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[Axiom-developer] 20080425.01.tpd.patch (CATS Schaums-Axiom equivalence
|
From: |
daly |
|
Subject: |
[Axiom-developer] 20080425.01.tpd.patch (CATS Schaums-Axiom equivalence testing (24-25)) |
|
Date: |
Sat, 26 Apr 2008 00:36:36 -0500 |
There are some differences between Schaums and Axiom but these
seem to occur in the complex plane. The likely problem is a
difference between Schaum's and Axiom's choice of branch cuts.
Alternatively, the rules used for reduction have an implicit
branch cut which does not match.
This needs to be investigated further.
schaum24.input.pamphlet
14:471 Axiom cannot simplify this expression
14:472 Axiom cannot simplify this expression
14:473 Axiom cannot simplify this expression
14:474 Axiom cannot compute this integral
14:475 Axiom cannot simplify this expression
14:476 Axiom cannot simplify this expression
14:477 Axiom cannot simplify this expression
14:478 Axiom cannot simplify this expression
14:479 Axiom cannot simplify this expression
14:480 Axiom cannot compute this integral
14:481 Axiom cannot simplify this expression
14:482 Axiom cannot simplify this expression
14:483 SCHAUMS AND AXIOM DIFFER? (BRANCH CUTS?)
14:484 Axiom cannot compute this integral
14:485 Axiom cannot simplify this expression
14:486 Axiom cannot compute this integral
14:487 SCHAUMS AND AXIOM DIFFER? (branch cuts?)
14:488 Axiom and Schaums agree
14:489 Axiom and Schaums agree
14:490 Axiom and Schaums agree
14:491 Axiom cannot compute this integral
14:492 Schaums and Axiom agree
14:493 Axiom cannot simplify these expressions
14:494 Axiom cannot simplify these expressions
14:495 Axiom cannot simplify these expressions
14:496 Axiom cannot compute this integral
14:497 Axiom cannot simplify these expressions
14:498 Axiom cannot simplify these expressions
14:499 Axiom cannot simplify these expressions
14:500 Axiom cannot simplify this expression
14:501 Axiom cannot compute this integral
14:502 Axiom cannot simplify this expression
14:503 Axiom cannot compute this integral
14:504 Axiom cannot compute this integral
14:505 SCHAUMS AND AXIOM DISAGREE? (branch cuts?)
14:506 Axiom cannot compute this integral
14:507 Axiom cannot compute this integral
14:508 Axiom cannot compute this integral
schaum25.input.pamphlet
14:509 Schaums and Axiom agree
14:510 Schaums and Axiom agree
14:511 Schaums and Axiom agree
14:512 Axiom cannot compute this integral
14:513 Schaums and Axiom agree by definition
14:514 Axiom cannot compute this integral
14:515 Schaums and Axiom agree
14:516 Schaums and Axiom agree
14:517 Axiom cannot simplify these expressions
14:518 Schaums and Axiom agree
14:519 Schaums and Axiom agree
14:520 Schaums and Axiom agree
14:521 Schaums and Axiom agree
14:522 Schaums and Axiom agree by definition
14:523 Axiom cannot compute this integral
14:524 Axiom cannot compute this integral
========================================================================
diff --git a/changelog b/changelog
index 314b281..3956a7b 100644
--- a/changelog
+++ b/changelog
@@ -1,3 +1,5 @@
+20080425 tpd src/input/schaum25.input show Schaums-Axiom equivalence
+20080425 tpd src/input/schaum24.input show Schaums-Axiom equivalence
20080424 tpd src/input/schaum23.input show Schaums-Axiom equivalence
20080423 tpd src/input/schaum22.input show Schaums-Axiom equivalence
20080423 tpd src/input/schaum21.input show Schaums-Axiom equivalence
diff --git a/src/input/schaum24.input.pamphlet
b/src/input/schaum24.input.pamphlet
index 1a6b60a..9c034da 100644
--- a/src/input/schaum24.input.pamphlet
+++ b/src/input/schaum24.input.pamphlet
@@ -18,7 +18,7 @@ $$
)set message auto off
)clear all
---S 1 of 38
+--S 1
aa:=integrate(asin(x/a),x)
--R
--R
@@ -32,6 +32,31 @@ aa:=integrate(asin(x/a),x)
--R 2
--R Type: Union(Expression
Integer,...)
--E
+
+--S 2
+bb:=s+asin(x/a)+sqrt(a^2-x^2)
+--R
+--R +---------+
+--R | 2 2 x
+--R (2) \|- x + a + asin(-) + s
+--R a
+--R Type: Expression
Integer
+--E
+
+--S 3 14:471 Axiom cannot simplify this expression
+cc:=aa-bb
+--R
+--R +---------+
+--R | 2 2
+--R 2x\|- x + a x
+--R - x atan(--------------) - 2asin(-) - 2s
+--R 2 2 a
+--R 2x - a
+--R (3) ----------------------------------------
+--R 2
+--R Type: Expression
Integer
+--E
+
@
\section{\cite{1}:14.472~~~~~$\displaystyle
@@ -43,7 +68,7 @@ $$
<<*>>=
)clear all
---S 2 of 38
+--S 4
aa:=integrate(x*asin(x/a),x)
--R
--R
@@ -57,6 +82,32 @@ aa:=integrate(x*asin(x/a),x)
--R 8
--R Type: Union(Expression
Integer,...)
--E
+
+--S 5
+bb:=(x^2/2-a^2/4)*asin(x/a)+(x*sqrt(a^2-x^2))/4
+--R
+--R +---------+
+--R | 2 2 2 2 x
+--R x\|- x + a + (2x - a )asin(-)
+--R a
+--R (2) ---------------------------------
+--R 4
+--R Type: Expression
Integer
+--E
+
+--S 6 14:472 Axiom cannot simplify this expression
+cc:=aa-bb
+--R
+--R +---------+
+--R | 2 2
+--R 2 2 2x\|- x + a 2 2 x
+--R (- 2x + a )atan(--------------) + (- 4x + 2a )asin(-)
+--R 2 2 a
+--R 2x - a
+--R (3) -------------------------------------------------------
+--R 8
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.473~~~~~$\displaystyle
@@ -67,7 +118,7 @@ $$
<<*>>=
)clear all
---S 3 of 38
+--S 7
aa:=integrate(x^2*asin(x/a),x)
--R
--R
@@ -81,6 +132,32 @@ aa:=integrate(x^2*asin(x/a),x)
--R 18
--R Type: Union(Expression
Integer,...)
--E
+
+--S 8
+bb:=x^3/3*asin(x/a)+((x^2+2*a^2)*sqrt(a^2-x^2))/9
+--R
+--R +---------+
+--R 2 2 | 2 2 3 x
+--R (x + 2a )\|- x + a + 3x asin(-)
+--R a
+--R (2) -----------------------------------
+--R 9
+--R Type: Expression
Integer
+--E
+
+--S 9 14:473 Axiom cannot simplify this expression
+cc:=aa-bb
+--R
+--R +---------+
+--R | 2 2
+--R 3 2x\|- x + a 3 x
+--R - x atan(--------------) - 2x asin(-)
+--R 2 2 a
+--R 2x - a
+--R (3) -------------------------------------
+--R 6
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.474~~~~~$\displaystyle
@@ -93,7 +170,7 @@ $$
<<*>>=
)clear all
---S 4 of 38
+--S 10 14:474 Axiom cannot compute this integral
aa:=integrate(asin(x/a)/x,x)
--R
--R
@@ -115,7 +192,7 @@ $$
<<*>>=
)clear all
---S 5 of 38
+--S 11
aa:=integrate(asin(x/a)/x^2,x)
--R
--R
@@ -130,6 +207,38 @@ aa:=integrate(asin(x/a)/x^2,x)
--R 2a x
--R Type: Union(Expression
Integer,...)
--E
+
+--S 12
+bb:=-asin(x/a)/x-1/a*log((a+sqrt(a^2-x^2))/x)
+--R
+--R +---------+
+--R | 2 2
+--R \|- x + a + a x
+--R - x log(----------------) - a asin(-)
+--R x a
+--R (2) -------------------------------------
+--R a x
+--R Type: Expression
Integer
+--E
+
+--S 13 14:475 Axiom cannot simplify this expression
+cc:=aa-bb
+--R
+--R (3)
+--R +---------+ +---------+
+--R | 2 2 | 2 2
+--R - x log(\|- x + a + a) + x log(\|- x + a - a)
+--R +
+--R +---------+ +---------+
+--R | 2 2 | 2 2
+--R \|- x + a + a 2x\|- x + a x
+--R 2x log(----------------) + a atan(--------------) + 2a asin(-)
+--R x 2 2 a
+--R 2x - a
+--R /
+--R 2a x
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.476~~~~~$\displaystyle
@@ -140,7 +249,7 @@ $$
<<*>>=
)clear all
---S 6 of 38
+--S 14
aa:=integrate(asin(x/a)^2,x)
--R
--R
@@ -154,6 +263,36 @@ aa:=integrate(asin(x/a)^2,x)
--R 4
--R Type: Union(Expression
Integer,...)
--E
+
+--S 15
+bb:=x*asin(x/a)^2-2*x+2*sqrt(a^2-x^2)*asin(x/a)
+--R
+--R +---------+
+--R x | 2 2 x 2
+--R (2) 2asin(-)\|- x + a + x asin(-) - 2x
+--R a a
+--R Type: Expression
Integer
+--E
+
+--S 16 14:476 Axiom cannot simplify this expression
+cc:=aa-bb
+--R
+--R (3)
+--R +---------+ 2 +---------+
+--R | 2 2 +---------+ | 2 2
+--R 2x\|- x + a | 2 2 2x\|- x + a
+--R x atan(--------------) - 4\|- x + a atan(--------------)
+--R 2 2 2 2
+--R 2x - a 2x - a
+--R +
+--R +---------+
+--R x | 2 2 x 2
+--R - 8asin(-)\|- x + a - 4x asin(-)
+--R a a
+--R /
+--R 4
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.477~~~~~$\displaystyle
@@ -164,7 +303,7 @@ $$
<<*>>=
)clear all
---S 7 of 38
+--S 17
aa:=integrate(acos(x/a),x)
--R
--R
@@ -178,6 +317,30 @@ aa:=integrate(acos(x/a),x)
--R 2
--R Type: Union(Expression
Integer,...)
--E
+
+--S 18
+bb:=x*acos(x/a)-sqrt(a^2-x^2)
+--R
+--R +---------+
+--R | 2 2 x
+--R (2) - \|- x + a + x acos(-)
+--R a
+--R Type: Expression
Integer
+--E
+
+--S 19 14:477 Axiom cannot simplify this expression
+cc:=aa-bb
+--R
+--R +---------+
+--R | 2 2
+--R 2x\|- x + a x
+--R x atan(--------------) - 2x acos(-)
+--R 2 2 a
+--R 2x - a
+--R (3) -----------------------------------
+--R 2
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.478~~~~~$\displaystyle
@@ -189,7 +352,7 @@ $$
<<*>>=
)clear all
---S 8 of 38
+--S 20
aa:=integrate(x*acos(x/a),x)
--R
--R
@@ -203,6 +366,32 @@ aa:=integrate(x*acos(x/a),x)
--R 8
--R Type: Union(Expression
Integer,...)
--E
+
+--S 21
+bb:=(x^2/2-a^2/4)*acos(x/a)-(x*sqrt(a^2-x^2))/4
+--R
+--R +---------+
+--R | 2 2 2 2 x
+--R - x\|- x + a + (2x - a )acos(-)
+--R a
+--R (2) -----------------------------------
+--R 4
+--R Type: Expression
Integer
+--E
+
+--S 22 14:478 Axiom cannot simplify this expression
+cc:=aa-bb
+--R
+--R +---------+
+--R | 2 2
+--R 2 2 2x\|- x + a 2 2 x
+--R (2x - a )atan(--------------) + (- 4x + 2a )acos(-)
+--R 2 2 a
+--R 2x - a
+--R (3) -----------------------------------------------------
+--R 8
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.479~~~~~$\displaystyle
@@ -213,7 +402,7 @@ $$
<<*>>=
)clear all
---S 9 of 38
+--S 23
aa:=integrate(x^2*acos(x/a),x)
--R
--R
@@ -227,6 +416,32 @@ aa:=integrate(x^2*acos(x/a),x)
--R 18
--R Type: Union(Expression
Integer,...)
--E
+
+--S 24
+bb:=x^3/3*acos(x/a)-((x^2+2*a^2)*sqrt(a^2-x^2))/9
+--R
+--R +---------+
+--R 2 2 | 2 2 3 x
+--R (- x - 2a )\|- x + a + 3x acos(-)
+--R a
+--R (2) -------------------------------------
+--R 9
+--R Type: Expression
Integer
+--E
+
+--S 25 14:479 Axiom cannot simplify this expression
+cc:=aa-bb
+--R
+--R +---------+
+--R | 2 2
+--R 3 2x\|- x + a 3 x
+--R x atan(--------------) - 2x acos(-)
+--R 2 2 a
+--R 2x - a
+--R (3) -----------------------------------
+--R 6
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.480~~~~~$\displaystyle
@@ -237,7 +452,7 @@ $$
<<*>>=
)clear all
---S 10 of 38
+--S 26 14:480 Axiom cannot compute this integral
aa:=integrate(acos(x/a)/x,x)
--R
--R
@@ -258,7 +473,7 @@ $$
<<*>>=
)clear all
---S 11 of 38
+--S 27
aa:=integrate(acos(x/a)/x^2,x)
--R
--R
@@ -273,6 +488,38 @@ aa:=integrate(acos(x/a)/x^2,x)
--R 2a x
--R Type: Union(Expression
Integer,...)
--E
+
+--S 28
+bb:=-acos(x/a)/x+1/a*log((a+sqrt(a^2-x^2))/x)
+--R
+--R +---------+
+--R | 2 2
+--R \|- x + a + a x
+--R x log(----------------) - a acos(-)
+--R x a
+--R (2) -----------------------------------
+--R a x
+--R Type: Expression
Integer
+--E
+
+--S 29 14:481 Axiom cannot simplify this expression
+cc:=aa-bb
+--R
+--R (3)
+--R +---------+ +---------+
+--R | 2 2 | 2 2
+--R x log(\|- x + a + a) - x log(\|- x + a - a)
+--R +
+--R +---------+ +---------+
+--R | 2 2 | 2 2
+--R \|- x + a + a 2x\|- x + a x
+--R - 2x log(----------------) - a atan(--------------) + 2a acos(-)
+--R x 2 2 a
+--R 2x - a
+--R /
+--R 2a x
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.482~~~~~$\displaystyle
@@ -283,7 +530,7 @@ $$
<<*>>=
)clear all
---S 12 of 38
+--S 30
aa:=integrate(acos(x/a)^2,x)
--R
--R
@@ -296,7 +543,37 @@ aa:=integrate(acos(x/a)^2,x)
--R (1) ----------------------------------------------------------------
--R 4
--R Type: Union(Expression
Integer,...)
---E
+--E
+
+--S 31
+bb:=x*acos(x/a)^2-2*x-2*sqrt(a^2-x^2)*acos(x/a)
+--R
+--R +---------+
+--R x | 2 2 x 2
+--R (2) - 2acos(-)\|- x + a + x acos(-) - 2x
+--R a a
+--R Type: Expression
Integer
+--E
+
+--S 32 14:482 Axiom cannot simplify this expression
+cc:=aa-bb
+--R
+--R (3)
+--R +---------+ 2 +---------+
+--R | 2 2 +---------+ | 2 2
+--R 2x\|- x + a | 2 2 2x\|- x + a
+--R x atan(--------------) - 4\|- x + a atan(--------------)
+--R 2 2 2 2
+--R 2x - a 2x - a
+--R +
+--R +---------+
+--R x | 2 2 x 2
+--R 8acos(-)\|- x + a - 4x acos(-)
+--R a a
+--R /
+--R 4
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.483~~~~~$\displaystyle
@@ -307,7 +584,7 @@ $$
<<*>>=
)clear all
---S 13 of 38
+--S 33
aa:=integrate(atan(x/a),x)
--R
--R
@@ -319,6 +596,62 @@ aa:=integrate(atan(x/a),x)
--R 2
--R Type: Union(Expression
Integer,...)
--E
+
+--S 34
+bb:=x*atan(x/a)-a/2*log(x^2+a^2)
+--R
+--R 2 2 x
+--R - a log(x + a ) + 2x atan(-)
+--R a
+--R (2) -----------------------------
+--R 2
+--R Type: Expression
Integer
+--E
+
+--S 35
+cc:=aa-bb
+--R
+--R x 2a x
+--R - 2x atan(-) - x atan(-------)
+--R a 2 2
+--R x - a
+--R (3) ------------------------------
+--R 2
+--R Type: Expression
Integer
+--E
+
+--S 36
+atanrule:=rule(atan(x) == -%i/2*log((1+%i*x)/(1-%i*x)))
+--R
+--R - x + %i
+--R %i log(--------)
+--R x + %i
+--R (4) atan(x) == - ----------------
+--R 2
+--R Type: RewriteRule(Integer,Complex Integer,Expression Complex
Integer)
+--E
+
+--S 37
+dd:=atanrule cc
+--R
+--R 2 2
+--R x + 2%i a x - a - x + %i a
+--R %i x log(-----------------) + 2%i x log(----------)
+--R 2 2 x + %i a
+--R x - 2%i a x - a
+--R (5) ---------------------------------------------------
+--R 4
+--R Type: Expression Complex
Integer
+--E
+
+--S 38 14:483 SCHAUMS AND AXIOM DIFFER? (BRANCH CUTS?)
+ee:=expandLog dd
+--R
+--R %i x log(- 1)
+--R (6) -------------
+--R 2
+--R Type: Expression Complex
Integer
+--E
@
\section{\cite{1}:14.484~~~~~$\displaystyle
@@ -329,7 +662,7 @@ $$
<<*>>=
)clear all
---S 14 of 38
+--S 39 14:484 Axiom cannot compute this integral
aa:=integrate(x*tan(x/a),x)
--R
--R
@@ -349,7 +682,7 @@ $$
<<*>>=
)clear all
---S 15 of 38
+--S 40
aa:=integrate(x^2*atan(x/a),x)
--R
--R
@@ -361,6 +694,29 @@ aa:=integrate(x^2*atan(x/a),x)
--R 6
--R Type: Union(Expression
Integer,...)
--E
+
+--S 41
+bb:=x^3/2*atan(x/a)-(a*x^2)/6+a^3/6*log(x^2+a^2)
+--R
+--R 3 2 2 3 x 2
+--R a log(x + a ) + 3x atan(-) - a x
+--R a
+--R (2) ----------------------------------
+--R 6
+--R Type: Expression
Integer
+--E
+
+--S 42 14:485 Axiom cannot simplify this expression
+cc:=aa-bb
+--R
+--R 3 x 3 2a x
+--R - 3x atan(-) - x atan(-------)
+--R a 2 2
+--R x - a
+--R (3) ------------------------------
+--R 6
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.486~~~~~$\displaystyle
@@ -371,7 +727,7 @@ $$
<<*>>=
)clear all
---S 16 of 38
+--S 43 14:486 Axiom cannot compute this integral
aa:=integrate(atan(x/a)/x,x)
--R
--R
@@ -393,7 +749,7 @@ $$
<<*>>=
)clear all
---S 17 of 38
+--S 44
aa:=integrate(atan(x/a)/x^2,x)
--R
--R
@@ -405,6 +761,73 @@ aa:=integrate(atan(x/a)/x^2,x)
--R 2a x
--R Type: Union(Expression
Integer,...)
--E
+
+--S 45
+bb:=-1/x*atan(x/a)-1/(2*a)*log((x^2+a^2)/x^2)
+--R
+--R 2 2
+--R x + a x
+--R - x log(-------) - 2a atan(-)
+--R 2 a
+--R x
+--R (2) -----------------------------
+--R 2a x
+--R Type: Expression
Integer
+--E
+
+--S 46
+cc:=aa-bb
+--R
+--R (3)
+--R 2 2
+--R 2 2 x + a x 2a
x
+--R - x log(x + a ) + 2x log(x) + x log(-------) + 2a atan(-) + a
atan(-------)
+--R 2 a 2
2
+--R x x -
a
+--R
----------------------------------------------------------------------------
+--R 2a x
+--R Type: Expression
Integer
+--E
+
+--S 47
+atanrule:=rule(atan(x) == -%i/2*log((1+%i*x)/(1-%i*x)))
+--R
+--R - x + %i
+--R %i log(--------)
+--R x + %i
+--R (4) atan(x) == - ----------------
+--R 2
+--R Type: RewriteRule(Integer,Complex Integer,Expression Complex
Integer)
+--E
+
+--S 48
+dd:=atanrule cc
+--R
+--R (5)
+--R 2 2
+--R 2 2 x + 2%i a x - a
+--R - 2x log(x + a ) + 4x log(x) - %i a log(-----------------)
+--R 2 2
+--R x - 2%i a x - a
+--R +
+--R 2 2
+--R x + a - x + %i a
+--R 2x log(-------) - 2%i a log(----------)
+--R 2 x + %i a
+--R x
+--R /
+--R 4a x
+--R Type: Expression Complex
Integer
+--E
+
+--S 49 14:487 SCHAUMS AND AXIOM DIFFER? (branch cuts?)
+ee:=expandLog dd
+--R
+--R %i log(- 1)
+--R (6) - -----------
+--R 2x
+--R Type: Expression Complex
Integer
+--E
@
\section{\cite{1}:14.488~~~~~$\displaystyle
@@ -415,7 +838,7 @@ $$
<<*>>=
)clear all
---S 18 of 38
+--S 50
aa:=integrate(acot(x/a),x)
--R
--R
@@ -426,7 +849,85 @@ aa:=integrate(acot(x/a),x)
--R (1) --------------------------------
--R 2
--R Type: Union(Expression
Integer,...)
+--E
+
+--S 51
+bb:=x*acot(x/a)+a/2*log(x^2+a^2)
+--R
+--R 2 2 x
+--R a log(x + a ) + 2x acot(-)
+--R a
+--R (2) ---------------------------
+--R 2
+--R Type: Expression
Integer
--E
+
+--S 52
+cc:=aa-bb
+--R
+--R 2a x x
+--R x atan(-------) - 2x acot(-)
+--R 2 2 a
+--R x - a
+--R (3) ----------------------------
+--R 2
+--R Type: Expression
Integer
+--E
+
+--S 53
+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 54
+dd:=atanrule cc
+--R
+--R 2 2
+--R x + 2%i a x - a x
+--R - %i x log(-----------------) - 4x acot(-)
+--R 2 2 a
+--R x - 2%i a x - a
+--R (5) ------------------------------------------
+--R 4
+--R Type: Expression Complex
Integer
+--E
+
+--S 55
+acotrule:=rule(acot(x) == -%i/2*log((%i*x-1)/(%i*x+1)))
+--R
+--R x + %i
+--R %i log(------)
+--R x - %i
+--R (6) acot(x) == - --------------
+--R 2
+--R Type: RewriteRule(Integer,Complex Integer,Expression Complex
Integer)
+--E
+
+--S 56
+ee:=acotrule dd
+--R
+--R 2 2
+--R x + 2%i a x - a x + %i a
+--R - %i x log(-----------------) + 2%i x log(--------)
+--R 2 2 x - %i a
+--R x - 2%i a x - a
+--R (7) ---------------------------------------------------
+--R 4
+--R Type: Expression Complex
Integer
+--E
+
+--S 57 14:488 Axiom and Schaums agree
+ff:=expandLog %
+--R
+--R (8) 0
+--R Type: Expression Complex
Integer
+--E
@
\section{\cite{1}:14.489~~~~~$\displaystyle
@@ -437,7 +938,7 @@ $$
<<*>>=
)clear all
---S 19 of 38
+--S 58
aa:=integrate(x*acot(x/a),x)
--R
--R
@@ -449,6 +950,84 @@ aa:=integrate(x*acot(x/a),x)
--R 4
--R Type: Union(Expression
Integer,...)
--E
+
+--S 59
+bb:=1/2*(x^2+a^2)*acot(x/a)+(a*x)/2
+--R
+--R 2 2 x
+--R (x + a )acot(-) + a x
+--R a
+--R (2) ----------------------
+--R 2
+--R Type: Expression
Integer
+--E
+
+--S 60
+cc:=aa-bb
+--R
+--R 2 2 2a x 2 2 x
+--R (x + a )atan(-------) + (- 2x - 2a )acot(-)
+--R 2 2 a
+--R x - a
+--R (3) ---------------------------------------------
+--R 4
+--R Type: Expression
Integer
+--E
+
+--S 61
+acotrule:=rule(acot(x) == -%i/2*log((%i*x-1)/(%i*x+1)))
+--R
+--R x + %i
+--R %i log(------)
+--R x - %i
+--R (4) acot(x) == - --------------
+--R 2
+--R Type: RewriteRule(Integer,Complex Integer,Expression Complex
Integer)
+--E
+
+--S 62
+dd:=acotrule cc
+--R
+--R 2 2 x + %i a 2 2 2a x
+--R (%i x + %i a )log(--------) + (x + a )atan(-------)
+--R x - %i a 2 2
+--R x - a
+--R (5) -----------------------------------------------------
+--R 4
+--R Type: Expression Complex
Integer
+--E
+
+--S 63
+atanrule:=rule(atan(x) == -%i/2*log((1+%i*x)/(1-%i*x)))
+--R
+--R - x + %i
+--R %i log(--------)
+--R x + %i
+--R (6) atan(x) == - ----------------
+--R 2
+--R Type: RewriteRule(Integer,Complex Integer,Expression Complex
Integer)
+--E
+
+--S 64
+ee:=atanrule dd
+--R
+--R (7)
+--R 2 2
+--R 2 2 x + 2%i a x - a 2 2 x + %i a
+--R (- %i x - %i a )log(-----------------) + (2%i x + 2%i a )log(--------)
+--R 2 2 x - %i a
+--R x - 2%i a x - a
+--R ------------------------------------------------------------------------
+--R 8
+--R Type: Expression Complex
Integer
+--E
+
+--S 65 14:489 Axiom and Schaums agree
+ff:=expandLog ee
+--R
+--R (8) 0
+--R Type: Expression Complex
Integer
+--E
@
\section{\cite{1}:14.490~~~~~$\displaystyle
@@ -459,7 +1038,7 @@ $$
<<*>>=
)clear all
---S 20 of 38
+--S 66
aa:=integrate(x^2*acot(x/a),x)
--R
--R
@@ -470,7 +1049,85 @@ aa:=integrate(x^2*acot(x/a),x)
--R (1) -----------------------------------------
--R 6
--R Type: Union(Expression
Integer,...)
+--E
+
+--S 67
+bb:=x^3/3*acot(x/a)+(a*x^2)/6-a^3/6*log(x^2+a^2)
+--R
+--R 3 2 2 3 x 2
+--R - a log(x + a ) + 2x acot(-) + a x
+--R a
+--R (2) ------------------------------------
+--R 6
+--R Type: Expression
Integer
--E
+
+--S 68
+cc:=aa-bb
+--R
+--R 3 2a x 3 x
+--R x atan(-------) - 2x acot(-)
+--R 2 2 a
+--R x - a
+--R (3) ----------------------------
+--R 6
+--R Type: Expression
Integer
+--E
+
+--S 69
+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 70
+dd:=atanrule cc
+--R
+--R 2 2
+--R 3 x + 2%i a x - a 3 x
+--R - %i x log(-----------------) - 4x acot(-)
+--R 2 2 a
+--R x - 2%i a x - a
+--R (5) ------------------------------------------
+--R 12
+--R Type: Expression Complex
Integer
+--E
+
+--S 71
+acotrule:=rule(acot(x) == -%i/2*log((%i*x-1)/(%i*x+1)))
+--R
+--R x + %i
+--R %i log(------)
+--R x - %i
+--R (6) acot(x) == - --------------
+--R 2
+--R Type: RewriteRule(Integer,Complex Integer,Expression Complex
Integer)
+--E
+
+--S 72
+ee:=acotrule dd
+--R
+--R 2 2
+--R 3 x + 2%i a x - a 3 x + %i a
+--R - %i x log(-----------------) + 2%i x log(--------)
+--R 2 2 x - %i a
+--R x - 2%i a x - a
+--R (7) ---------------------------------------------------
+--R 12
+--R Type: Expression Complex
Integer
+--E
+
+--S 73 14:490 Axiom and Schaums agree
+ff:=expandLog ee
+--R
+--R (8) 0
+--R Type: Expression Complex
Integer
+--E
@
\section{\cite{1}:14.491~~~~~$\displaystyle
@@ -481,7 +1138,7 @@ $$
<<*>>=
)clear all
---S 21 of 38
+--S 74 14:491 Axiom cannot compute this integral
aa:=integrate(acot(x/a)/x,x)
--R
--R
@@ -502,7 +1159,7 @@ $$
<<*>>=
)clear all
---S 22 of 38
+--S 75
aa:=integrate(acot(x/a)/x^2,x)
--R
--R
@@ -514,6 +1171,101 @@ aa:=integrate(acot(x/a)/x^2,x)
--R 2a x
--R Type: Union(Expression
Integer,...)
--E
+
+--S 76
+bb:=-acot(x/a)/x+1/(2*a)*log((x^2+a^2)/x^2)
+--R
+--R 2 2
+--R x + a x
+--R x log(-------) - 2a acot(-)
+--R 2 a
+--R x
+--R (2) ---------------------------
+--R 2a x
+--R Type: Expression
Integer
+--E
+
+--S 77
+cc:=aa-bb
+--R
+--R (3)
+--R 2 2
+--R 2 2 x + a 2a x x
+--R x log(x + a ) - 2x log(x) - x log(-------) - a atan(-------) + 2a
acot(-)
+--R 2 2 2 a
+--R x x - a
+--R
--------------------------------------------------------------------------
+--R 2a x
+--R Type: Expression
Integer
+--E
+
+--S 78
+acotrule:=rule(acot(x) == -%i/2*log((%i*x-1)/(%i*x+1)))
+--R
+--R x + %i
+--R %i log(------)
+--R x - %i
+--R (4) acot(x) == - --------------
+--R 2
+--R Type: RewriteRule(Integer,Complex Integer,Expression Complex
Integer)
+--E
+
+--S 79
+dd:=acotrule cc
+--R
+--R (5)
+--R 2 2
+--R 2 2 x + %i a x + a
+--R x log(x + a ) - 2x log(x) - %i a log(--------) - x log(-------)
+--R x - %i a 2
+--R x
+--R +
+--R 2a x
+--R - a atan(-------)
+--R 2 2
+--R x - a
+--R /
+--R 2a x
+--R Type: Expression Complex
Integer
+--E
+
+--S 80
+atanrule:=rule(atan(x) == -%i/2*log((1+%i*x)/(1-%i*x)))
+--R
+--R - x + %i
+--R %i log(--------)
+--R x + %i
+--R (6) atan(x) == - ----------------
+--R 2
+--R Type: RewriteRule(Integer,Complex Integer,Expression Complex
Integer)
+--E
+
+--S 81
+ee:=atanrule dd
+--R
+--R (7)
+--R 2 2
+--R 2 2 x + 2%i a x - a
+--R 2x log(x + a ) - 4x log(x) + %i a log(-----------------)
+--R 2 2
+--R x - 2%i a x - a
+--R +
+--R 2 2
+--R x + %i a x + a
+--R - 2%i a log(--------) - 2x log(-------)
+--R x - %i a 2
+--R x
+--R /
+--R 4a x
+--R Type: Expression Complex
Integer
+--E
+
+--S 82 14:492 Schaums and Axiom agree
+ff:=expandLog ee
+--R
+--R (8) 0
+--R Type: Expression Complex
Integer
+--E
@
\section{\cite{1}:14.493~~~~~$\displaystyle
@@ -534,7 +1286,7 @@ $$
<<*>>=
)clear all
---S 23 of 38
+--S 83
aa:=integrate(asec(x/a),x)
--R
--R
@@ -555,6 +1307,72 @@ aa:=integrate(asec(x/a),x)
--R 2
--R Type: Union(Expression
Integer,...)
--E
+
+--S 84
+bb1:=x*asec(x/a)-a*log(x+sqrt(x^2-a^2))
+--R
+--R +-------+
+--R | 2 2 x
+--R (2) - a log(\|x - a + x) + x asec(-)
+--R a
+--R Type: Expression
Integer
+--E
+
+--S 85
+bb2:=x*asec(x/a)+a*log(x+sqrt(x^2-a^2))
+--R
+--R +-------+
+--R | 2 2 x
+--R (3) a log(\|x - a + x) + x asec(-)
+--R a
+--R Type: Expression
Integer
+--E
+
+--S 86
+cc1:=aa-bb1
+--R
+--R (4)
+--R +---------+
+--R +-------+ +-+ | 2 2
+--R | 2 2 +-+ 2x\|2 \|- x + a
+--R 2a log(\|x - a + x) - a\|2 atan(------------------)
+--R 2 2
+--R 3x - 2a
+--R +
+--R +---------+
+--R | 2 2
+--R 2a\|- x + a x x
+--R x atan(--------------) - 2a atan(------------) - 2x asec(-)
+--R 2 +---------+ a
+--R x | 2 2
+--R \|- x + a
+--R /
+--R 2
+--R Type: Expression
Integer
+--E
+
+--S 87 14:493 Axiom cannot simplify these expressions
+cc2:=aa-bb2
+--R
+--R (5)
+--R +---------+
+--R +-------+ +-+ | 2 2
+--R | 2 2 +-+ 2x\|2 \|- x + a
+--R - 2a log(\|x - a + x) - a\|2 atan(------------------)
+--R 2 2
+--R 3x - 2a
+--R +
+--R +---------+
+--R | 2 2
+--R 2a\|- x + a x x
+--R x atan(--------------) - 2a atan(------------) - 2x asec(-)
+--R 2 +---------+ a
+--R x | 2 2
+--R \|- x + a
+--R /
+--R 2
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.494~~~~~$\displaystyle
@@ -575,7 +1393,7 @@ $$
<<*>>=
)clear all
---S 24 of 38
+--S 88
aa:=integrate(x*asec(x/a),x)
--R
--R
@@ -589,6 +1407,60 @@ aa:=integrate(x*asec(x/a),x)
--R 4
--R Type: Union(Expression
Integer,...)
--E
+
+--S 89
+bb1:=x^2/2*asec(x/a)-(a*sqrt(x^2-a^2))/2
+--R
+--R +-------+
+--R | 2 2 2 x
+--R - a\|x - a + x asec(-)
+--R a
+--R (2) -------------------------
+--R 2
+--R Type: Expression
Integer
+--E
+
+--S 90
+bb2:=x^2/2*asec(x/a)+(a*sqrt(x^2-a^2))/2
+--R
+--R +-------+
+--R | 2 2 2 x
+--R a\|x - a + x asec(-)
+--R a
+--R (3) -----------------------
+--R 2
+--R Type: Expression
Integer
+--E
+
+--S 91
+cc1:=aa-bb1
+--R
+--R (4)
+--R +---------+
+--R | 2 2 +-------+ +---------+
+--R 2 2 2a\|- x + a | 2 2 | 2 2 2
x
+--R (x - 2a )atan(--------------) + 2a\|x - a + 2a\|- x + a - 2x
asec(-)
+--R 2
a
+--R x
+--R
---------------------------------------------------------------------------
+--R 4
+--R Type: Expression
Integer
+--E
+
+--S 92 14:494 Axiom cannot simplify these expressions
+cc2:=aa-bb2
+--R
+--R (5)
+--R +---------+
+--R | 2 2 +-------+ +---------+
+--R 2 2 2a\|- x + a | 2 2 | 2 2 2
x
+--R (x - 2a )atan(--------------) - 2a\|x - a + 2a\|- x + a - 2x
asec(-)
+--R 2
a
+--R x
+--R
---------------------------------------------------------------------------
+--R 4
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.495~~~~~$\displaystyle
@@ -616,7 +1488,7 @@ $$
<<*>>=
)clear all
---S 25 of 38
+--S 93
aa:=integrate(x^2*asec(x/a),x)
--R
--R
@@ -638,6 +1510,87 @@ aa:=integrate(x^2*asec(x/a),x)
--R 6
--R Type: Union(Expression
Integer,...)
--E
+
+--S 94
+bb1:=x^3/3*asec(x/a)-(a*x*sqrt(x^2-a^2))/6-a^3/6*log(x+sqrt(x^2-a^2))
+--R
+--R +-------+ +-------+
+--R 3 | 2 2 | 2 2 3 x
+--R - a log(\|x - a + x) - a x\|x - a + 2x asec(-)
+--R a
+--R (2) ----------------------------------------------------
+--R 6
+--R Type: Expression
Integer
+--E
+
+--S 95
+bb2:=x^3/3*asec(x/a)+(a*x*sqrt(x^2-a^2))/6+a^3/6*log(x+sqrt(x^2-a^2))
+--R
+--R +-------+ +-------+
+--R 3 | 2 2 | 2 2 3 x
+--R a log(\|x - a + x) + a x\|x - a + 2x asec(-)
+--R a
+--R (3) --------------------------------------------------
+--R 6
+--R Type: Expression
Integer
+--E
+
+--S 96
+cc1:=aa-bb1
+--R
+--R (4)
+--R +---------+
+--R +-------+ +-+ | 2 2
+--R 3 | 2 2 3 +-+ 2x\|2 \|- x + a
+--R a log(\|x - a + x) - 2a \|2 atan(------------------)
+--R 2 2
+--R 3x - 2a
+--R +
+--R +---------+
+--R | 2 2 +-------+
+--R 3 2a\|- x + a 3 x | 2 2
+--R x atan(--------------) - 5a atan(------------) + a x\|x - a
+--R 2 +---------+
+--R x | 2 2
+--R \|- x + a
+--R +
+--R +---------+
+--R | 2 2 3 x
+--R a x\|- x + a - 2x asec(-)
+--R a
+--R /
+--R 6
+--R Type: Expression
Integer
+--E
+
+--S 97 14:495 Axiom cannot simplify these expressions
+cc2:=aa-bb2
+--R
+--R (5)
+--R +---------+
+--R +-------+ +-+ | 2 2
+--R 3 | 2 2 3 +-+ 2x\|2 \|- x + a
+--R - a log(\|x - a + x) - 2a \|2 atan(------------------)
+--R 2 2
+--R 3x - 2a
+--R +
+--R +---------+
+--R | 2 2 +-------+
+--R 3 2a\|- x + a 3 x | 2 2
+--R x atan(--------------) - 5a atan(------------) - a x\|x - a
+--R 2 +---------+
+--R x | 2 2
+--R \|- x + a
+--R +
+--R +---------+
+--R | 2 2 3 x
+--R a x\|- x + a - 2x asec(-)
+--R a
+--R /
+--R 6
+--R Type: Expression
Integer
+--E
+
@
\section{\cite{1}:14.496~~~~~$\displaystyle
@@ -650,7 +1603,7 @@ $$
<<*>>=
)clear all
---S 26 of 38
+--S 98 14:496 Axiom cannot compute this integral
aa:=integrate(asec(x/a)/x,x)
--R
--R
@@ -681,7 +1634,7 @@ $$
<<*>>=
)clear all
---S 27 of 38
+--S 99
aa:=integrate(asec(x/a)/x^2,x)
--R
--R
@@ -696,6 +1649,70 @@ aa:=integrate(asec(x/a)/x^2,x)
--R 2a x\|2
--R Type: Union(Expression
Integer,...)
--E
+
+--S 100
+bb1:=-asec(x/a)/x+sqrt(x^2-a^2)/(a*x)
+--R
+--R +-------+
+--R | 2 2 x
+--R \|x - a - a asec(-)
+--R a
+--R (2) ----------------------
+--R a x
+--R Type: Expression
Integer
+--E
+
+--S 101
+bb2:=-asec(x/a)/x-sqrt(x^2-a^2)/(a*x)
+--R
+--R +-------+
+--R | 2 2 x
+--R - \|x - a - a asec(-)
+--R a
+--R (3) ------------------------
+--R a x
+--R Type: Expression
Integer
+--E
+
+--S 102
+cc1:=aa-bb1
+--R
+--R (4)
+--R +---------+ +---------+
+--R +-+ | 2 2 | 2 2
+-------+
+--R 2x\|2 \|- x + a +-+ 2a\|- x + a +-+ | 2
2
+--R x atan(------------------) - a\|2 atan(--------------) - 2\|2 \|x -
a
+--R 2 2 2
+--R 3x - 2a x
+--R +
+--R +-+ x
+--R 2a\|2 asec(-)
+--R a
+--R /
+--R +-+
+--R 2a x\|2
+--R Type: Expression
Integer
+--E
+
+--S 103 14:497 Axiom cannot simplify these expressions
+cc2:=aa-bb2
+--R
+--R (5)
+--R +---------+ +---------+
+--R +-+ | 2 2 | 2 2
+-------+
+--R 2x\|2 \|- x + a +-+ 2a\|- x + a +-+ | 2
2
+--R x atan(------------------) - a\|2 atan(--------------) + 2\|2 \|x -
a
+--R 2 2 2
+--R 3x - 2a x
+--R +
+--R +-+ x
+--R 2a\|2 asec(-)
+--R a
+--R /
+--R +-+
+--R 2a x\|2
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.498~~~~~$\displaystyle
@@ -716,7 +1733,7 @@ $$
<<*>>=
)clear all
---S 28 of 38
+--S 104
aa:=integrate(acsc(x/a),x)
--R
--R
@@ -737,6 +1754,73 @@ aa:=integrate(acsc(x/a),x)
--R 2
--R Type: Union(Expression
Integer,...)
--E
+
+--S 105
+bb1:=x*acsc(x/a)+a*log(x+sqrt(x^2-a^2))
+--R
+--R +-------+
+--R | 2 2 x
+--R (2) a log(\|x - a + x) + x acsc(-)
+--R a
+--R Type: Expression
Integer
+--E
+
+--S 106
+bb2:=x*acsc(x/a)-a*log(x+sqrt(x^2-a^2))
+--R
+--R +-------+
+--R | 2 2 x
+--R (3) - a log(\|x - a + x) + x acsc(-)
+--R a
+--R Type: Expression
Integer
+--E
+
+--S 107
+cc1:=aa-bb1
+--R
+--R (4)
+--R +---------+
+--R +-------+ +-+ | 2 2
+--R | 2 2 +-+ 2x\|2 \|- x + a
+--R - 2a log(\|x - a + x) + a\|2 atan(------------------)
+--R 2 2
+--R 3x - 2a
+--R +
+--R +---------+
+--R | 2 2
+--R 2a\|- x + a x x
+--R - x atan(--------------) + 2a atan(------------) - 2x acsc(-)
+--R 2 +---------+ a
+--R x | 2 2
+--R \|- x + a
+--R /
+--R 2
+--R Type: Expression
Integer
+--E
+
+--S 108 14:498 Axiom cannot simplify these expressions
+cc2:=aa-bb2
+--R
+--R (5)
+--R +---------+
+--R +-------+ +-+ | 2 2
+--R | 2 2 +-+ 2x\|2 \|- x + a
+--R 2a log(\|x - a + x) + a\|2 atan(------------------)
+--R 2 2
+--R 3x - 2a
+--R +
+--R +---------+
+--R | 2 2
+--R 2a\|- x + a x x
+--R - x atan(--------------) + 2a atan(------------) - 2x acsc(-)
+--R 2 +---------+ a
+--R x | 2 2
+--R \|- x + a
+--R /
+--R 2
+--R Type: Expression
Integer
+--E
+
@
\section{\cite{1}:14.499~~~~~$\displaystyle
@@ -757,7 +1841,7 @@ $$
<<*>>=
)clear all
---S 29 of 38
+--S 109
aa:=integrate(x*acsc(x/a),x)
--R
--R
@@ -771,6 +1855,60 @@ aa:=integrate(x*acsc(x/a),x)
--R 4
--R Type: Union(Expression
Integer,...)
--E
+
+--S 110
+bb1:=x^2/2*acsc(x/a)+(a*sqrt(x^2-a^2))/2
+--R
+--R +-------+
+--R | 2 2 2 x
+--R a\|x - a + x acsc(-)
+--R a
+--R (2) -----------------------
+--R 2
+--R Type: Expression
Integer
+--E
+
+--S 111
+bb2:=x^2/2*acsc(x/a)-(a*sqrt(x^2-a^2))/2
+--R
+--R +-------+
+--R | 2 2 2 x
+--R - a\|x - a + x acsc(-)
+--R a
+--R (3) -------------------------
+--R 2
+--R Type: Expression
Integer
+--E
+
+--S 112
+cc1:=aa-bb1
+--R
+--R (4)
+--R +---------+
+--R | 2 2 +-------+ +---------+
+--R 2 2 2a\|- x + a | 2 2 | 2 2 2
x
+--R (- x + 2a )atan(--------------) - 2a\|x - a - 2a\|- x + a - 2x
acsc(-)
+--R 2
a
+--R x
+--R
-----------------------------------------------------------------------------
+--R 4
+--R Type: Expression
Integer
+--E
+
+--S 113 14:499 Axiom cannot simplify these expressions
+cc2:=aa-bb2
+--R
+--R (5)
+--R +---------+
+--R | 2 2 +-------+ +---------+
+--R 2 2 2a\|- x + a | 2 2 | 2 2 2
x
+--R (- x + 2a )atan(--------------) + 2a\|x - a - 2a\|- x + a - 2x
acsc(-)
+--R 2
a
+--R x
+--R
-----------------------------------------------------------------------------
+--R 4
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.500~~~~~$\displaystyle
@@ -798,7 +1936,7 @@ $$
<<*>>=
)clear all
---S 30 of 38
+--S 114
aa:=integrate(x^2*acsc(x/a),x)
--R
--R
@@ -820,6 +1958,86 @@ aa:=integrate(x^2*acsc(x/a),x)
--R 6
--R Type: Union(Expression
Integer,...)
--E
+
+--S 115
+bb1:=x^3/3*acsc(x/a)+(a*x*sqrt(x^2-a^2))/6+a^3/6*log(x+sqrt(x^2-a^2))
+--R
+--R +-------+ +-------+
+--R 3 | 2 2 | 2 2 3 x
+--R a log(\|x - a + x) + a x\|x - a + 2x acsc(-)
+--R a
+--R (2) --------------------------------------------------
+--R 6
+--R Type: Expression
Integer
+--E
+
+--S 116
+bb2:=x^3/3*acsc(x/a)-(a*x*sqrt(x^2-a^2))/6-a^3/6*log(x+sqrt(x^2-a^2))
+--R
+--R +-------+ +-------+
+--R 3 | 2 2 | 2 2 3 x
+--R - a log(\|x - a + x) - a x\|x - a + 2x acsc(-)
+--R a
+--R (3) ----------------------------------------------------
+--R 6
+--R Type: Expression
Integer
+--E
+
+--S 117
+cc1:=aa-bb1
+--R
+--R (4)
+--R +---------+
+--R +-------+ +-+ | 2 2
+--R 3 | 2 2 3 +-+ 2x\|2 \|- x + a
+--R - a log(\|x - a + x) + 2a \|2 atan(------------------)
+--R 2 2
+--R 3x - 2a
+--R +
+--R +---------+
+--R | 2 2 +-------+
+--R 3 2a\|- x + a 3 x | 2 2
+--R - x atan(--------------) + 5a atan(------------) - a x\|x - a
+--R 2 +---------+
+--R x | 2 2
+--R \|- x + a
+--R +
+--R +---------+
+--R | 2 2 3 x
+--R - a x\|- x + a - 2x acsc(-)
+--R a
+--R /
+--R 6
+--R Type: Expression
Integer
+--E
+
+--S 118 14:500 Axiom cannot simplify this expression
+cc2:=aa-bb2
+--R
+--R (5)
+--R +---------+
+--R +-------+ +-+ | 2 2
+--R 3 | 2 2 3 +-+ 2x\|2 \|- x + a
+--R a log(\|x - a + x) + 2a \|2 atan(------------------)
+--R 2 2
+--R 3x - 2a
+--R +
+--R +---------+
+--R | 2 2 +-------+
+--R 3 2a\|- x + a 3 x | 2 2
+--R - x atan(--------------) + 5a atan(------------) + a x\|x - a
+--R 2 +---------+
+--R x | 2 2
+--R \|- x + a
+--R +
+--R +---------+
+--R | 2 2 3 x
+--R - a x\|- x + a - 2x acsc(-)
+--R a
+--R /
+--R 6
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.501~~~~~$\displaystyle
@@ -832,7 +2050,7 @@ $$
<<*>>=
)clear all
---S 31 of 38
+--S 119 14:501 Axiom cannot compute this integral
aa:=integrate(acsc(x/a)/x,x)
--R
--R
@@ -863,7 +2081,7 @@ $$
<<*>>=
)clear all
---S 32 of 38
+--S 120
aa:=integrate(acsc(x/a)/x^2,x)
--R
--R
@@ -878,6 +2096,72 @@ aa:=integrate(acsc(x/a)/x^2,x)
--R 2a x\|2
--R Type: Union(Expression
Integer,...)
--E
+
+--S 121
+bb1:=-acsc(x/a)/x-sqrt(x^2-a^2)/(a*x)
+--R
+--R +-------+
+--R | 2 2 x
+--R - \|x - a - a acsc(-)
+--R a
+--R (2) ------------------------
+--R a x
+--R Type: Expression
Integer
+--E
+
+--S 122
+bb2:=-acsc(x/a)/x+sqrt(x^2-a^2)/(a*x)
+--R
+--R +-------+
+--R | 2 2 x
+--R \|x - a - a acsc(-)
+--R a
+--R (3) ----------------------
+--R a x
+--R Type: Expression
Integer
+--E
+
+--S 123
+cc1:=aa-bb1
+--R
+--R (4)
+--R +---------+ +---------+
+--R +-+ | 2 2 | 2 2
+--R 2x\|2 \|- x + a +-+ 2a\|- x + a
+--R - x atan(------------------) + a\|2 atan(--------------)
+--R 2 2 2
+--R 3x - 2a x
+--R +
+--R +-------+
+--R +-+ | 2 2 +-+ x
+--R 2\|2 \|x - a + 2a\|2 acsc(-)
+--R a
+--R /
+--R +-+
+--R 2a x\|2
+--R Type: Expression
Integer
+--E
+
+--S 124 14:502 Axiom cannot simplify this expression
+cc2:=aa-bb2
+--R
+--R (5)
+--R +---------+ +---------+
+--R +-+ | 2 2 | 2 2
+--R 2x\|2 \|- x + a +-+ 2a\|- x + a
+--R - x atan(------------------) + a\|2 atan(--------------)
+--R 2 2 2
+--R 3x - 2a x
+--R +
+--R +-------+
+--R +-+ | 2 2 +-+ x
+--R - 2\|2 \|x - a + 2a\|2 acsc(-)
+--R a
+--R /
+--R +-+
+--R 2a x\|2
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.503~~~~~$\displaystyle
@@ -888,7 +2172,7 @@ $$
<<*>>=
)clear all
---S 33 of 38
+--S 125 14:503 Axiom cannot compute this integral
aa:=integrate(x^m*asin(x/a),x)
--R
--R
@@ -908,7 +2192,7 @@ $$
<<*>>=
)clear all
---S 34 of 38
+--S 126 14:504 Axiom cannot compute this integral
aa:=integrate(x^m*acos(x/a),x)
--R
--R
@@ -926,10 +2210,15 @@ $$\int{x^m\tan^{-1}\frac{x}{a}}=
\frac{x^{m_1}}{m+1}\tan^{-1}\frac{x}{a}
-\frac{a}{m+1}\int{\frac{x^{m+1}}{x^2+a^2}}
$$
+This appears to be an interesting integral. Axiom found a closed
+form solution to the problem. However, the t1 integral below does
+not have a closed form solution. Note that we did not return a
+result for the prior two integrals, nor for the next integral. They
+have the same form but are expressed in terms of asin, acos, and acot.
<<*>>=
)clear all
---S 35 of 38
+--S 127
aa:=integrate(x*m*atan(x/a),x)
--R
--R
@@ -941,6 +2230,97 @@ aa:=integrate(x*m*atan(x/a),x)
--R 4
--R Type: Union(Expression
Integer,...)
--E
+
+--S 128
+t1:=integrate(x^(m+1)/(x^2+a^2),x)
+--E
+@
+Since we cannot get a closed form version of the prior integral
+we proceed to try to prove that Axiom got a correct answer. We
+do this by computing the derivate of 'aa' above and finding the
+difference from the original formula.
+
+So first we generate the derivative:
+<<*>>=
+
+--S 129
+bb:=D(aa,x)
+--R
+--R 2a x
+--R m x atan(-------)
+--R 2 2
+--R x - a
+--R (3) - -----------------
+--R 2
+--R Type: Expression
Integer
+--E
+@
+Then we input the original expression
+<<*>>=
+--S 130
+aa1:=x*m*atan(x/a)
+--R
+--R x
+--R (4) m x atan(-)
+--R a
+--R Type: Expression
Integer
+--E
+@
+Now we take their difference
+<<*>>=
+--S 131
+dd:=aa1-bb
+--R
+--R x 2a x
+--R 2m x atan(-) + m x atan(-------)
+--R a 2 2
+--R x - a
+--R (5) --------------------------------
+--R 2
+--R Type: Expression
Integer
+--E
+@
+Now we input the atan transformation
+<<*>>=
+--S 132
+atanrule:=rule(atan(x) == -%i/2*log((1+%i*x)/(1-%i*x)))
+--R
+--R - x + %i
+--R %i log(--------)
+--R x + %i
+--R (6) atan(x) == - ----------------
+--R 2
+--R Type: RewriteRule(Integer,Complex Integer,Expression Complex
Integer)
+--E
+@
+And apply the transformation to the difference
+<<*>>=
+--S 133
+ee:=atanrule dd
+--R
+--R 2 2
+--R x + 2%i a x - a - x + %i a
+--R - %i m x log(-----------------) - 2%i m x log(----------)
+--R 2 2 x + %i a
+--R x - 2%i a x - a
+--R (7) ---------------------------------------------------------
+--R 4
+--R Type: Expression Complex
Integer
+--E
+@
+And now we simplify
+<<*>>=
+--S 134 14:505 SCHAUMS AND AXIOM DISAGREE? (branch cuts?)
+ff:=expandLog ee
+--R
+--R %i m x log(- 1)
+--R (8) - ---------------
+--R 2
+--R Type: Expression Complex
Integer
+--E
+And we get the surprising result that they are not equal.
+In fact, they differ by a complex value depending on x.
+Likely there is a branch-cut issue lurking somewhere.
@
\section{\cite{1}:14.506~~~~~$\displaystyle
@@ -952,7 +2332,7 @@ $$
<<*>>=
)clear all
---S 36 of 38
+--S 135 14:506 Axiom cannot compute this integral
aa:=integrate(x^m*acot(x/a),x)
--R
--R
@@ -982,7 +2362,7 @@ $$
<<*>>=
)clear all
---S 37 of 38
+--S 136 14:507 Axiom cannot compute this integral
aa:=integrate(x^m*asec(x/a),x)
--R
--R
@@ -1012,7 +2392,7 @@ $$
<<*>>=
)clear all
---S 38 of 38
+--S 137 14:508 Axiom cannot compute this integral
aa:=integrate(x^m*acsc(x/a),x)
--R
--R
diff --git a/src/input/schaum25.input.pamphlet
b/src/input/schaum25.input.pamphlet
index 250e7c3..0a55f34 100644
--- a/src/input/schaum25.input.pamphlet
+++ b/src/input/schaum25.input.pamphlet
@@ -18,16 +18,32 @@ $$
)set message auto off
)clear all
---S 1 of 16
-aa:=integrate(e^(a*x),x)
---R
+--S 1
+aa:=integrate(%e^(a*x),x)
--R
---R a x log(e)
+--R a x
--R %e
---R (1) ------------
---R a log(e)
+--R (1) -----
+--R a
--R Type: Union(Expression
Integer,...)
--E
+
+--S 2
+bb:=%e^(a*x)/a
+--R
+--R a x
+--R %e
+--R (2) -----
+--R a
+--R Type: Expression
Integer
+--E
+
+--S 3 14:509 Schaums and Axiom agree
+cc:=aa-bb
+--R
+--R (3) 0
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.510~~~~~$\displaystyle
@@ -38,17 +54,34 @@ $$
<<*>>=
)clear all
---S 2 of 16
-aa:=integrate(x*e^(a*x),x)
---R
+--S 4
+aa:=integrate(x*%e^(a*x),x)
--R
---R a x log(e)
---R (a x log(e) - 1)%e
---R (1) ----------------------------
---R 2 2
---R a log(e)
+--R a x
+--R (a x - 1)%e
+--R (1) --------------
+--R 2
+--R a
--R Type: Union(Expression
Integer,...)
--E
+
+--S 5
+bb:=%e^(a*x)/a*(x-1/a)
+--R
+--R a x
+--R (a x - 1)%e
+--R (2) --------------
+--R 2
+--R a
+--R Type: Expression
Integer
+--E
+
+--S 6 14:510 Schaums and Axiom agree
+cc:=aa-bb
+--R
+--R (3) 0
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.511~~~~~$\displaystyle
@@ -59,17 +92,34 @@ $$
<<*>>=
)clear all
---S 3 of 16
-aa:=integrate(x^2*e^(a*x),x)
---R
+--S 7
+aa:=integrate(x^2*%e^(a*x),x)
--R
---R 2 2 2 a x log(e)
---R (a x log(e) - 2a x log(e) + 2)%e
---R (1) -------------------------------------------
---R 3 3
---R a log(e)
+--R 2 2 a x
+--R (a x - 2a x + 2)%e
+--R (1) ----------------------
+--R 3
+--R a
--R Type: Union(Expression
Integer,...)
--E
+
+--S 8
+bb:=%e^(a*x)/a*(x^2-(2*x)/a+2/a^2)
+--R
+--R 2 2 a x
+--R (a x - 2a x + 2)%e
+--R (2) ----------------------
+--R 3
+--R a
+--R Type: Expression
Integer
+--E
+
+--S 9 14:511 Schaums and Axiom agree
+cc:=aa-bb
+--R
+--R (3) 0
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.512~~~~~$\displaystyle
@@ -89,16 +139,16 @@ $$
<<*>>=
)clear all
---S 4 of 16
-aa:=integrate(x^n*e^(a*x),x)
---R
+--S 10 14:512 Axiom cannot compute this integral
+aa:=integrate(x^n*%e^(a*x),x)
--R
--R x
---I ++ n %I a
---I (1) | %I e d%I
+--I ++ %I a n
+--I (1) | %e %I d%I
--R ++
--R Type: Union(Expression
Integer,...)
---E
+--E
+
@
\section{\cite{1}:14.513~~~~~$\displaystyle
@@ -110,11 +160,10 @@ $$
<<*>>=
)clear all
---S 5 of 16
-aa:=integrate(e^(a*x)/x,x)
---R
+--S 11 14:513 Schaums and Axiom agree by definition
+aa:=integrate(%e^(a*x)/x,x)
--R
---R (1) Ei(a x log(e))
+--R (1) Ei(a x)
--R Type: Union(Expression
Integer,...)
--E
@
@@ -127,15 +176,14 @@ $$
<<*>>=
)clear all
---S 6 of 16
-aa:=integrate(e^(a*x)/x^n,x)
---R
+--S 12 14:514 Axiom cannot compute this integral
+aa:=integrate(%e^(a*x)/x^n,x)
--R
---I x %I a
---R ++ e
---I (1) | ----- d%I
---R ++ n
---I %I
+--I x %I a
+--R ++ %e
+--I (1) | ------ d%I
+--R ++ n
+--I %I
--R Type: Union(Expression
Integer,...)
--E
@
@@ -148,16 +196,32 @@ $$
<<*>>=
)clear all
---S 7 of 16
-aa:=integrate(1/(p+q*e^(a*x)),x)
---R
+--S 13
+aa:=integrate(1/(p+q*%e^(a*x)),x)
--R
---R a x log(e)
---R - log(q %e + p) + a x log(e)
---R (1) --------------------------------------
---R a p log(e)
+--R a x
+--R - log(q %e + p) + a x
+--R (1) ------------------------
+--R a p
--R Type: Union(Expression
Integer,...)
--E
+
+--S 14
+bb:=x/p-1/(a*p)*log(p+q*%e^(a*x))
+--R
+--R a x
+--R - log(q %e + p) + a x
+--R (2) ------------------------
+--R a p
+--R Type: Expression
Integer
+--E
+
+--S 15 14:515 Schaums and Axiom agree
+cc:=aa-bb
+--R
+--R (3) 0
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.516~~~~~$\displaystyle
@@ -168,16 +232,34 @@ $$
<<*>>=
)clear all
---S 8 of 16
-aa:=integrate(1/(p+q*e^(a*x)),x)
---R
+--S 16
+aa:=integrate(1/(p+q*%e^(a*x))^2,x)
--R
---R a x log(e)
---R - log(q %e + p) + a x log(e)
---R (1) --------------------------------------
---R a p log(e)
+--R a x a x a x
+--R (- q %e - p)log(q %e + p) + a q x %e + a p x + p
+--R (1) ---------------------------------------------------------
+--R 2 a x 3
+--R a p q %e + a p
--R Type: Union(Expression
Integer,...)
--E
+
+--S 17
+bb:=x/p^2+1/(a*p*(p+q*%e^(a*x)))-1/(a*p^2)*log(p+q*%e^(a*x))
+--R
+--R a x a x a x
+--R (- q %e - p)log(q %e + p) + a q x %e + a p x + p
+--R (2) ---------------------------------------------------------
+--R 2 a x 3
+--R a p q %e + a p
+--R Type: Expression
Integer
+--E
+
+--S 18 14:516 Schaums and Axiom agree
+cc:=aa-bb
+--R
+--R (3) 0
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.517~~~~~$\displaystyle
@@ -197,16 +279,114 @@ $$
<<*>>=
)clear all
---S 9 of 16
-aa:=integrate(1/(p*e^(a*x)+q*e^(a*x)),x)
---R
+--S 19
+aa:=integrate(1/(p*%e^(a*x)+q*%e^-(a*x)),x)
--R
---R 1
---R (1) - -----------------------------
---R a x log(e)
---R (a q + a p)log(e)%e
---R Type: Union(Expression
Integer,...)
+--R a x 2 +-----+ a x
+--R (p (%e ) - q)\|- p q + 2p q %e a x +---+
+--R log(-------------------------------------) %e \|p q
+--R a x 2 atan(-----------)
+--R p (%e ) + q q
+--R (1) [------------------------------------------,-----------------]
+--R +-----+ +---+
+--R 2a\|- p q a\|p q
+--R Type: Union(List Expression
Integer,...)
--E
+
+--S 20
+bb1:=1/(a*sqrt(p*q))*atan(sqrt(p/q)*%e^(a*x))
+--R
+--R +-+
+--R a x |p
+--R atan(%e |- )
+--R \|q
+--R (2) ---------------
+--R +---+
+--R a\|p q
+--R Type: Expression
Integer
+--E
+
+--S 21
+bb2:=1/(2*a*sqrt(-p*q))*log((%e^(a*x)-sqrt(-q/p))/(%e^(a*x)+sqrt(-q/p)))
+--R
+--R +---+
+--R | q a x
+--R - |- - + %e
+--R \| p
+--R log(----------------)
+--R +---+
+--R | q a x
+--R |- - + %e
+--R \| p
+--R (3) ---------------------
+--R +-----+
+--R 2a\|- p q
+--R Type: Expression
Integer
+--E
+
+--S 22
+cc1:=aa.1-bb1
+--R
+--R (4)
+--R a x 2 +-----+ a x
+-+
+--R +---+ (p (%e ) - q)\|- p q + 2p q %e +-----+ a x |p
+--R \|p q log(-------------------------------------) - 2\|- p q atan(%e
|- )
+--R a x 2 \|q
+--R p (%e ) + q
+--R
---------------------------------------------------------------------------
+--R +-----+ +---+
+--R 2a\|- p q \|p q
+--R Type: Expression
Integer
+--E
+
+--S 23
+cc2:=aa.2-bb1
+--R
+--R a x +---+ +-+
+--R %e \|p q a x |p
+--R atan(-----------) - atan(%e |- )
+--R q \|q
+--R (5) -----------------------------------
+--R +---+
+--R a\|p q
+--R Type: Expression
Integer
+--E
+
+--S 24
+cc3:=aa.1-bb2
+--R
+--R +---+
+--R | q a x
+--R a x 2 +-----+ a x - |- - + %e
+--R (p (%e ) - q)\|- p q + 2p q %e \| p
+--R log(-------------------------------------) - log(----------------)
+--R a x 2 +---+
+--R p (%e ) + q | q a x
+--R |- - + %e
+--R \| p
+--R (6) ------------------------------------------------------------------
+--R +-----+
+--R 2a\|- p q
+--R Type: Expression
Integer
+--E
+
+--S 25 14:517 Axiom cannot simplify these expressions
+cc4:=aa.2-bb2
+--R
+--R +---+
+--R | q a x
+--R - |- - + %e a x +---+
+--R +---+ \| p +-----+ %e \|p q
+--R - \|p q log(----------------) + 2\|- p q atan(-----------)
+--R +---+ q
+--R | q a x
+--R |- - + %e
+--R \| p
+--R (7) ----------------------------------------------------------
+--R +-----+ +---+
+--R 2a\|- p q \|p q
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.518~~~~~$\displaystyle
@@ -217,17 +397,34 @@ $$
<<*>>=
)clear all
---S 10 of 16
-aa:=integrate(e^(a*x)*sin(b*x),x)
---R
+--S 26
+aa:=integrate(%e^(a*x)*sin(b*x),x)
--R
---R a x log(e)
---R (a log(e)sin(b x) - b cos(b x))%e
---R (1) -------------------------------------------
---R 2 2 2
---R a log(e) + b
+--R a x a x
+--R a %e sin(b x) - b cos(b x)%e
+--R (1) ---------------------------------
+--R 2 2
+--R b + a
--R Type: Union(Expression
Integer,...)
--E
+
+--S 27
+bb:=((%e^(a*x))*(a*sin(b*x)-b*cos(b*x)))/(a^2+b^2)
+--R
+--R a x a x
+--R a %e sin(b x) - b cos(b x)%e
+--R (2) ---------------------------------
+--R 2 2
+--R b + a
+--R Type: Expression
Integer
+--E
+
+--S 28 14:518 Schaums and Axiom agree
+cc:=aa-bb
+--R
+--R (3) 0
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.519~~~~~$\displaystyle
@@ -238,17 +435,34 @@ $$
<<*>>=
)clear all
---S 11 of 16
-aa:=integrate(e^(a*x)*cos(b*x),x)
---R
+--S 29
+aa:=integrate(%e^(a*x)*cos(b*x),x)
--R
---R a x log(e)
---R (b sin(b x) + a cos(b x)log(e))%e
---R (1) -------------------------------------------
---R 2 2 2
---R a log(e) + b
+--R a x a x
+--R b %e sin(b x) + a cos(b x)%e
+--R (1) ---------------------------------
+--R 2 2
+--R b + a
--R Type: Union(Expression
Integer,...)
--E
+
+--S 30
+bb:=((%e^(a*x))*(a*cos(b*x)+b*sin(b*x)))/(a^2+b^2)
+--R
+--R a x a x
+--R b %e sin(b x) + a cos(b x)%e
+--R (2) ---------------------------------
+--R 2 2
+--R b + a
+--R Type: Expression
Integer
+--E
+
+--S 31 14:519 Schaums and Axiom agree
+cc:=aa-bb
+--R
+--R (3) 0
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.520~~~~~$\displaystyle
@@ -260,24 +474,36 @@ $$
<<*>>=
)clear all
---S 12 of 16
-aa:=integrate(x*e^(a*x)*sin(b*x),x)
---R
+--S 32
+aa:=integrate(x*%e^(a*x)*sin(b*x),x)
--R
--R (1)
---R 3 3 2 2 2 2
---R (a x log(e) - a log(e) + a b x log(e) + b )sin(b x)
---R +
---R 2 2 3
---R - a b x cos(b x)log(e) + 2a b cos(b x)log(e) - b x cos(b x)
---R *
---R a x log(e)
---R %e
---R /
---R 4 4 2 2 2 4
---R a log(e) + 2a b log(e) + b
+--R 2 3 2 2 a x 3 2
a x
+--R ((a b + a )x + b - a )%e sin(b x) + ((- b - a b)x + 2a b)cos(b x)%e
+--R
---------------------------------------------------------------------------
+--R 4 2 2 4
+--R b + 2a b + a
--R Type: Union(Expression
Integer,...)
--E
+
+--S 33
+bb:=(x*%e^(a*x)*(a*sin(b*x)-b*cos(b*x)))/(a^2+b^2)-(%e^(a*x)*((a^2-b^2)*sin(b*x)-2*a*b*cos(b*x)))/(a^2+b^2)^2
+--R
+--R (2)
+--R 2 3 2 2 a x 3 2
a x
+--R ((a b + a )x + b - a )%e sin(b x) + ((- b - a b)x + 2a b)cos(b x)%e
+--R
---------------------------------------------------------------------------
+--R 4 2 2 4
+--R b + 2a b + a
+--R Type: Expression
Integer
+--E
+
+--S 34 14:520 Schaums and Axiom agree
+cc:=aa-bb
+--R
+--R (3) 0
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.521~~~~~$\displaystyle
@@ -289,24 +515,36 @@ $$
<<*>>=
)clear all
---S 13 of 16
-aa:=integrate(x*e^(a*x)*cos(b*x),x)
---R
+--S 35
+aa:=integrate(x*%e^(a*x)*cos(b*x),x)
--R
--R (1)
---R 2 2 3 3 3
---R (a b x log(e) - 2a b log(e) + b x)sin(b x) + a x cos(b x)log(e)
---R +
---R 2 2 2 2
---R - a cos(b x)log(e) + a b x cos(b x)log(e) + b cos(b x)
---R *
---R a x log(e)
---R %e
---R /
---R 4 4 2 2 2 4
---R a log(e) + 2a b log(e) + b
+--R 3 2 a x 2 3 2 2 a x
+--R ((b + a b)x - 2a b)%e sin(b x) + ((a b + a )x + b - a )cos(b x)%e
+--R -------------------------------------------------------------------------
+--R 4 2 2 4
+--R b + 2a b + a
--R Type: Union(Expression
Integer,...)
--E
+
+--S 36
+bb:=(x*%e^(a*x)*(a*cos(b*x)+b*sin(b*x)))/(a^2+b^2)-(%e^(a*x)*((a^2-b^2)*cos(b*x)+2*a*b*sin(b*x)))/(a^2+b^2)^2
+--R
+--R (2)
+--R 3 2 a x 2 3 2 2 a x
+--R ((b + a b)x - 2a b)%e sin(b x) + ((a b + a )x + b - a )cos(b x)%e
+--R -------------------------------------------------------------------------
+--R 4 2 2 4
+--R b + 2a b + a
+--R Type: Expression
Integer
+--E
+
+--S 37 14:521 Schaums and Axiom agree
+cc:=aa-bb
+--R
+--R (3) 0
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.522~~~~~$\displaystyle
@@ -317,14 +555,13 @@ $$
<<*>>=
)clear all
---S 14 of 16
-aa:=integrate(e^(a*x)*log(x),x)
---R
+--S 38 14:522 Schaums and Axiom agree by definition
+aa:=integrate(%e^(a*x)*log(x),x)
--R
---R x
---I ++ %I a
---I (1) | log(%I)e d%I
---R ++
+--R a x
+--R %e log(x) - Ei(a x)
+--R (1) ---------------------
+--R a
--R Type: Union(Expression
Integer,...)
--E
@
@@ -338,13 +575,12 @@ $$
<<*>>=
)clear all
---S 15 of 16
-aa:=integrate(e^(a*x)*sin(b*x)^n,x)
---R
+--S 39 14:523 Axiom cannot compute this integral
+aa:=integrate(%e^(a*x)*sin(b*x)^n,x)
--R
--R x
---I ++ %I a n
---I (1) | e sin(%I b) d%I
+--I ++ %I a n
+--I (1) | %e sin(%I b) d%I
--R ++
--R Type: Union(Expression
Integer,...)
--E
@@ -359,13 +595,12 @@ $$
<<*>>=
)clear all
---S 16 of 16
-aa:=integrate(e^(a*x)*cos(b*x)^n,x)
---R
+--S 40 14:524 Axiom cannot compute this integral
+aa:=integrate(%e^(a*x)*cos(b*x)^n,x)
--R
--R x
---I ++ %I a n
---I (1) | e cos(%I b) d%I
+--I ++ %I a n
+--I (1) | %e cos(%I b) d%I
--R ++
--R Type: Union(Expression
Integer,...)
--E
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- [Axiom-developer] 20080425.01.tpd.patch (CATS Schaums-Axiom equivalence testing (24-25)),
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