +--R + +--R 3 2 3 2 2 3 +--R (2q - 2p q)sinh(a x) + (2q - 2p q)cosh(a x) + 2p q - 2p +--R / +--R 2 2 +--R q sinh(a x) + (2q cosh(a x) + 2p)sinh(a x) + q cosh(a x) +--R + +--R 2p cosh(a x) + q +--R + +--R +-------+ +--R +---------+ | 2 2 +--R | 2 2 (q sinh(a x) + q cosh(a x) + p)\|q - p +--R - 2p\|- q + p atan(-----------------------------------------) +--R 2 2 +--R q - p +--R / +--R +---------+ +-------+ +--R | 2 2 | 2 2 +--R a q\|- q + p \|q - p +--R Type: Expression Integer +--E + +--S 50 +cc3:=aa.1-bb2 +--R +--R (7) +--R +-------+ +--R | 2 2 +--R p\|q - p +--R * +--R log +--R 2 2 2 2 2 +--R q sinh(a x) + (2q cosh(a x) + 2p q)sinh(a x) + q cosh(a x) +--R + +--R 2 2 +--R 2p q cosh(a x) - q + 2p +--R * +--R +---------+ +--R | 2 2 +--R \|- q + p +--R + +--R 3 2 3 2 2 3 +--R (- 2q + 2p q)sinh(a x) + (- 2q + 2p q)cosh(a x) - 2p q + 2p +--R / +--R 2 2 +--R q sinh(a x) + (2q cosh(a x) + 2p)sinh(a x) + q cosh(a x) +--R + +--R 2p cosh(a x) + q +--R + +--R +-------+ +--R +---------+ | 2 2 +--R | 2 2 (q sinh(a x) + q cosh(a x) + p)\|q - p +--R 2p\|- q + p atan(-----------------------------------------) +--R 2 2 +--R q - p +--R / +--R +---------+ +-------+ +--R | 2 2 | 2 2 +--R a q\|- q + p \|q - p +--R Type: Expression Integer +--E + +--S 51 14:634 Schaums and Axiom agree +cc4:=aa.2-bb2 +--R +--R (8) 0 +--R Type: Expression Integer +--E @ \section{\cite{1}:14.635~~~~~\displaystyle @@ -270,7 +975,7 @@$$ <<*>>= )clear all ---S 10 of 10 +--S 52 14:635 Axiom cannot compute this integral aa:=integrate(sech(a*x)^n,x) --R --R diff --git a/src/input/schaum33.input.pamphlet b/src/input/schaum33.input.pamphlet index c5ee6a2..74e0c8b 100644 --- a/src/input/schaum33.input.pamphlet +++ b/src/input/schaum33.input.pamphlet @@ -18,7 +18,7 @@ $$)set message auto off )clear all ---S 1 of 10 +--S 1 aa:=integrate(csch(a*x),x) --R --R @@ -27,6 +27,38 @@ aa:=integrate(csch(a*x),x) --R a --R Type: Union(Expression Integer,...) --E + +--S 2 +bb:=1/a*log(tanh((a*x)/2)) +--R +--R a x +--R log(tanh(---)) +--R 2 +--R (2) -------------- +--R a +--R Type: Expression Integer +--E + +--S 3 +cc:=aa-bb +--R +--R (3) +--R a x +--R - log(tanh(---)) - log(sinh(a x) + cosh(a x) + 1) +--R 2 +--R + +--R log(sinh(a x) + cosh(a x) - 1) +--R / +--R a +--R Type: Expression Integer +--E + +--S 4 14:636 Schaums and Axiom agree +dd:=complexNormalize cc +--R +--R (4) 0 +--R Type: Expression Integer +--E @ \section{\cite{1}:14.637~~~~~\displaystyle @@ -37,7 +69,7 @@$$ <<*>>= )clear all ---S 2 of 10 +--S 5 aa:=integrate(csch(a*x)^2,x) --R --R @@ -47,6 +79,30 @@ aa:=integrate(csch(a*x)^2,x) --R a sinh(a x) + 2a cosh(a x)sinh(a x) + a cosh(a x) - a --R Type: Union(Expression Integer,...) --E + +--S 6 +bb:=-coth(a*x)/a +--R +--R coth(a x) +--R (2) - --------- +--R a +--R Type: Expression Integer +--E + +--S 7 14:637 Axiom cannot simplify this expression +cc:=aa-bb +--R +--R (3) +--R 2 +--R coth(a x)sinh(a x) + 2cosh(a x)coth(a x)sinh(a x) +--R + +--R 2 +--R (cosh(a x) - 1)coth(a x) - 2 +--R / +--R 2 2 +--R a sinh(a x) + 2a cosh(a x)sinh(a x) + a cosh(a x) - a +--R Type: Expression Integer +--E @ \section{\cite{1}:14.638~~~~~$\displaystyle
@@ -57,7 +113,7 @@ $$<<*>>= )clear all ---S 3 of 10 +--S 8 aa:=integrate(csch(a*x)^3,x) --R --R @@ -93,6 +149,72 @@ aa:=integrate(csch(a*x)^3,x) --R 2a --R Type: Union(Expression Integer,...) --E + +--S 9 +bb:=-(csch(a*x)*coth(a*x))/(2*a)-1/(2*a)*log(tanh((a*x)/2)) +--R +--R a x +--R - log(tanh(---)) - coth(a x)csch(a x) +--R 2 +--R (2) ------------------------------------- +--R 2a +--R Type: Expression Integer +--E + +--S 10 14:638 Axiom cannot simplify this expression +cc:=aa-bb +--R +--R (3) +--R 4 3 2 2 +--R sinh(a x) + 4cosh(a x)sinh(a x) + (6cosh(a x) - 2)sinh(a x) +--R + +--R 3 4 2 +--R (4cosh(a x) - 4cosh(a x))sinh(a x) + cosh(a x) - 2cosh(a x) + 1 +--R * +--R a x +--R log(tanh(---)) +--R 2 +--R + +--R 4 3 2 2 +--R sinh(a x) + 4cosh(a x)sinh(a x) + (6cosh(a x) - 2)sinh(a x) +--R + +--R 3 4 2 +--R (4cosh(a x) - 4cosh(a x))sinh(a x) + cosh(a x) - 2cosh(a x) + 1 +--R * +--R log(sinh(a x) + cosh(a x) + 1) +--R + +--R 4 3 2 2 +--R - sinh(a x) - 4cosh(a x)sinh(a x) + (- 6cosh(a x) + 2)sinh(a x) +--R + +--R 3 4 2 +--R (- 4cosh(a x) + 4cosh(a x))sinh(a x) - cosh(a x) + 2cosh(a x) - 1 +--R * +--R log(sinh(a x) + cosh(a x) - 1) +--R + +--R 4 +--R coth(a x)csch(a x)sinh(a x) +--R + +--R 3 +--R (4cosh(a x)coth(a x)csch(a x) - 2)sinh(a x) +--R + +--R 2 2 +--R ((6cosh(a x) - 2)coth(a x)csch(a x) - 6cosh(a x))sinh(a x) +--R + +--R 3 2 +--R ((4cosh(a x) - 4cosh(a x))coth(a x)csch(a x) - 6cosh(a x) - 2)sinh(a x) +--R + +--R 4 2 3 +--R (cosh(a x) - 2cosh(a x) + 1)coth(a x)csch(a x) - 2cosh(a x) - 2cosh(a x) +--R / +--R 4 3 2 2 +--R 2a sinh(a x) + 8a cosh(a x)sinh(a x) + (12a cosh(a x) - 4a)sinh(a x) +--R + +--R 3 4 2 +--R (8a cosh(a x) - 8a cosh(a x))sinh(a x) + 2a cosh(a x) - 4a cosh(a x) +--R + +--R 2a +--R Type: Expression Integer +--E @ \section{\cite{1}:14.639~~~~~\displaystyle @@ -103,7 +225,7 @@$$
<<*>>=
)clear all

---S 4 of 10
+--S 11
aa:=integrate(csch(a*x)^n*coth(a*x),x)
--R
--R
@@ -121,6 +243,202 @@ aa:=integrate(csch(a*x)^n*coth(a*x),x)
--R     a n
--R                                          Type: Union(Expression
Integer,...)
--E
+
+--S 12
+bb:=-csch(a*x)^n/(n*a)
+--R
+--R                   n
+--R          csch(a x)
+--R   (2)  - ----------
+--R              a n
+--R                                                     Type: Expression
Integer
+--E
+
+--S 13
+cc:=aa-bb
+--R
+--R   (3)
+--R                                 2sinh(a x) + 2cosh(a x)
+--R       - sinh(n log(-------------------------------------------------))
+--R                             2                                  2
+--R                    sinh(a x)  + 2cosh(a x)sinh(a x) + cosh(a x)  - 1
+--R     +
+--R                                 2sinh(a x) + 2cosh(a x)
+--R       - cosh(n log(-------------------------------------------------))
+--R                             2                                  2
+--R                    sinh(a x)  + 2cosh(a x)sinh(a x) + cosh(a x)  - 1
+--R     +
+--R                n
+--R       csch(a x)
+--R  /
+--R     a n
+--R                                                     Type: Expression
Integer
+--E
+
+--S 14
+cschrule:=rule(csch(x) == 1/sinh(x))
+--R
+--R                      1
+--R   (4)  csch(x) == -------
+--R                   sinh(x)
+--R                        Type: RewriteRule(Integer,Integer,Expression
Integer)
+--E
+
+--S 15
+dd:=cschrule cc
+--R
+--R   (5)
+--R                                 2sinh(a x) + 2cosh(a x)
+--R       - sinh(n log(-------------------------------------------------))
+--R                             2                                  2
+--R                    sinh(a x)  + 2cosh(a x)sinh(a x) + cosh(a x)  - 1
+--R     +
+--R                                 2sinh(a x) + 2cosh(a x)
+--R       - cosh(n log(-------------------------------------------------))
+--R                             2                                  2
+--R                    sinh(a x)  + 2cosh(a x)sinh(a x) + cosh(a x)  - 1
+--R     +
+--R            1     n
+--R       (---------)
+--R        sinh(a x)
+--R  /
+--R     a n
+--R                                                     Type: Expression
Integer
+--E
+
+--S 16
+ee:=expandLog dd
+--R
+--R   (6)
+--R       sinh
+--R                           2                                  2
+--R            n log(sinh(a x)  + 2cosh(a x)sinh(a x) + cosh(a x)  - 1)
+--R          +
+--R            - n log(sinh(a x) + cosh(a x)) - n log(2)
+--R     +
+--R       -
+--R          cosh
+--R                              2                                  2
+--R               n log(sinh(a x)  + 2cosh(a x)sinh(a x) + cosh(a x)  - 1)
+--R             +
+--R               - n log(sinh(a x) + cosh(a x)) - n log(2)
+--R     +
+--R            1     n
+--R       (---------)
+--R        sinh(a x)
+--R  /
+--R     a n
+--R                                                     Type: Expression
Integer
+--E
+
+--S 17
+sinhsqrrule:=rule(sinh(x)^2 == 1/2*cosh(2*x)-1/2)
+--R
+--R               2    cosh(2x) - 1
+--R   (7)  sinh(x)  == ------------
+--R                          2
+--R                        Type: RewriteRule(Integer,Integer,Expression
Integer)
+--E
+
+--S 18
+ff:=sinhsqrrule ee
+--R
+--R   (8)
+--R       sinh
+--R                                                               2
+--R                  4cosh(a x)sinh(a x) + cosh(2a x) + 2cosh(a x)  - 3
+--R            n log(--------------------------------------------------)
+--R                                           2
+--R          +
+--R            - n log(sinh(a x) + cosh(a x)) - n log(2)
+--R     +
+--R       -
+--R          cosh
+--R                                                                  2
+--R                     4cosh(a x)sinh(a x) + cosh(2a x) + 2cosh(a x)  - 3
+--R               n log(--------------------------------------------------)
+--R                                              2
+--R             +
+--R               - n log(sinh(a x) + cosh(a x)) - n log(2)
+--R     +
+--R            1     n
+--R       (---------)
+--R        sinh(a x)
+--R  /
+--R     a n
+--R                                                     Type: Expression
Integer
+--E
+
+--S 19
+coshsqrrule:=rule(cosh(x)^2 == 1/2*cosh(2*x)+1/2)
+--R
+--R               2    cosh(2x) + 1
+--R   (9)  cosh(x)  == ------------
+--R                          2
+--R                        Type: RewriteRule(Integer,Integer,Expression
Integer)
+--E
+
+--S 20
+gg:=coshsqrrule ff
+--R
+--R   (10)
+--R       sinh
+--R            n log(2cosh(a x)sinh(a x) + cosh(2a x) - 1)
+--R          +
+--R            - n log(sinh(a x) + cosh(a x)) - n log(2)
+--R     +
+--R       -
+--R          cosh
+--R               n log(2cosh(a x)sinh(a x) + cosh(2a x) - 1)
+--R             +
+--R               - n log(sinh(a x) + cosh(a x)) - n log(2)
+--R     +
+--R            1     n
+--R       (---------)
+--R        sinh(a x)
+--R  /
+--R     a n
+--R                                                     Type: Expression
Integer
+--E
+
+--S 21
+sinhcoshrule:=rule(sinh(x)*cosh(y) == 1/2*(sinh(x+y)+sinh(x-y)))
+--R
+--I                              %O sinh(y + x) - %O sinh(y - x)
+--I   (11)  %O cosh(y)sinh(x) == -------------------------------
+--R                                             2
+--R                        Type: RewriteRule(Integer,Integer,Expression
Integer)
+--E
+
+--S 22
+hh:=sinhcoshrule gg
+--R
+--R   (12)
+--R       sinh
+--R            n log(sinh(2a x) + cosh(2a x) - 1) - n log(sinh(a x) + cosh(a
x))
+--R          +
+--R            - n log(2)
+--R     +
+--R       -
+--R          cosh
+--R               n log(sinh(2a x) + cosh(2a x) - 1) - n log(sinh(a x) +
cosh(a x))
+--R             +
+--R               - n log(2)
+--R     +
+--R            1     n
+--R       (---------)
+--R        sinh(a x)
+--R  /
+--R     a n
+--R                                                     Type: Expression
Integer
+--E
+
+--S 23     14:639 Schaums and Axiom agree
+ii:=complexNormalize hh
+--R
+--R   (13)  0
+--R                                                     Type: Expression
Integer
+--E
@

\section{\cite{1}:14.640~~~~~$\displaystyle @@ -131,7 +449,7 @@ $$<<*>>= )clear all ---S 5 of 10 +--S 24 aa:=integrate(1/csch(a*x),x) --R --R @@ -140,6 +458,22 @@ aa:=integrate(1/csch(a*x),x) --R a --R Type: Union(Expression Integer,...) --E + +--S 25 +bb:=1/a*cosh(a*x) +--R +--R cosh(a x) +--R (2) --------- +--R a +--R Type: Expression Integer +--E + +--S 26 14:640 Schaums and Axiom agree +cc:=aa-bb +--R +--R (3) 0 +--R Type: Expression Integer +--E @ \section{\cite{1}:14.641~~~~~\displaystyle @@ -152,7 +486,7 @@$$ <<*>>= )clear all ---S 6 of 10 +--S 27 14:641 Axiom cannot compute this integral aa:=integrate(x*csch(a*x),x) --R --R @@ -172,7 +506,7 @@ $$<<*>>= )clear all ---S 7 of 10 +--S 28 aa:=integrate(x*csch(a*x)^2,x) --R --R @@ -191,6 +525,156 @@ aa:=integrate(x*csch(a*x)^2,x) --R a sinh(a x) + 2a cosh(a x)sinh(a x) + a cosh(a x) - a --R Type: Union(Expression Integer,...) --E + +--S 29 +bb:=-(x*coth(a*x))/a+1/a^2*log(sinh(a*x)) +--R +--R log(sinh(a x)) - a x coth(a x) +--R (2) ------------------------------ +--R 2 +--R a +--R Type: Expression Integer +--E + +--S 30 +cc:=aa-bb +--R +--R (3) +--R 2 2 +--R (- sinh(a x) - 2cosh(a x)sinh(a x) - cosh(a x) + 1)log(sinh(a x)) +--R + +--R 2 2 +--R (sinh(a x) + 2cosh(a x)sinh(a x) + cosh(a x) - 1) +--R * +--R 2sinh(a x) +--R log(- ---------------------) +--R sinh(a x) - cosh(a x) +--R + +--R 2 +--R (a x coth(a x) - 2a x)sinh(a x) +--R + +--R (2a x cosh(a x)coth(a x) - 4a x cosh(a x))sinh(a x) +--R + +--R 2 2 +--R (a x cosh(a x) - a x)coth(a x) - 2a x cosh(a x) +--R / +--R 2 2 2 2 2 2 +--R a sinh(a x) + 2a cosh(a x)sinh(a x) + a cosh(a x) - a +--R Type: Expression Integer +--E + +--S 31 +dd:=expandLog cc +--R +--R (4) +--R 2 2 +--R (- sinh(a x) - 2cosh(a x)sinh(a x) - cosh(a x) + 1) +--R * +--R log(sinh(a x) - cosh(a x)) +--R + +--R 2 +--R (a x coth(a x) + log(- 2) - 2a x)sinh(a x) +--R + +--R (2a x cosh(a x)coth(a x) + (2log(- 2) - 4a x)cosh(a x))sinh(a x) +--R + +--R 2 2 +--R (a x cosh(a x) - a x)coth(a x) + (log(- 2) - 2a x)cosh(a x) - log(- 2) +--R / +--R 2 2 2 2 2 2 +--R a sinh(a x) + 2a cosh(a x)sinh(a x) + a cosh(a x) - a +--R Type: Expression Integer +--E + +--S 32 +sinhsqrrule:=rule(sinh(x)^2 == 1/2*cosh(2*x)-1/2) +--R +--R 2 cosh(2x) - 1 +--R (5) sinh(x) == ------------ +--R 2 +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 33 +ee:=sinhsqrrule dd +--R +--R (6) +--R 2 +--R (- 4cosh(a x)sinh(a x) - cosh(2a x) - 2cosh(a x) + 3) +--R * +--R log(sinh(a x) - cosh(a x)) +--R + +--R (4a x cosh(a x)coth(a x) + (4log(- 2) - 8a x)cosh(a x))sinh(a x) +--R + +--R 2 +--R (a x cosh(2a x) + 2a x cosh(a x) - 3a x)coth(a x) +--R + +--R 2 +--R (log(- 2) - 2a x)cosh(2a x) + (2log(- 2) - 4a x)cosh(a x) - 3log(- 2) +--R + +--R 2a x +--R / +--R 2 2 2 2 2 +--R 4a cosh(a x)sinh(a x) + a cosh(2a x) + 2a cosh(a x) - 3a +--R Type: Expression Integer +--E + +--S 34 +coshsqrrule:=rule(cosh(x)^2 == 1/2*cosh(2*x)+1/2) +--R +--R 2 cosh(2x) + 1 +--R (7) cosh(x) == ------------ +--R 2 +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 35 +ff:=coshsqrrule ee +--R +--R (8) +--R (- 2cosh(a x)sinh(a x) - cosh(2a x) + 1)log(sinh(a x) - cosh(a x)) +--R + +--R (2a x cosh(a x)coth(a x) + (2log(- 2) - 4a x)cosh(a x))sinh(a x) +--R + +--R (a x cosh(2a x) - a x)coth(a x) + (log(- 2) - 2a x)cosh(2a x) - log(- 2) +--R / +--R 2 2 2 +--R 2a cosh(a x)sinh(a x) + a cosh(2a x) - a +--R Type: Expression Integer +--E + +--S 36 +sinhcoshrule:=rule(sinh(x)*cosh(y) == 1/2*(sinh(x+y)+sinh(x-y))) +--R +--I %P sinh(y + x) - %P sinh(y - x) +--I (9) %P cosh(y)sinh(x) == ------------------------------- +--R 2 +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 37 +gg:=sinhcoshrule ff +--R +--R (10) +--R (- sinh(2a x) - cosh(2a x) + 1)log(sinh(a x) - cosh(a x)) +--R + +--R (a x coth(a x) + log(- 2) - 2a x)sinh(2a x) +--R + +--R (a x cosh(2a x) - a x)coth(a x) + (log(- 2) - 2a x)cosh(2a x) - log(- 2) +--R / +--R 2 2 2 +--R a sinh(2a x) + a cosh(2a x) - a +--R Type: Expression Integer +--E + +--S 38 14:642 Axiom cannot simplify this expression +hh:=complexNormalize gg +--R +--R - log(- 1) + log(- 2) +--R (11) --------------------- +--R 2 +--R a +--R Type: Expression Integer +--E @ \section{\cite{1}:14.643~~~~~\displaystyle @@ -202,7 +686,7 @@$$ <<*>>= )clear all ---S 8 of 10 +--S 39 14:643 Axiom cannot compute this integral aa:=integrate(csch(a*x)/x,x) --R --R @@ -222,7 +706,7 @@ $$<<*>>= )clear all ---S 9 of 10 +--S 40 aa:=integrate(1/(q+p*csch(a*x)),x) --R --R @@ -257,6 +741,251 @@ aa:=integrate(1/(q+p*csch(a*x)),x) --R a q\|q + p --R Type: Union(Expression Integer,...) --E + +--S 41 +t1:=integrate(1/(p+q*sinh(a*x)),x) +--R +--R (2) +--R log +--R 2 2 2 2 2 +--R q sinh(a x) + (2q cosh(a x) + 2p q)sinh(a x) + q cosh(a x) +--R + +--R 2 2 +--R 2p q cosh(a x) + q + 2p +--R * +--R +-------+ +--R | 2 2 +--R \|q + p +--R + +--R 3 2 3 2 2 3 +--R (- 2q - 2p q)sinh(a x) + (- 2q - 2p q)cosh(a x) - 2p q - 2p +--R / +--R 2 2 +--R q sinh(a x) + (2q cosh(a x) + 2p)sinh(a x) + q cosh(a x) +--R + +--R 2p cosh(a x) - q +--R / +--R +-------+ +--R | 2 2 +--R a\|q + p +--R Type: Union(Expression Integer,...) +--E + +--S 42 +bb:=x/q-p/q*t1 +--R +--R (3) +--R - +--R p +--R * +--R log +--R 2 2 2 +--R q sinh(a x) + (2q cosh(a x) + 2p q)sinh(a x) +--R + +--R 2 2 2 2 +--R q cosh(a x) + 2p q cosh(a x) + q + 2p +--R * +--R +-------+ +--R | 2 2 +--R \|q + p +--R + +--R 3 2 3 2 2 3 +--R (- 2q - 2p q)sinh(a x) + (- 2q - 2p q)cosh(a x) - 2p q - 2p +--R / +--R 2 2 +--R q sinh(a x) + (2q cosh(a x) + 2p)sinh(a x) + q cosh(a x) +--R + +--R 2p cosh(a x) - q +--R + +--R +-------+ +--R | 2 2 +--R a x\|q + p +--R / +--R +-------+ +--R | 2 2 +--R a q\|q + p +--R Type: Expression Integer +--E + +--S 43 +cc:=aa-bb +--R +--R (4) +--R p +--R * +--R log +--R 2 2 2 2 2 +--R q sinh(a x) + (2q cosh(a x) + 2p q)sinh(a x) + q cosh(a x) +--R + +--R 2 2 +--R 2p q cosh(a x) + q + 2p +--R * +--R +-------+ +--R | 2 2 +--R \|q + p +--R + +--R 3 2 3 2 2 3 +--R (2q + 2p q)sinh(a x) + (2q + 2p q)cosh(a x) + 2p q + 2p +--R / +--R 2 2 +--R q sinh(a x) + (2q cosh(a x) + 2p)sinh(a x) + q cosh(a x) +--R + +--R 2p cosh(a x) - q +--R + +--R p +--R * +--R log +--R 2 2 2 2 2 +--R q sinh(a x) + (2q cosh(a x) + 2p q)sinh(a x) + q cosh(a x) +--R + +--R 2 2 +--R 2p q cosh(a x) + q + 2p +--R * +--R +-------+ +--R | 2 2 +--R \|q + p +--R + +--R 3 2 3 2 2 3 +--R (- 2q - 2p q)sinh(a x) + (- 2q - 2p q)cosh(a x) - 2p q - 2p +--R / +--R 2 2 +--R q sinh(a x) + (2q cosh(a x) + 2p)sinh(a x) + q cosh(a x) +--R + +--R 2p cosh(a x) - q +--R / +--R +-------+ +--R | 2 2 +--R a q\|q + p +--R Type: Expression Integer +--E + +--S 44 +sinhsqrrule:=rule(sinh(x)^2 == 1/2*cosh(2*x)-1/2) +--R +--R 2 cosh(2x) - 1 +--R (5) sinh(x) == ------------ +--R 2 +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 45 +dd:=sinhsqrrule cc +--R +--R (6) +--R p +--R * +--R log +--R 2 2 +--R (4q cosh(a x) + 4p q)sinh(a x) + q cosh(2a x) +--R + +--R 2 2 2 2 +--R 2q cosh(a x) + 4p q cosh(a x) + q + 4p +--R * +--R +-------+ +--R | 2 2 +--R \|q + p +--R + +--R 3 2 3 2 2 3 +--R (4q + 4p q)sinh(a x) + (4q + 4p q)cosh(a x) + 4p q + 4p +--R / +--R 2 +--R (4q cosh(a x) + 4p)sinh(a x) + q cosh(2a x) + 2q cosh(a x) +--R + +--R 4p cosh(a x) - 3q +--R + +--R p +--R * +--R log +--R 2 2 +--R (4q cosh(a x) + 4p q)sinh(a x) + q cosh(2a x) +--R + +--R 2 2 2 2 +--R 2q cosh(a x) + 4p q cosh(a x) + q + 4p +--R * +--R +-------+ +--R | 2 2 +--R \|q + p +--R + +--R 3 2 3 2 2 3 +--R (- 4q - 4p q)sinh(a x) + (- 4q - 4p q)cosh(a x) - 4p q - 4p +--R / +--R 2 +--R (4q cosh(a x) + 4p)sinh(a x) + q cosh(2a x) + 2q cosh(a x) +--R + +--R 4p cosh(a x) - 3q +--R / +--R +-------+ +--R | 2 2 +--R a q\|q + p +--R Type: Expression Integer +--E + +--S 46 +coshsqrrule:=rule(cosh(x)^2 == 1/2*cosh(2*x)+1/2) +--R +--R 2 cosh(2x) + 1 +--R (7) cosh(x) == ------------ +--R 2 +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 47 +ee:=coshsqrrule dd +--R +--R (8) +--R p +--R * +--R log +--R 2 2 +--R (2q cosh(a x) + 2p q)sinh(a x) + q cosh(2a x) +--R + +--R 2 2 +--R 2p q cosh(a x) + q + 2p +--R * +--R +-------+ +--R | 2 2 +--R \|q + p +--R + +--R 3 2 3 2 2 3 +--R (2q + 2p q)sinh(a x) + (2q + 2p q)cosh(a x) + 2p q + 2p +--R / +--R (2q cosh(a x) + 2p)sinh(a x) + q cosh(2a x) + 2p cosh(a x) - q +--R + +--R p +--R * +--R log +--R 2 2 +--R (2q cosh(a x) + 2p q)sinh(a x) + q cosh(2a x) +--R + +--R 2 2 +--R 2p q cosh(a x) + q + 2p +--R * +--R +-------+ +--R | 2 2 +--R \|q + p +--R + +--R 3 2 3 2 2 3 +--R (- 2q - 2p q)sinh(a x) + (- 2q - 2p q)cosh(a x) - 2p q - 2p +--R / +--R (2q cosh(a x) + 2p)sinh(a x) + q cosh(2a x) + 2p cosh(a x) - q +--R / +--R +-------+ +--R | 2 2 +--R a q\|q + p +--R Type: Expression Integer +--E + +--S 48 14:644 Schaums and Axiom differ by a constant +ff:=complexNormalize ee +--R +--R 4 2 2 +--R p log(q + p q ) +--R (9) ---------------- +--R +-------+ +--R | 2 2 +--R a q\|q + p +--R Type: Expression Integer +--E @ \section{\cite{1}:14.645~~~~~\displaystyle @@ -268,7 +997,7 @@$$ <<*>>= )clear all ---S 10 of 10 +--S 49 14:645 Axiom cannot compute this integral aa:=integrate(csch(a*x)^n,x) --R --R diff --git a/src/input/schaum34.input.pamphlet b/src/input/schaum34.input.pamphlet index 1a5c359..83d7061 100644 --- a/src/input/schaum34.input.pamphlet +++ b/src/input/schaum34.input.pamphlet @@ -18,7 +18,7 @@ $$)set message auto off )clear all ---S 1 of 32 +--S 1 aa:=integrate(asinh(x/a),x) --R --R @@ -33,6 +33,69 @@ aa:=integrate(asinh(x/a),x) --R \|x + a - x --R Type: Union(Expression Integer,...) --E + +--S 2 +bb:=x*asinh(x/a)-sqrt(x^2+a^2) +--R +--R +-------+ +--R | 2 2 x +--R (2) - \|x + a + x asinh(-) +--R a +--R Type: Expression Integer +--E + +--S 3 +cc:=aa-bb +--R +--R +-------+ +--R | 2 2 +--R \|x + a + x x +--R (3) x log(--------------) - x asinh(-) +--R a a +--R Type: Expression Integer +--E + +--S 4 +asinhlogrule:=rule(asinh(x) == log(x+sqrt(x^2+1))) +--R +--R +------+ +--R | 2 +--R (4) asinh(x) == log(\|x + 1 + x) +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 5 +dd:=asinhlogrule cc +--R +--R +-------+ +--R | 2 2 +--R |x + a +--R +-------+ a |------- + x +--R | 2 2 | 2 +--R \|x + a + x \| a +--R (5) x log(--------------) - x log(---------------) +--R a a +--R Type: Expression Integer +--E + +--S 6 +ee:=expandLog dd +--R +--R +-------+ +--R +-------+ | 2 2 +--R | 2 2 |x + a +--R (6) x log(\|x + a + x) - x log(a |------- + x) +--R | 2 +--R \| a +--R Type: Expression Integer +--E + +--S 7 14:646 Schaums and Axiom agree +ff:=rootSimp ee +--R +--R (7) 0 +--R Type: Expression Integer +--E @ \section{\cite{1}:14.647~~~~~\displaystyle @@ -44,7 +107,7 @@$$ <<*>>= )clear all ---S 2 of 32 +--S 8 aa:=integrate(x*asinh(x/a),x) --R --R @@ -64,6 +127,77 @@ aa:=integrate(x*asinh(x/a),x) --R 8x\|x + a - 8x - 4a --R Type: Union(Expression Integer,...) --E + +--S 9 +bb:=(x^2/2+a^2/4)*asinh(x/a)-(x*sqrt(x^2+a^2))/4 +--R +--R +-------+ +--R | 2 2 2 2 x +--R - x\|x + a + (2x + a )asinh(-) +--R a +--R (2) ---------------------------------- +--R 4 +--R Type: Expression Integer +--E + +--S 10 +cc:=aa-bb +--R +--R +-------+ +--R | 2 2 +--R 2 2 \|x + a + x 2 2 x +--R (2x + a )log(--------------) + (- 2x - a )asinh(-) +--R a a +--R (3) ---------------------------------------------------- +--R 4 +--R Type: Expression Integer +--E + +--S 11 +asinhlogrule:=rule(asinh(x) == log(x+sqrt(x^2+1))) +--R +--R +------+ +--R | 2 +--R (4) asinh(x) == log(\|x + 1 + x) +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 12 +dd:=asinhlogrule cc +--R +--R +-------+ +--R | 2 2 +--R |x + a +--R +-------+ a |------- + x +--R | 2 2 | 2 +--R 2 2 \|x + a + x 2 2 \| a +--R (2x + a )log(--------------) + (- 2x - a )log(---------------) +--R a a +--R (5) ---------------------------------------------------------------- +--R 4 +--R Type: Expression Integer +--E + +--S 13 +ee:=expandLog dd +--R +--R +-------+ +--R +-------+ | 2 2 +--R 2 2 | 2 2 2 2 |x + a +--R (2x + a )log(\|x + a + x) + (- 2x - a )log(a |------- + x) +--R | 2 +--R \| a +--R (6) ---------------------------------------------------------------- +--R 4 +--R Type: Expression Integer +--E + +--S 14 14:647 Schaums and Axiom agree +ff:=rootSimp ee +--R +--R (7) 0 +--R Type: Expression Integer +--E @ \section{\cite{1}:14.648~~~~~$\displaystyle
@@ -74,7 +208,7 @@ $$<<*>>= )clear all ---S 3 of 32 +--S 15 aa:=integrate(x^2*asinh(x/a),x) --R --R @@ -94,6 +228,77 @@ aa:=integrate(x^2*asinh(x/a),x) --R (36x + 9a )\|x + a - 36x - 27a x --R Type: Union(Expression Integer,...) --E + +--S 16 +bb:=x^3/3*asinh(x/a)+((2*a^2-x^2)*sqrt(x^2+a^2))/9 +--R +--R +-------+ +--R 2 2 | 2 2 3 x +--R (- x + 2a )\|x + a + 3x asinh(-) +--R a +--R (2) ------------------------------------ +--R 9 +--R Type: Expression Integer +--E + +--S 17 +cc:=aa-bb +--R +--R +-------+ +--R | 2 2 +--R 3 \|x + a + x 3 x +--R x log(--------------) - x asinh(-) +--R a a +--R (3) ---------------------------------- +--R 3 +--R Type: Expression Integer +--E + +--S 18 +asinhlogrule:=rule(asinh(x) == log(x+sqrt(x^2+1))) +--R +--R +------+ +--R | 2 +--R (4) asinh(x) == log(\|x + 1 + x) +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 19 +dd:=asinhlogrule cc +--R +--R +-------+ +--R | 2 2 +--R |x + a +--R +-------+ a |------- + x +--R | 2 2 | 2 +--R 3 \|x + a + x 3 \| a +--R x log(--------------) - x log(---------------) +--R a a +--R (5) ---------------------------------------------- +--R 3 +--R Type: Expression Integer +--E + +--S 20 +ee:=expandLog dd +--R +--R +-------+ +--R +-------+ | 2 2 +--R 3 | 2 2 3 |x + a +--R x log(\|x + a + x) - x log(a |------- + x) +--R | 2 +--R \| a +--R (6) ---------------------------------------------- +--R 3 +--R Type: Expression Integer +--E + +--S 21 14:648 Schaums and Axiom agree +ff:=rootSimp ee +--R +--R (7) 0 +--R Type: Expression Integer +--E @ \section{\cite{1}:14.649~~~~~\displaystyle @@ -124,7 +329,7 @@$$
<<*>>=
)clear all

---S 4 of 32
+--S 22     14:649 Axiom cannot compute this integral
aa:=integrate(asinh(x/a)/x,x)
--R
--R
@@ -146,7 +351,7 @@ $$<<*>>= )clear all ---S 5 of 32 +--S 23 aa:=integrate(asinh(x/a)/x^2,x) --R --R @@ -164,6 +369,113 @@ aa:=integrate(asinh(x/a)/x^2,x) --R a x --R Type: Union(Expression Integer,...) --E + +--S 24 +bb:=-asinh(x/a)/x-1/a*log((a+sqrt(x^2+a^2))/x) +--R +--R +-------+ +--R | 2 2 +--R \|x + a + a x +--R - x log(--------------) - a asinh(-) +--R x a +--R (2) ------------------------------------ +--R a x +--R Type: Expression Integer +--E + +--S 25 +cc:=aa-bb +--R +--R (3) +--R +-------+ +-------+ +--R | 2 2 | 2 2 +--R - x log(\|x + a - x + a) + x log(\|x + a - x - a) +--R + +--R +-------+ +-------+ +--R | 2 2 | 2 2 +--R \|x + a + x \|x + a + a x +--R - a log(--------------) + x log(--------------) + a asinh(-) +--R a x a +--R / +--R a x +--R Type: Expression Integer +--E + +--S 26 +asinhlogrule:=rule(asinh(x) == log(x+sqrt(x^2+1))) +--R +--R +------+ +--R | 2 +--R (4) asinh(x) == log(\|x + 1 + x) +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 27 +dd:=asinhlogrule cc +--R +--R (5) +--R +-------+ +-------+ +--R | 2 2 | 2 2 +--R - x log(\|x + a - x + a) + x log(\|x + a - x - a) +--R + +--R +-------+ +--R | 2 2 +--R |x + a +--R +-------+ +-------+ a |------- + x +--R | 2 2 | 2 2 | 2 +--R \|x + a + x \|x + a + a \| a +--R - a log(--------------) + x log(--------------) + a log(---------------) +--R a x a +--R / +--R a x +--R Type: Expression Integer +--E + +--S 28 +ee:=expandLog dd +--R +--R (6) +--R +-------+ +-------+ +--R | 2 2 | 2 2 +--R - a log(\|x + a + x) + x log(\|x + a + a) +--R + +--R +-------+ +-------+ +--R | 2 2 | 2 2 +--R - x log(\|x + a - x + a) + x log(\|x + a - x - a) +--R + +--R +-------+ +--R | 2 2 +--R |x + a +--R a log(a |------- + x) - x log(x) +--R | 2 +--R \| a +--R / +--R a x +--R Type: Expression Integer +--E + +--S 29 +ff:=rootSimp ee +--R +--R (7) +--R +-------+ +-------+ +-------+ +--R | 2 2 | 2 2 | 2 2 +--R log(\|x + a + a) - log(\|x + a - x + a) + log(\|x + a - x - a) +--R + +--R - log(x) +--R / +--R a +--R Type: Expression Integer +--E + +--S 30 14:650 Schaums and Axiom differ by a constant +gg:=complexNormalize ff +--R +--R log(- 1) +--R (8) - -------- +--R a +--R Type: Expression Integer +--E @ \section{\cite{1}:14.651~~~~~\displaystyle @@ -182,7 +494,7 @@$$
<<*>>=
)clear all

---S 6 of 32
+--S 31
aa:=integrate(acosh(x/a),x)
--R
--R
@@ -197,6 +509,100 @@ aa:=integrate(acosh(x/a),x)
--R                                \|x  - a   - x
--R                                          Type: Union(Expression
Integer,...)
--E
+
+--S 32
+bb1:=x*acosh(x/a)-sqrt(x^2-a^2)
+--R
+--R           +-------+
+--R           | 2    2            x
+--R   (2)  - \|x  - a   + x acosh(-)
+--R                               a
+--R                                                     Type: Expression
Integer
+--E
+
+--S 33
+bb2:=x*acosh(x/a)+sqrt(x^2-a^2)
+--R
+--R         +-------+
+--R         | 2    2            x
+--R   (3)  \|x  - a   + x acosh(-)
+--R                             a
+--R                                                     Type: Expression
Integer
+--E
+
+--S 34
+cc1:=aa-bb1
+--R
+--R               +-------+
+--R               | 2    2
+--R              \|x  - a   + x            x
+--R   (4)  x log(--------------) - x acosh(-)
+--R                     a                  a
+--R                                                     Type: Expression
Integer
+--E
+
+--S 35
+cc2:=aa-bb2
+--R
+--R   (5)
+--R                              +-------+
+--R          +-------+           | 2    2                             +-------+
+--R          | 2    2     2     \|x  - a   + x               x        | 2    2
+--R       (x\|x  - a   - x )log(--------------) + (- x acosh(-) + 2x)\|x  - a
+--R                                    a                     a
+--R     +
+--R        2      x      2     2
+--R       x acosh(-) - 2x  + 2a
+--R               a
+--R  /
+--R      +-------+
+--R      | 2    2
+--R     \|x  - a   - x
+--R                                                     Type: Expression
Integer
+--E
+
+--S 36
+acoshlogrule:=rule(acosh(x) == log(x+sqrt(x^2-1)))
+--R
+--R                         +------+
+--R                         | 2
+--R   (6)  acosh(x) == log(\|x  - 1  + x)
+--R                        Type: RewriteRule(Integer,Integer,Expression
Integer)
+--E
+
+--S 37
+dd1:=acoshlogrule cc1
+--R
+--R                                        +-------+
+--R                                        | 2    2
+--R                                        |x  - a
+--R               +-------+              a |-------  + x
+--R               | 2    2                 |    2
+--R              \|x  - a   + x           \|   a
+--R   (7)  x log(--------------) - x log(---------------)
+--R                     a                       a
+--R                                                     Type: Expression
Integer
+--E
+
+--S 38
+ee1:=expandLog dd1
+--R
+--R                                        +-------+
+--R               +-------+                | 2    2
+--R               | 2    2                 |x  - a
+--R   (8)  x log(\|x  - a   + x) - x log(a |-------  + x)
+--R                                        |    2
+--R                                       \|   a
+--R                                                     Type: Expression
Integer
+--E
+
+--S 39     14:651 Schaums and Axiom agree
+ff1:=rootSimp ee1
+--R
+--R   (9)  0
+--R                                                     Type: Expression
Integer
+--E
+
@

\section{\cite{1}:14.652~~~~~$\displaystyle @@ -217,7 +623,7 @@ $$<<*>>= )clear all ---S 7 of 32 +--S 40 aa:=integrate(x*acosh(x/a),x) --R --R @@ -237,6 +643,114 @@ aa:=integrate(x*acosh(x/a),x) --R 8x\|x - a - 8x + 4a --R Type: Union(Expression Integer,...) --E + +--S 41 +bb1:=1/4*(2*x^2-a^2)*acosh(x/a)-1/4*x*sqrt(x^2-a^2) +--R +--R +-------+ +--R | 2 2 2 2 x +--R - x\|x - a + (2x - a )acosh(-) +--R a +--R (2) ---------------------------------- +--R 4 +--R Type: Expression Integer +--E + +--S 42 +bb2:=1/4*(2*x^2-a^2)*acosh(x/a)+1/4*x*sqrt(x^2-a^2) +--R +--R +-------+ +--R | 2 2 2 2 x +--R x\|x - a + (2x - a )acosh(-) +--R a +--R (3) -------------------------------- +--R 4 +--R Type: Expression Integer +--E + +--S 43 +cc1:=aa-bb1 +--R +--R +-------+ +--R | 2 2 +--R 2 2 \|x - a + x 2 2 x +--R (2x - a )log(--------------) + (- 2x + a )acosh(-) +--R a a +--R (4) ---------------------------------------------------- +--R 4 +--R Type: Expression Integer +--E + +--S 44 +cc2:=aa-bb2 +--R +--R (5) +--R +-------+ +--R +-------+ | 2 2 +--R 3 2 | 2 2 4 2 2 4 \|x - a + x +--R ((4x - 2a x)\|x - a - 4x + 4a x - a )log(--------------) +--R a +--R + +--R +-------+ +--R 3 2 x 3 2 | 2 2 +--R ((- 4x + 2a x)acosh(-) + 4x - 2a x)\|x - a +--R a +--R + +--R 4 2 2 4 x 4 2 2 +--R (4x - 4a x + a )acosh(-) - 4x + 4a x +--R a +--R / +--R +-------+ +--R | 2 2 2 2 +--R 8x\|x - a - 8x + 4a +--R Type: Expression Integer +--E + +--S 45 +acoshlogrule:=rule(acosh(x) == log(x+sqrt(x^2-1))) +--R +--R +------+ +--R | 2 +--R (6) acosh(x) == log(\|x - 1 + x) +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 46 +dd1:=acoshlogrule cc1 +--R +--R +-------+ +--R | 2 2 +--R |x - a +--R +-------+ a |------- + x +--R | 2 2 | 2 +--R 2 2 \|x - a + x 2 2 \| a +--R (2x - a )log(--------------) + (- 2x + a )log(---------------) +--R a a +--R (7) ---------------------------------------------------------------- +--R 4 +--R Type: Expression Integer +--E + +--S 47 +ee1:=expandLog dd1 +--R +--R +-------+ +--R +-------+ | 2 2 +--R 2 2 | 2 2 2 2 |x - a +--R (2x - a )log(\|x - a + x) + (- 2x + a )log(a |------- + x) +--R | 2 +--R \| a +--R (8) ---------------------------------------------------------------- +--R 4 +--R Type: Expression Integer +--E + +--S 48 14:652 Schaums and Axiom agree +ff1:=rootSimp ee1 +--R +--R (9) 0 +--R Type: Expression Integer +--E @ \section{\cite{1}:14.653~~~~~\displaystyle @@ -257,7 +771,7 @@$$ <<*>>= )clear all ---S 8 of 32 +--S 49 aa:=integrate(x^2*acosh(x/a),x) --R --R @@ -277,6 +791,114 @@ aa:=integrate(x^2*acosh(x/a),x) --R (36x - 9a )\|x - a - 36x + 27a x --R Type: Union(Expression Integer,...) --E + +--S 50 +bb1:=1/3*x^3*acosh(x/a)-1/9*(x^2+2*a^2)*sqrt(x^2-a^2) +--R +--R +-------+ +--R 2 2 | 2 2 3 x +--R (- x - 2a )\|x - a + 3x acosh(-) +--R a +--R (2) ------------------------------------ +--R 9 +--R Type: Expression Integer +--E + +--S 51 +bb2:=1/3*x^3*acosh(x/a)+1/9*(x^2+2*a^2)*sqrt(x^2-a^2) +--R +--R +-------+ +--R 2 2 | 2 2 3 x +--R (x + 2a )\|x - a + 3x acosh(-) +--R a +--R (3) ---------------------------------- +--R 9 +--R Type: Expression Integer +--E + +--S 52 +cc1:=aa-bb1 +--R +--R +-------+ +--R | 2 2 +--R 3 \|x - a + x 3 x +--R x log(--------------) - x acosh(-) +--R a a +--R (4) ---------------------------------- +--R 3 +--R Type: Expression Integer +--E + +--S 53 +cc2:=aa-bb2 +--R +--R (5) +--R +-------+ +--R +-------+ | 2 2 +--R 5 2 3 | 2 2 6 2 4 \|x - a + x +--R ((12x - 3a x )\|x - a - 12x + 9a x )log(--------------) +--R a +--R + +--R +-------+ +--R 5 2 3 x 5 2 3 4 | 2 2 +--R ((- 12x + 3a x )acosh(-) + 8x + 10a x - 12a x)\|x - a +--R a +--R + +--R 6 2 4 x 6 2 4 4 2 6 +--R (12x - 9a x )acosh(-) - 8x - 6a x + 18a x - 4a +--R a +--R / +--R +-------+ +--R 2 2 | 2 2 3 2 +--R (36x - 9a )\|x - a - 36x + 27a x +--R Type: Expression Integer +--E + +--S 54 +acoshlogrule:=rule(acosh(x) == log(x+sqrt(x^2-1))) +--R +--R +------+ +--R | 2 +--R (6) acosh(x) == log(\|x - 1 + x) +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 55 +dd1:=acoshlogrule cc1 +--R +--R +-------+ +--R | 2 2 +--R |x - a +--R +-------+ a |------- + x +--R | 2 2 | 2 +--R 3 \|x - a + x 3 \| a +--R x log(--------------) - x log(---------------) +--R a a +--R (7) ---------------------------------------------- +--R 3 +--R Type: Expression Integer +--E + +--S 56 +ee1:=expandLog dd1 +--R +--R +-------+ +--R +-------+ | 2 2 +--R 3 | 2 2 3 |x - a +--R x log(\|x - a + x) - x log(a |------- + x) +--R | 2 +--R \| a +--R (8) ---------------------------------------------- +--R 3 +--R Type: Expression Integer +--E + +--S 57 14:653 Schaums and Axiom agree +ff1:=rootSimp ee1 +--R +--R (9) 0 +--R Type: Expression Integer +--E @ \section{\cite{1}:14.654~~~~~$\displaystyle
@@ -296,7 +918,7 @@ $$<<*>>= )clear all ---S 9 of 32 +--S 58 14:654 Axiom cannot compute this integral aa:=integrate(acosh(x/a)/x,x) --R --R @@ -325,7 +947,7 @@$$
<<*>>=
)clear all

---S 10 of 32
+--S 59
aa:=integrate(acosh(x/a)/x^2,x)
--R
--R
@@ -338,6 +960,68 @@ aa:=integrate(acosh(x/a)/x^2,x)
--R                               a x
--R                                          Type: Union(Expression
Integer,...)
--E
+
+--S 60
+bb1:=-acosh(x/a)/x-1/a*log((a+sqrt(x^2+a^2))/x)
+--R
+--R                 +-------+
+--R                 | 2    2
+--R                \|x  + a   + a            x
+--R        - x log(--------------) - a acosh(-)
+--R                       x                  a
+--R   (2)  ------------------------------------
+--R                         a x
+--R                                                     Type: Expression
Integer
+--E
+
+--S 61
+bb2:=-acosh(x/a)/x+1/a*log((a+sqrt(x^2+a^2))/x)
+--R
+--R               +-------+
+--R               | 2    2
+--R              \|x  + a   + a            x
+--R        x log(--------------) - a acosh(-)
+--R                     x                  a
+--R   (3)  ----------------------------------
+--R                        a x
+--R                                                     Type: Expression
Integer
+--E
+
+--S 62
+cc1:=aa-bb1
+--R
+--R   (4)
+--R              +-------+               +-------+                 +-------+
+--R              | 2    2                | 2    2                  | 2    2
+--R             \|x  + a   + a          \|x  - a   + x            \|x  - a   -
x
+--R       x log(--------------) - a log(--------------) + 2x
atan(--------------)
+--R                    x                       a                         a
+--R     +
+--R               x
+--R       a acosh(-)
+--R               a
+--R  /
+--R     a x
+--R                                                     Type: Expression
Integer
+--E
+
+--S 63     14:655 Axiom cannot simplify these expressions
+cc2:=aa-bb2
+--R
+--R   (5)
+--R                +-------+               +-------+                 +-------+
+--R                | 2    2                | 2    2                  | 2    2
+--R               \|x  + a   + a          \|x  - a   + x            \|x  - a
- x
+--R       - x log(--------------) - a log(--------------) + 2x
atan(--------------)
+--R                      x                       a                         a
+--R     +
+--R               x
+--R       a acosh(-)
+--R               a
+--R  /
+--R     a x
+--R                                                     Type: Expression
Integer
+--E
@

\section{\cite{1}:14.656~~~~~$\displaystyle @@ -348,7 +1032,7 @@ $$<<*>>= )clear all ---S 11 of 32 +--S 64 aa:=integrate(atanh(x/a),x) --R --R @@ -359,6 +1043,58 @@ aa:=integrate(atanh(x/a),x) --R 2 --R Type: Union(Expression Integer,...) --E + +--S 65 +bb:=x*atanh(x/a)+a/2*log(a^2-x^2) +--R +--R 2 2 x +--R a log(- x + a ) + 2x atanh(-) +--R a +--R (2) ------------------------------ +--R 2 +--R Type: Expression Integer +--E + +--S 66 +cc:=aa-bb +--R +--R 2 2 - x - a 2 2 x +--R a log(x - a ) + x log(-------) - a log(- x + a ) - 2x atanh(-) +--R x - a a +--R (3) ---------------------------------------------------------------- +--R 2 +--R Type: Expression Integer +--E + +--S 67 +atanhrule:=rule(atanh(x) == 1/2*log((1+x)/(1-x))) +--R +--R - x - 1 +--R log(-------) +--R x - 1 +--R (4) atanh(x) == ------------ +--R 2 +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 68 +dd:=atanhrule cc +--R +--R 2 2 2 2 +--R a log(x - a ) - a log(- x + a ) +--R (5) --------------------------------- +--R 2 +--R Type: Expression Integer +--E + +--S 69 14:656 Schaums and Axiom differ by a constant +ee:=complexNormalize dd +--R +--R a log(- 1) +--R (6) ---------- +--R 2 +--R Type: Expression Integer +--E @ \section{\cite{1}:14.657~~~~~\displaystyle @@ -369,7 +1105,7 @@$$ <<*>>= )clear all ---S 12 of 32 +--S 70 aa:=integrate(x*atanh(x/a),x) --R --R @@ -380,6 +1116,46 @@ aa:=integrate(x*atanh(x/a),x) --R 4 --R Type: Union(Expression Integer,...) --E + +--S 71 +bb:=(a*x)/2+1/2*(x^2-a^2)*atanh(x/a) +--R +--R 2 2 x +--R (x - a )atanh(-) + a x +--R a +--R (2) ----------------------- +--R 2 +--R Type: Expression Integer +--E + +--S 72 +cc:=aa-bb +--R +--R 2 2 - x - a 2 2 x +--R (x - a )log(-------) + (- 2x + 2a )atanh(-) +--R x - a a +--R (3) --------------------------------------------- +--R 4 +--R Type: Expression Integer +--E + +--S 73 +atanhrule:=rule(atanh(x) == 1/2*log((1+x)/(1-x))) +--R +--R - x - 1 +--R log(-------) +--R x - 1 +--R (4) atanh(x) == ------------ +--R 2 +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 74 14:657 Schaums and Axiom agree +dd:=atanhrule cc +--R +--R (5) 0 +--R Type: Expression Integer +--E @ \section{\cite{1}:14.658~~~~~$\displaystyle
@@ -391,7 +1167,7 @@ $$<<*>>= )clear all ---S 13 of 32 +--S 75 aa:=integrate(x^2*atanh(x/a),x) --R --R @@ -402,6 +1178,59 @@ aa:=integrate(x^2*atanh(x/a),x) --R 6 --R Type: Union(Expression Integer,...) --E + +--S 76 +bb:=(a*x^2)/6+x^3/3*atanh(x/a)+a^3/6*log(a^2-x^2) +--R +--R 3 2 2 3 x 2 +--R a log(- x + a ) + 2x atanh(-) + a x +--R a +--R (2) ------------------------------------- +--R 6 +--R Type: Expression Integer +--E + +--S 77 +cc:=aa-bb +--R +--R 3 2 2 3 - x - a 3 2 2 3 x +--R a log(x - a ) + x log(-------) - a log(- x + a ) - 2x atanh(-) +--R x - a a +--R (3) ---------------------------------------------------------------- +--R 6 +--R Type: Expression Integer +--E + +--S 78 +atanhrule:=rule(atanh(x) == 1/2*log((1+x)/(1-x))) +--R +--R - x - 1 +--R log(-------) +--R x - 1 +--R (4) atanh(x) == ------------ +--R 2 +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 79 +dd:=atanhrule cc +--R +--R 3 2 2 3 2 2 +--R a log(x - a ) - a log(- x + a ) +--R (5) --------------------------------- +--R 6 +--R Type: Expression Integer +--E + +--S 80 14:658 Schaums and Axiom differ by a constant +ee:=complexNormalize dd +--R +--R 3 +--R a log(- 1) +--R (6) ---------- +--R 6 +--R Type: Expression Integer +--E @ \section{\cite{1}:14.659~~~~~\displaystyle @@ -412,7 +1241,7 @@$$
<<*>>=
)clear all

---S 14 of 32
+--S 81     14:659 Axiom cannot compute this integral
aa:=integrate(atanh(x/a)/x,x)
--R
--R
@@ -433,7 +1262,7 @@ $$<<*>>= )clear all ---S 15 of 32 +--S 82 aa:=integrate(atanh(x/a)/x^2,x) --R --R @@ -444,6 +1273,70 @@ aa:=integrate(atanh(x/a)/x^2,x) --R 2a x --R Type: Union(Expression Integer,...) --E + +--S 83 +bb:=-atanh(x/a)/x+1/(2*a)*log(x^2/(a^2-x^2)) +--R +--R 2 +--R x x +--R x log(- -------) - 2a atanh(-) +--R 2 2 a +--R x - a +--R (2) ------------------------------ +--R 2a x +--R Type: Expression Integer +--E + +--S 84 +cc:=aa-bb +--R +--R (3) +--R 2 +--R 2 2 x - x - a +--R - x log(x - a ) + 2x log(x) - x log(- -------) - a log(-------) +--R 2 2 x - a +--R x - a +--R + +--R x +--R 2a atanh(-) +--R a +--R / +--R 2a x +--R Type: Expression Integer +--E + +--S 85 +atanhrule:=rule(atanh(x) == 1/2*log((1+x)/(1-x))) +--R +--R - x - 1 +--R log(-------) +--R x - 1 +--R (4) atanh(x) == ------------ +--R 2 +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 86 +dd:=atanhrule cc +--R +--R 2 +--R 2 2 x +--R - log(x - a ) + 2log(x) - log(- -------) +--R 2 2 +--R x - a +--R (5) ----------------------------------------- +--R 2a +--R Type: Expression Integer +--E + +--S 87 14:660 Schaums and Axiom agree +ee:=expandLog dd +--R +--R log(- 1) +--R (6) - -------- +--R 2a +--R Type: Expression Integer +--E @ \section{\cite{1}:14.661~~~~~\displaystyle @@ -451,10 +1344,15 @@ aa:=integrate(atanh(x/a)/x^2,x)$$\int{\coth^{-1}\frac{x}{a}}=
x\coth^{-1}{x}+\frac{a}{2}\ln(x^2-a^2)
$$+ +Note that it appears there is a typo in Schaums (1968 printing 4). +$$\int{\coth^{-1}\frac{x}{a}}=
+x\coth^{-1}{x/a}+\frac{a}{2}\ln(x^2-a^2)
+$$<<*>>= )clear all ---S 16 of 32 +--S 88 aa:=integrate(acoth(x/a),x) --R --R @@ -465,6 +1363,46 @@ aa:=integrate(acoth(x/a),x) --R 2 --R Type: Union(Expression Integer,...) --E + +--S 89 +bb:=x*acoth(x/a)+a/2*log(x^2-a^2) +--R +--R 2 2 x +--R a log(x - a ) + 2x acoth(-) +--R a +--R (2) ---------------------------- +--R 2 +--R Type: Expression Integer +--E + +--S 90 +cc:=aa-bb +--R +--R x + a x +--R x log(-----) - 2x acoth(-) +--R x - a a +--R (3) -------------------------- +--R 2 +--R Type: Expression Integer +--E + +--S 91 +acothrule:=rule(acoth(x) == 1/2*log((x+1)/(x-1))) +--R +--R x + 1 +--R log(-----) +--R x - 1 +--R (4) acoth(x) == ---------- +--R 2 +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 92 14:661 Schaums and Axiom agree +dd:=acothrule cc +--R +--R (5) 0 +--R Type: Expression Integer +--E @ \section{\cite{1}:14.662~~~~~\displaystyle @@ -475,7 +1413,7 @@$$
<<*>>=
)clear all

---S 17 of 32
+--S 93
aa:=integrate(x*acoth(x/a),x)
--R
--R
@@ -486,6 +1424,46 @@ aa:=integrate(x*acoth(x/a),x)
--R                     4
--R                                          Type: Union(Expression
Integer,...)
--E
+
+--S 94
+bb:=(a*x)/2+1/2*(x^2-a^2)*acoth(x/a)
+--R
+--R          2    2       x
+--R        (x  - a )acoth(-) + a x
+--R                       a
+--R   (2)  -----------------------
+--R                   2
+--R                                                     Type: Expression
Integer
+--E
+
+--S 95
+cc:=aa-bb
+--R
+--R          2    2     x + a         2     2       x
+--R        (x  - a )log(-----) + (- 2x  + 2a )acoth(-)
+--R                     x - a                       a
+--R   (3)  -------------------------------------------
+--R                             4
+--R                                                     Type: Expression
Integer
+--E
+
+--S 96
+acothrule:=rule(acoth(x) == 1/2*log((x+1)/(x-1)))
+--R
+--R                        x + 1
+--R                    log(-----)
+--R                        x - 1
+--R   (4)  acoth(x) == ----------
+--R                         2
+--R                        Type: RewriteRule(Integer,Integer,Expression
Integer)
+--E
+
+--S 97     14:662 Schaums and Axiom agree
+dd:=acothrule cc
+--R
+--R   (5)  0
+--R                                                     Type: Expression
Integer
+--E
@

\section{\cite{1}:14.663~~~~~$\displaystyle @@ -497,7 +1475,7 @@ $$<<*>>= )clear all ---S 18 of 32 +--S 98 aa:=integrate(x^2*acoth(x/a),x) --R --R @@ -508,6 +1486,46 @@ aa:=integrate(x^2*acoth(x/a),x) --R 6 --R Type: Union(Expression Integer,...) --E + +--S 99 +bb:=(a*x^2)/6+x^3/3*acoth(x/a)+a^3/6*log(x^2-a^2) +--R +--R 3 2 2 3 x 2 +--R a log(x - a ) + 2x acoth(-) + a x +--R a +--R (2) ----------------------------------- +--R 6 +--R Type: Expression Integer +--E + +--S 100 +cc:=aa-bb +--R +--R 3 x + a 3 x +--R x log(-----) - 2x acoth(-) +--R x - a a +--R (3) -------------------------- +--R 6 +--R Type: Expression Integer +--E + +--S 101 +acothrule:=rule(acoth(x) == 1/2*log((x+1)/(x-1))) +--R +--R x + 1 +--R log(-----) +--R x - 1 +--R (4) acoth(x) == ---------- +--R 2 +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 102 14:663 Schaums and Axiom agree +dd:=acothrule cc +--R +--R (5) 0 +--R Type: Expression Integer +--E @ \section{\cite{1}:14.664~~~~~\displaystyle @@ -518,7 +1536,7 @@$$ <<*>>= )clear all ---S 19 of 32 +--S 103 14:664 Axiom cannot compute this integral aa:=integrate(acoth(x/a)/x,x) --R --R @@ -539,7 +1557,7 @@ $$<<*>>= )clear all ---S 20 of 32 +--S 104 aa:=integrate(acoth(x/a)/x^2,x) --R --R @@ -550,6 +1568,64 @@ aa:=integrate(acoth(x/a)/x^2,x) --R 2a x --R Type: Union(Expression Integer,...) --E + +--S 105 +bb:=-acoth(x/a)/x+1/(2*a)*log(x^2/(x^2-a^2)) +--R +--R 2 +--R x x +--R x log(-------) - 2a acoth(-) +--R 2 2 a +--R x - a +--R (2) ---------------------------- +--R 2a x +--R Type: Expression Integer +--E + +--S 106 +cc:=aa-bb +--R +--R (3) +--R 2 +--R 2 2 x + a x x +--R - x log(x - a ) + 2x log(x) - a log(-----) - x log(-------) + 2a acoth(-) +--R x - a 2 2 a +--R x - a +--R -------------------------------------------------------------------------- +--R 2a x +--R Type: Expression Integer +--E + +--S 107 +acothrule:=rule(acoth(x) == 1/2*log((x+1)/(x-1))) +--R +--R x + 1 +--R log(-----) +--R x - 1 +--R (4) acoth(x) == ---------- +--R 2 +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 108 +dd:=acothrule cc +--R +--R 2 +--R 2 2 x +--R - log(x - a ) + 2log(x) - log(-------) +--R 2 2 +--R x - a +--R (5) --------------------------------------- +--R 2a +--R Type: Expression Integer +--E + +--S 109 14:665 Schaums and Axiom agree +ee:=expandLog dd +--R +--R (6) 0 +--R Type: Expression Integer +--E @ \section{\cite{1}:14.666~~~~~\displaystyle @@ -568,7 +1644,7 @@$$ <<*>>= )clear all ---S 21 of 32 +--S 110 aa:=integrate(asech(x/a),x) --R --R @@ -579,7 +1655,308 @@ aa:=integrate(asech(x/a),x) --R x x --R Type: Union(Expression Integer,...) --E + +--S 111 +bb1:=x*asech(x/a)+a*asin(x/a) +--R +--R x x +--R (2) a asin(-) + x asech(-) +--R a a +--R Type: Expression Integer +--E + +--S 112 +bb2:=x*asech(x/a)-a*asin(x/a) +--R +--R x x +--R (3) - a asin(-) + x asech(-) +--R a a +--R Type: Expression Integer +--E + +--S 113 +cc1:=aa-bb1 +--R +--R (4) +--R +---------+ +---------+ +--R | 2 2 | 2 2 +--R \|- x + a + a \|- x + a - a x x +--R x log(----------------) - 2a atan(----------------) - a asin(-) - x asech(-) +--R x x a a +--R Type: Expression Integer +--E + +--S 114 +cc2:=aa-bb2 +--R +--R (5) +--R +---------+ +---------+ +--R | 2 2 | 2 2 +--R \|- x + a + a \|- x + a - a x x +--R x log(----------------) - 2a atan(----------------) + a asin(-) - x asech(-) +--R x x a a +--R Type: Expression Integer +--E + +--S 115 +asechrule:=rule(asech(x) == log(1/x+sqrt(1/x^2-1))) +--R +--R +--------+ +--R | 2 +--R |- x + 1 +--R x |-------- + 1 +--R | 2 +--R \| x +--R (6) asech(x) == log(----------------) +--R x +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 116 +dd1:=asechrule cc1 +--R +--R (7) +--R +---------+ +--R | 2 2 +--R |- x + a +--R x |--------- + a +---------+ +--R | 2 | 2 2 +--R \| x \|- x + a + a +--R - x log(-----------------) + x log(----------------) +--R x x +--R + +--R +---------+ +--R | 2 2 +--R \|- x + a - a x +--R - 2a atan(----------------) - a asin(-) +--R x a +--R Type: Expression Integer +--E + +--S 117 +asinrule:=rule(asin(x) == %i*log(-%i*x+sqrt(1-x^2))) +--R +--R +--------+ +--R | 2 +--R (8) asin(x) == %i log(\|- x + 1 - %i x) +--R Type: RewriteRule(Integer,Complex Integer,Expression Complex Integer) +--E + +--S 118 +ee1:=asinrule dd1 +--R +--R (9) +--R +---------+ +---------+ +--R | 2 2 | 2 2 +--R |- x + a |- x + a +--R x |--------- + a a |--------- - %i x +--R | 2 | 2 +--R \| x \| a +--R - x log(-----------------) - %i a log(--------------------) +--R x a +--R + +--R +---------+ +---------+ +--R | 2 2 | 2 2 +--R \|- x + a + a \|- x + a - a +--R x log(----------------) - 2a atan(----------------) +--R x x +--R Type: Expression Complex Integer +--E + +--S 119 +atanrule:=rule(atan(x) == -%i/2*log((1+%i*x)/(1-%i*x))) +--R +--R - x + %i +--R %i log(--------) +--R x + %i +--R (10) atan(x) == - ---------------- +--R 2 +--R Type: RewriteRule(Integer,Complex Integer,Expression Complex Integer) +--E + +--S 120 +ff1:=atanrule ee1 +--R +--R (11) +--R +---------+ +---------+ +--R | 2 2 | 2 2 +--R |- x + a |- x + a +--R x |--------- + a a |--------- - %i x +--R | 2 | 2 +--R \| x \| a +--R - x log(-----------------) - %i a log(--------------------) +--R x a +--R + +--R +---------+ +---------+ +--R | 2 2 | 2 2 +--R \|- x + a + a - \|- x + a + %i x + a +--R x log(----------------) + %i a log(-------------------------) +--R x +---------+ +--R | 2 2 +--R \|- x + a + %i x - a +--R Type: Expression Complex Integer +--E + +--S 121 +gg1:=expandLog ff1 +--R +--R (12) +--R +---------+ +---------+ +--R | 2 2 | 2 2 +--R |- x + a |- x + a +--R - x log(x |--------- + a) - %i a log(a |--------- - %i x) +--R | 2 | 2 +--R \| x \| a +--R + +--R +---------+ +---------+ +--R | 2 2 | 2 2 +--R - %i a log(\|- x + a + %i x - a) + x log(\|- x + a + a) +--R + +--R +---------+ +--R | 2 2 +--R %i a log(\|- x + a - %i x - a) + %i a log(a) + %i a log(- 1) +--R Type: Expression Complex Integer +--E + +--S 122 +hh1:=rootSimp gg1 +--R +--R (13) +--R +-------+ +-------+ +--R | 2 2 | 2 2 +--R - %i a log(%i\|x - a + %i x - a) - %i a log(%i\|x - a - %i x) +--R + +--R +-------+ +--R | 2 2 +--R %i a log(%i\|x - a - %i x - a) + %i a log(a) + %i a log(- 1) +--R Type: Expression Complex Integer +--E + +--S 123 14:666 Schaums and Axiom agree +ii1:=complexNormalize hh1 +--R +--R (14) 0 +--R Type: Expression Complex Integer +--E + +@ +Note that Axiom has a built-in assumption about the sign of asech(x/a). +We can see this if we simplify the cc2 value and show that it differs +by a complex value of x. +<<*>>= +--S 124 +dd2:=asechrule cc2 +--R +--R (15) +--R +---------+ +--R | 2 2 +--R |- x + a +--R x |--------- + a +---------+ +--R | 2 | 2 2 +--R \| x \|- x + a + a +--R - x log(-----------------) + x log(----------------) +--R x x +--R + +--R +---------+ +--R | 2 2 +--R \|- x + a - a x +--R - 2a atan(----------------) + a asin(-) +--R x a +--R Type: Expression Integer +--E + +--S 125 +ee2:=asinrule dd2 +--R +--R (16) +--R +---------+ +---------+ +--R | 2 2 | 2 2 +--R |- x + a |- x + a +--R x |--------- + a a |--------- - %i x +--R | 2 | 2 +--R \| x \| a +--R - x log(-----------------) + %i a log(--------------------) +--R x a +--R + +--R +---------+ +---------+ +--R | 2 2 | 2 2 +--R \|- x + a + a \|- x + a - a +--R x log(----------------) - 2a atan(----------------) +--R x x +--R Type: Expression Complex Integer +--E + +--S 126 +ff2:=atanrule ee2 +--R +--R (17) +--R +---------+ +---------+ +--R | 2 2 | 2 2 +--R |- x + a |- x + a +--R x |--------- + a a |--------- - %i x +--R | 2 | 2 +--R \| x \| a +--R - x log(-----------------) + %i a log(--------------------) +--R x a +--R + +--R +---------+ +---------+ +--R | 2 2 | 2 2 +--R \|- x + a + a - \|- x + a + %i x + a +--R x log(----------------) + %i a log(-------------------------) +--R x +---------+ +--R | 2 2 +--R \|- x + a + %i x - a +--R Type: Expression Complex Integer +--E + +--S 127 +gg2:=expandLog ff2 +--R +--R (18) +--R +---------+ +---------+ +--R | 2 2 | 2 2 +--R |- x + a |- x + a +--R - x log(x |--------- + a) + %i a log(a |--------- - %i x) +--R | 2 | 2 +--R \| x \| a +--R + +--R +---------+ +---------+ +--R | 2 2 | 2 2 +--R - %i a log(\|- x + a + %i x - a) + x log(\|- x + a + a) +--R + +--R +---------+ +--R | 2 2 +--R %i a log(\|- x + a - %i x - a) - %i a log(a) + %i a log(- 1) +--R Type: Expression Complex Integer +--E + +--S 128 +hh2:=rootSimp gg2 +--R +--R (19) +--R +-------+ +-------+ +--R | 2 2 | 2 2 +--R - %i a log(%i\|x - a + %i x - a) + %i a log(%i\|x - a - %i x) +--R + +--R +-------+ +--R | 2 2 +--R %i a log(%i\|x - a - %i x - a) - %i a log(a) + %i a log(- 1) +--R Type: Expression Complex Integer +--E + +--S 129 +ii2:=complexNormalize hh2 +--R +--R +-------+ +--R | 2 2 +--R (20) 2%i a log(%i\|x - a - %i x) - 2%i a log(a) +--R Type: Expression Complex Integer +--E + @ +Thus we can conjecture that solutions that show up with x in only the +imaginary part do so when the assumption of the sign of an inverse +function differs. \section{\cite{1}:14.667~~~~~$\displaystyle
\int{x{\rm ~sech}^{-1}\frac{x}{a}}~dx$} @@ -599,7 +1976,7 @@ $$<<*>>= )clear all ---S 22 of 32 +--S 130 aa:=integrate(x*asech(x/a),x) --R --R @@ -614,6 +1991,118 @@ aa:=integrate(x*asech(x/a),x) --R 2\|- x + a - 2a --R Type: Union(Expression Integer,...) --E + +--S 131 +bb1:=1/2*x^2*asech(x/a)-1/2*a*sqrt(a^2-x^2) +--R +--R +---------+ +--R | 2 2 2 x +--R - a\|- x + a + x asech(-) +--R a +--R (2) ---------------------------- +--R 2 +--R Type: Expression Integer +--E + +--S 132 +bb2:=1/2*x^2*asech(x/a)+1/2*a*sqrt(a^2-x^2) +--R +--R +---------+ +--R | 2 2 2 x +--R a\|- x + a + x asech(-) +--R a +--R (3) -------------------------- +--R 2 +--R Type: Expression Integer +--E + +--S 133 +cc1:=aa-bb1 +--R +--R +---------+ +--R | 2 2 +--R 2 \|- x + a + a 2 x 2 +--R x log(----------------) - x asech(-) - a +--R x a +--R (4) ----------------------------------------- +--R 2 +--R Type: Expression Integer +--E + +--S 134 +cc2:=aa-bb2 +--R +--R (5) +--R +---------+ +--R +---------+ | 2 2 +--R 2 | 2 2 2 \|- x + a + a +--R (x \|- x + a - a x )log(----------------) +--R x +--R + +--R +---------+ +--R 2 x 2 | 2 2 2 x 2 3 +--R (- x asech(-) + a )\|- x + a + a x asech(-) + 2a x - a +--R a a +--R / +--R +---------+ +--R | 2 2 +--R 2\|- x + a - 2a +--R Type: Expression Integer +--E + +--S 135 +asechrule:=rule(asech(x) == log(1/x+sqrt(1/x^2-1))) +--R +--R +--------+ +--R | 2 +--R |- x + 1 +--R x |-------- + 1 +--R | 2 +--R \| x +--R (6) asech(x) == log(----------------) +--R x +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 136 +dd1:=asechrule cc1 +--R +--R +---------+ +--R | 2 2 +--R |- x + a +--R x |--------- + a +---------+ +--R | 2 | 2 2 +--R 2 \| x 2 \|- x + a + a 2 +--R - x log(-----------------) + x log(----------------) - a +--R x x +--R (7) --------------------------------------------------------- +--R 2 +--R Type: Expression Integer +--E + +--S 137 +ee1:=expandLog dd1 +--R +--R +---------+ +--R | 2 2 +---------+ +--R 2 |- x + a 2 | 2 2 2 +--R - x log(x |--------- + a) + x log(\|- x + a + a) - a +--R | 2 +--R \| x +--R (8) --------------------------------------------------------- +--R 2 +--R Type: Expression Integer +--E + +--S 138 14:667 Schaums and Axiom differ by a constant +ff1:=rootSimp ee1 +--R +--R 2 +--R a +--R (9) - -- +--R 2 +--R Type: Expression Integer +--E @ \section{\cite{1}:14.668~~~~~\displaystyle @@ -633,10 +2122,13 @@$$\int{\frac{{\rm sech}^{-1}(x/a)}{x}}= \end{array} \right. $$+ +This is a interesting result since Axiom gives a closed form +solution to the problem but Schaums gives a series solution. <<*>>= )clear all ---S 23 of 32 +--S 139 14:668 SCHAUMS AND AXIOM DIFFER (Axiom has closed form) aa:=integrate(asech(x/a)/x,x) --R --R @@ -660,7 +2152,7 @@$$ <<*>>= )clear all ---S 24 of 32 +--S 140 aa:=integrate(acsch(x/a),x) --R --R @@ -671,6 +2163,48 @@ aa:=integrate(acsch(x/a),x) --R x --R Type: Union(Expression Integer,...) --E + +--S 141 +bb1:=x*acsch(x/a)+a*asinh(x/a) +--R +--R x x +--R (2) a asinh(-) + x acsch(-) +--R a a +--R Type: Expression Integer +--E + +--S 142 +bb2:=x*acsch(x/a)-a*asinh(x/a) +--R +--R x x +--R (3) - a asinh(-) + x acsch(-) +--R a a +--R Type: Expression Integer +--E + +--S 143 +cc1:=aa-bb1 +--R +--R (4) +--R +-------+ +--R +-------+ | 2 2 +--R | 2 2 \|x + a + a x x +--R - a log(\|x + a - x) + x log(--------------) - a asinh(-) - x acsch(-) +--R x a a +--R Type: Expression Integer +--E + +--S 144 14:669 Axiom cannot simplify these expressions +cc2:=aa-bb2 +--R +--R (5) +--R +-------+ +--R +-------+ | 2 2 +--R | 2 2 \|x + a + a x x +--R - a log(\|x + a - x) + x log(--------------) + a asinh(-) - x acsch(-) +--R x a a +--R Type: Expression Integer +--E @ \section{\cite{1}:14.670~~~~~$\displaystyle
@@ -682,7 +2216,7 @@ $$<<*>>= )clear all ---S 25 of 32 +--S 145 aa:=integrate(x*acsch(x/a),x) --R --R @@ -697,6 +2231,63 @@ aa:=integrate(x*acsch(x/a),x) --R 2\|x + a - 2x --R Type: Union(Expression Integer,...) --E + +--S 146 +bb1:=x^2/2*acsch(x/a)+(a*sqrt(x^2+a^2))/2 +--R +--R +-------+ +--R | 2 2 2 x +--R a\|x + a + x acsch(-) +--R a +--R (2) ------------------------ +--R 2 +--R Type: Expression Integer +--E + +--S 147 +bb2:=x^2/2*acsch(x/a)-(a*sqrt(x^2+a^2))/2 +--R +--R +-------+ +--R | 2 2 2 x +--R - a\|x + a + x acsch(-) +--R a +--R (3) -------------------------- +--R 2 +--R Type: Expression Integer +--E + +--S 148 +cc1:=aa-bb1 +--R +--R +-------+ +--R | 2 2 +--R 2 \|x + a + a 2 x +--R x log(--------------) - x acsch(-) +--R x a +--R (4) ---------------------------------- +--R 2 +--R Type: Expression Integer +--E + +--S 149 14:670 Axiom cannot simplify these expressions +cc2:=aa-bb2 +--R +--R (5) +--R +-------+ +--R +-------+ | 2 2 +-------+ +--R 2 | 2 2 3 \|x + a + a 2 x | 2 2 +--R (x \|x + a - x )log(--------------) + (- x acsch(-) - 2a x)\|x + a +--R x a +--R + +--R 3 x 2 3 +--R x acsch(-) + 2a x + 2a +--R a +--R / +--R +-------+ +--R | 2 2 +--R 2\|x + a - 2x +--R Type: Expression Integer +--E @ \section{\cite{1}:14.671~~~~~\displaystyle @@ -721,10 +2312,13 @@$$\int{\frac{{\rm csch}^{-1}(x/a)}{x}}=
\end{array}
\right.
$$+ +Schaums gives 3 different series expansions for this integral +but Axiom has computed a closed form. <<*>>= )clear all ---S 26 of 32 +--S 150 14:671 SCHAUMS AND AXIOM DIFFER (Axiom has closed form) aa:=integrate(acsch(x/a)/x,x) --R --R @@ -737,6 +2331,7 @@ aa:=integrate(acsch(x/a)/x,x) --R 2a --R Type: Union(Expression Integer,...) --E + @ \section{\cite{1}:14.672~~~~~\displaystyle @@ -748,7 +2343,7 @@$$
<<*>>=
)clear all

---S 27 of 32
+--S 151    14:672 Axiom cannot compute this integral
aa:=integrate(x^m*asinh(x/a),x)
--R
--R
@@ -780,7 +2375,7 @@ $$<<*>>= )clear all ---S 28 of 32 +--S 152 14:673 Axiom cannot compute this integral aa:=integrate(x^m*acosh(x/a),x) --R --R @@ -801,7 +2396,7 @@$$
<<*>>=
)clear all

---S 29 of 32
+--S 153    14:674 Axiom cannot compute this integral
aa:=integrate(x^m*atanh(x/a),x)
--R
--R
@@ -822,7 +2417,7 @@ $$<<*>>= )clear all ---S 30 of 32 +--S 154 14:675 Axiom cannot compute this integral aa:=integrate(x^m*acoth(x/a),x) --R --R @@ -854,7 +2449,7 @@$$
<<*>>=
)clear all

---S 31 of 32
+--S 155    14:676 Axiom cannot compute this integral
aa:=integrate(x^m*asech(x/a),x)
--R
--R
@@ -877,7 +2472,7 @@ 
<<*>>=
)clear all

---S 32 of 32
+--S 156    14:677 Axiom cannot compute this integral
aa:=integrate(x^m*acsch(x/a),x)
--R
--R