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[Axiom-developer] 20080416.01.tpd.patch (CATS Schaums-Axiom equivalence
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From: |
daly |
|
Subject: |
[Axiom-developer] 20080416.01.tpd.patch (CATS Schaums-Axiom equivalence testing (2-7)) |
|
Date: |
Thu, 17 Apr 2008 01:03:47 -0500 |
Item 14:150 Schaums and Axiom DISAGREE BY A NON-CONSTANT
in schaum7.input.pamphlet is particularly interesting because
it appears that the derivative of Axiom's answer is the original
integrand but the derivative of Schaum's answer is not, implying
that Schaum's has a mistake. This will be verified using other
systems later.
Axiom is weak in handling certain simplifications. Future work is
planned to correct this.
Richard Fateman has given me permission to use his TILU pattern
integration database in Axiom. This should give us much broader
integration results. TILU has not been tested against Schaums
but this testing will occur during the merge.
schaum2.input.pamphet
14:84 Schaums and Axiom agree
14:85 Schaums and Axiom agree
14:86 Schaums and Axiom agree
14:87a Schaums and Axiom differ by a constant
14:87b Schaums and Axiom differ by a constant
14:88 Schaums and Axiom differ by a constant
14:89 Schaums and Axiom differ by a constant
14:90 Axiom cannot simplify this expression
14:91 Axiom cannot simplify this expression
14:92 Axiom cannot simplify this expression
14:93 Schaums and Axiom agree
14:94 Axiom cannot do this integral
14:95 Axiom cannot do this integral
14:96 Axiom cannot do this integral
14:97 Axiom cannot do this integral
14:98 Axiom cannot do this integral
14:99 Axiom cannot simplify this expression
14:100 Axiom cannot simplify this expression
14:101 Axiom cannot simplify this expression
14:102 Axiom cannot do this integral
14:103 Axiom cannot do this integral
14:104 Axiom cannot do this integral
schaum3.input.pamphlet
14:105 Schaums and Axiom agree
14:106 Schaums and Axiom agree
14:107 Schaums and Axiom agree
14:108 Schaums and Axiom agree
14:109 Schaums and Axiom agree
14:110 Axiom cannot do this integral
14:111 Schaums and Axiom agree
14:112 Axiom cannot do this integral
schaum4.input.pamphlet
14:113 Schaums and Axiom agree
14:114 Axiom cannot simplify these answers
14:115 Axiom cannot simplify these answers
14:116 Axiom cannot compute this integral
14:117 Axiom cannot compute this integral
14:118 Axiom cannot compute this integral
14:119 Axiom cannot compute this integral
schaum5.input.pamphlet
14:120 Axiom cannot simplify these answers
14:121 Axiom cannot simplify this answer
14:122 Axiom cannot simplify this answer
14:123 Axiom cannot simplify these results
14:124 Axiom cannot simplify this result
schaum6.input.pamphlet
14:125 Schaums and Axiom agree
14:126 Schaums and Axiom agree
14:127 Schaums and Axiom agree
14:128 Schaums and Axiom agree
14:129 Schaums and Axiom agree
14:130 Schaums and Axiom agree
14:131 Schaums and Axiom agree
14:132 Schaums and Axiom agree
14:133 Schaums and Axiom agree
14:134 Schaums and Axiom differ by a constant
14:135 Schaums and Axiom agree
14:136 Schaums and Axiom agree
14:137 Schaums and Axiom agree
14:138 Schaums and Axiom agree
14:139 Axiom cannot do this integral
14:140 Schaums and Axiom cannot simplify this expression
14:141 Axiom cannot do this integral
14:142 Axiom cannot do this integral
14:143 Axiom cannot do this integral
schaum7.input.pamphlet
14:144 Schaums and Axiom agree
14:145 Schaums and Axiom agree
14:146 Schaums and Axiom agree
14:147 Schaums and Axiom agree
14:148 Schaums and Axiom agree
14:149 Schaums and Axiom agree
14:150 Schaums and Axiom DISAGREE BY A NON-CONSTANT
14:151 Schaums and Axiom agree
14:152 Schaums and Axiom agree
14:153 Schaums and Axiom agree
14:154 Schaums and Axiom agree
14:155 Schaums and Axiom agree
14:156 Schaums and Axiom agree
14:157 Schaums and Axiom agree
14:158 Axiom cannot do this integral
14:159 Axiom cannot simplify this expression
14:160 Axiom cannot compute this integral
14:161 Axiom cannot compute this integral
14:162 Axiom cannot compute this integral
======================================================================
diff --git a/changelog b/changelog
index 346e9f2..eda7bfd 100644
--- a/changelog
+++ b/changelog
@@ -1,3 +1,9 @@
+20080416 tpd src/input/schaum7.input show Schaums-Axiom equivalence
+20080416 tpd src/input/schaum6.input show Schaums-Axiom equivalence
+20080416 tpd src/input/schaum5.input show Schaums-Axiom equivalence
+20080416 tpd src/input/schaum4.input show Schaums-Axiom equivalence
+20080416 tpd src/input/schaum3.input show Schaums-Axiom equivalence
+20080416 tpd src/input/schaum2.input show Schaums-Axiom equivalence
20080415 tpd src/input/schaum1.input show Schaums-Axiom equivalence
20080414 tpd src/input/Makefile add integration regression testing
20080414 tpd src/input/schaum34.input integrals of csch(ax)
diff --git a/src/input/schaum2.input.pamphlet b/src/input/schaum2.input.pamphlet
index ba16925..184122e 100644
--- a/src/input/schaum2.input.pamphlet
+++ b/src/input/schaum2.input.pamphlet
@@ -7,15 +7,18 @@
\eject
\tableofcontents
\eject
-\section{\cite{1}:14.84~~~~~$\displaystyle\int{\frac{dx}{\sqrt{ax+b}}}$}
-$$\int{\frac{1}{\sqrt{ax+b}}}=\frac{2\sqrt{ax+b}}{a}$$
+\section{\cite{1}:14.84~~~~~$\displaystyle
+\int{\frac{dx}{\sqrt{ax+b}}}$}
+$$\int{\frac{1}{\sqrt{ax+b}}}=
+\frac{2\sqrt{ax+b}}{a}
+$$
<<*>>=
)spool schaum2.output
)set message test on
)set message auto off
)clear all
---S 1 of 92
+--S 1
aa:=integrate(1/sqrt(a*x+b),x)
--R
--R
@@ -25,9 +28,8 @@ aa:=integrate(1/sqrt(a*x+b),x)
--R a
--R Type: Union(Expression
Integer,...)
--E
-@
-<<*>>=
---S 2 of 92
+
+--S 2
bb:=(2*sqrt(a*x+b))/a
--R
--R
@@ -37,9 +39,8 @@ bb:=(2*sqrt(a*x+b))/a
--R a
--R Type: Expression
Integer
--E
-@
-<<*>>=
---S 3 of 92
+
+--S 3 14:84 Schaums and Axiom agree
cc:=aa-bb
--R
--R
@@ -48,12 +49,15 @@ cc:=aa-bb
--E
@
-\section{\cite{1}:14.85~~~~~$\displaystyle\int{\frac{x~dx}{\sqrt{ax+b}}}$}
-$$\int{\frac{x}{\sqrt{ax+b}}}=\frac{2(ax-2b)}{3a^2}\sqrt{ax+b}$$
+\section{\cite{1}:14.85~~~~~$\displaystyle
+\int{\frac{x~dx}{\sqrt{ax+b}}}$}
+$$\int{\frac{x}{\sqrt{ax+b}}}=
+\frac{2(ax-2b)}{3a^2}\sqrt{ax+b}
+$$
<<*>>=
)clear all
---S 4 of 92
+--S 4
aa:=integrate(x/sqrt(a*x+b),x)
--R
--R
@@ -64,9 +68,8 @@ aa:=integrate(x/sqrt(a*x+b),x)
--R 3a
--R Type: Union(Expression
Integer,...)
--E
-@
-<<*>>=
---S 5 of 92
+
+--S 5
bb:=(2*(a*x-2*b))/(3*a^2)*sqrt(a*x+b)
--R
--R
@@ -77,9 +80,8 @@ bb:=(2*(a*x-2*b))/(3*a^2)*sqrt(a*x+b)
--R 3a
--R Type: Expression
Integer
--E
-@
-<<*>>=
---S 6 of 92
+
+--S 6 14:85 Schaums and Axiom agree
cc:=aa-bb
--R
--R
@@ -88,13 +90,15 @@ cc:=aa-bb
--E
@
-\section{\cite{1}:14.86~~~~~$\displaystyle\int{\frac{x^2~dx}{\sqrt{ax+b}}}$}
+\section{\cite{1}:14.86~~~~~$\displaystyle
+\int{\frac{x^2~dx}{\sqrt{ax+b}}}$}
$$\int{\frac{x}{\sqrt{ax+b}}}=
-\frac{2(3a^2x^2-4abx+8b^2)}{15a^2}\sqrt{ax+b}$$
+\frac{2(3a^2x^2-4abx+8b^2)}{15a^2}\sqrt{ax+b}
+$$
<<*>>=
)clear all
---S 7 of 92
+--S 7
aa:=integrate(x^2/sqrt(a*x+b),x)
--R
--R
@@ -105,9 +109,8 @@ aa:=integrate(x^2/sqrt(a*x+b),x)
--R 15a
--R Type: Union(Expression
Integer,...)
--E
-@
-<<*>>=
---S 8 of 92
+
+--S 8
bb:=(2*(3*a^2*x^2-4*a*b*x+8*b^2))/(15*a^3)*sqrt(a*x+b)
--R
--R
@@ -118,9 +121,8 @@ bb:=(2*(3*a^2*x^2-4*a*b*x+8*b^2))/(15*a^3)*sqrt(a*x+b)
--R 15a
--R Type: Expression
Integer
--E
-@
-<<*>>=
---S 9 of 92
+
+--S 9 14:86 Schaums and Axiom agree
cc:=aa-bb
--R
--R
@@ -129,7 +131,8 @@ cc:=aa-bb
--E
@
-\section{\cite{1}:14.87~~~~~$\displaystyle\int{\frac{dx}{x\sqrt{ax+b}}}$}
+\section{\cite{1}:14.87~~~~~$\displaystyle
+\int{\frac{dx}{x\sqrt{ax+b}}}$}
$$\int{\frac{1}{x\sqrt{ax+b}}}=
\left\{
\begin{array}{l}
@@ -139,13 +142,14 @@ $$\int{\frac{1}{x\sqrt{ax+b}}}=
\displaystyle
\frac{2}{\sqrt{-b}}~\tan^{-1}\sqrt{\frac{ax+b}{-b}}
\end{array}
-\right.$$
+\right.
+$$
Note: the first answer assumes $b > 0$ and the second assumes $b < 0$.
<<*>>=
)clear all
---S 10 of 92
+--S 10
aa:=integrate(1/(x*sqrt(a*x+b)),x)
--R
--R
@@ -161,7 +165,7 @@ aa:=integrate(1/(x*sqrt(a*x+b)),x)
@
Cleary Spiegel's first answer assumes $b > 0$:
<<*>>=
---S 11 of 92
+--S 11
bb1:=1/sqrt(b)*log((sqrt(a*x+b)-sqrt(b))/(sqrt(a*x+b)+sqrt(b)))
--R
--R
@@ -178,7 +182,7 @@
bb1:=1/sqrt(b)*log((sqrt(a*x+b)-sqrt(b))/(sqrt(a*x+b)+sqrt(b)))
@
So we try the difference of the two results
<<*>>=
---S 12 of 92
+--S 12
cc11:=aa.1-bb1
--R
--R +-------+ +-+ +-------+ +-+
@@ -197,7 +201,7 @@ But the results don't simplify to 0. So we try some other
tricks.
Since both functions are of the form log(f(x))/sqrt(b) we extract
the f(x) from each. First we get the function from Axiom's first answer:
<<*>>=
---S 13 of 92
+--S 13
ff:=exp(aa.1*sqrt(b))
--R
--R +-------+ +-+
@@ -209,7 +213,7 @@ ff:=exp(aa.1*sqrt(b))
@
and we get the same form from Spiegel's answer
<<*>>=
---S 14 of 92
+--S 14
gg:=exp(bb1*sqrt(b))
--R
--R +-------+ +-+
@@ -226,7 +230,7 @@ denominator by $1 == (sqrt(a*x+b) - sqrt(b))/(sqrt(a*x+b) -
sqrt(b))$.
First we multiply the numerator by $(sqrt(a*x+b) - sqrt(b))$
<<*>>=
---S 15 of 92
+--S 15
gg1:=gg*(sqrt(a*x+b) - sqrt(b))
--R
--R +-+ +-------+
@@ -239,7 +243,7 @@ gg1:=gg*(sqrt(a*x+b) - sqrt(b))
@
Now we multiply the denominator by $(sqrt(a*x+b) - sqrt(b))$
<<*>>=
---S 16 of 92
+--S 16
gg2:=gg1/(sqrt(a*x+b) - sqrt(b))
--R
--R +-+ +-------+
@@ -251,7 +255,7 @@ gg2:=gg1/(sqrt(a*x+b) - sqrt(b))
@
and now we multiply by the integration constant $a*sqrt(b)$
<<*>>=
---S 17 of 92
+--S 17
gg3:=gg2*(a*sqrt(b))
--R
--R +-------+ +-+
@@ -263,7 +267,7 @@ gg3:=gg2*(a*sqrt(b))
@
and when we difference this with ff, the Axiom answer we get:
<<*>>=
---S 18 of 92
+--S 18 14:87a Schaums and Axiom differ by a constant
ff-gg3
--R
--R (9) 0
@@ -275,7 +279,7 @@ So the constant of integration difference is $a*sqrt(b)$
Now we look at the second equations. We difference Axiom's second answer
from Spiegel's answer:
<<*>>=
---S 19 of 92
+--S 19
t1:=aa.2-bb1
--R
--R +-------+ +-+ +---+ +-------+
@@ -292,7 +296,7 @@ t1:=aa.2-bb1
and again they do not simplify to zero. But we can show that both answers
differ by a constant because the derivative is zero:
<<*>>=
---S 20 of 92
+--S 20
D(t1,x)
--R
--R (11) 0
@@ -303,7 +307,7 @@ D(t1,x)
Rather than find the constant this time we will differentiate both
answers and compare them with the original equation.
<<*>>=
---S 21 of 92
+--S 21
target:=1/(x*sqrt(a*x+b))
--R
--R 1
@@ -315,7 +319,7 @@ target:=1/(x*sqrt(a*x+b))
@
and we select the second Axiom solution
<<*>>=
---S 22 of 92
+--S 22
aa2:=aa.2
--R
--R +---+ +-------+
@@ -330,7 +334,7 @@ aa2:=aa.2
@
take its derivative
<<*>>=
---S 23 of 92
+--S 23
ad2:=D(aa2,x)
--R
--R 1
@@ -343,7 +347,7 @@ ad2:=D(aa2,x)
When we take the difference of Axiom's input and the derivative of the
output we see:
<<*>>=
---S 24 of 92
+--S 24
ad2-target
--R
--R (15) 0
@@ -355,7 +359,7 @@ Thus the original equation and Axiom's derivative of the
integral are equal.
Now we do the same with Spiegel's answer. We take the derivative of his
answer.
<<*>>=
---S 25 of 92
+--S 25
ab1:=D(bb1,x)
--R
--R +-------+ +-+
@@ -368,7 +372,7 @@ ab1:=D(bb1,x)
@
and we difference it from the original equation
<<*>>=
---S 26 of 92
+--S 26 14:87b Schaums and Axiom differ by a constant
ab1-target
--R
--R (17) 0
@@ -380,13 +384,15 @@ Thus the original equation and Spiegel's derivative of
the integral are equal.
So we can conclude that both second answers are correct although they differ
by a constant of integration.
-\section{\cite{1}:14.88~~~~~$\displaystyle\int{\frac{dx}{x^2\sqrt{ax+b}}}$}
+\section{\cite{1}:14.88~~~~~$\displaystyle
+\int{\frac{dx}{x^2\sqrt{ax+b}}}$}
$$\int{\frac{1}{x^2\sqrt{ax+b}}}=
--\frac{\sqrt{ax+b}}{bx}-\frac{a}{2b}~\int{\frac{1}{x\sqrt{ax+b}}}$$
+-\frac{\sqrt{ax+b}}{bx}-\frac{a}{2b}~\int{\frac{1}{x\sqrt{ax+b}}}
+$$
<<*>>=
)clear all
---S 27 of 92
+--S 27
aa:=integrate(1/(x^2*sqrt(a*x+b)),x)
--R
--R
@@ -412,7 +418,7 @@ aa:=integrate(1/(x^2*sqrt(a*x+b)),x)
In order to write down the book answer we need to first take the
integral which has two results
<<*>>=
---S 28 of 92
+--S 28
dd:=integrate(1/(x*sqrt(a*x+b)),x)
--R
--R
@@ -429,7 +435,7 @@ dd:=integrate(1/(x*sqrt(a*x+b)),x)
and derive two results for the book answer. The first result assumes
$b > 0$
<<*>>=
---S 29 of 92
+--S 29
bb1:=-sqrt(a*x+b)/(b*x)-a/(2*b)*dd.1
--R
--R
@@ -445,7 +451,7 @@ bb1:=-sqrt(a*x+b)/(b*x)-a/(2*b)*dd.1
@
and the second result assumes $b < 0$.
<<*>>=
---S 30 of 92
+--S 30
bb2:=-sqrt(a*x+b)/(b*x)-a/(2*b)*dd.2
--R
--R
@@ -463,7 +469,7 @@ bb2:=-sqrt(a*x+b)/(b*x)-a/(2*b)*dd.2
So we compute the difference of Axiom's first result with Spiegel's
first result
<<*>>=
---S 31 of 92
+--S 31
cc11:=bb1-aa.1
--R
--R (5)
@@ -484,7 +490,7 @@ cc11:=bb1-aa.1
@
we compute its derivative
<<*>>=
---S 32 of 92
+--S 32
D(cc11,x)
--R
--R (6) 0
@@ -495,7 +501,7 @@ and we can see that the answers differ by a constant, the
constant of
integration. So Axiom's first answer should differentiate back to the target
equation.
<<*>>=
---S 33 of 92
+--S 33
target:=1/(x^2*sqrt(a*x+b))
--R
--R 1
@@ -507,7 +513,7 @@ target:=1/(x^2*sqrt(a*x+b))
@
We differentiate Axiom's first answer
<<*>>=
---S 34 of 92
+--S 34
ad1:=D(aa.1,x)
--R
--R +-+ +-------+ 2
@@ -520,7 +526,7 @@ ad1:=D(aa.1,x)
@
and subtract it from the target equation
<<*>>=
---S 35 of 92
+--S 35
ad1-target
--R
--R (9) 0
@@ -529,7 +535,7 @@ ad1-target
@
and now we do the same with first Spiegel's answer:
<<*>>=
---S 36 of 92
+--S 36
bd1:=D(bb1,x)
--R
--R +-+ +-------+ 2
@@ -542,7 +548,7 @@ bd1:=D(bb1,x)
@
and we subtract it from the target
<<*>>=
---S 37 of 92
+--S 37
bd1-target
--R
--R (11) 0
@@ -555,7 +561,7 @@ integrals differ by a constant.
Now we look at the second answers. We difference the answers and can
see immediately that they are equal.
<<*>>=
---S 38 of 92
+--S 38 14:88 Schaums and Axiom differ by a constant
cc22:=bb2-aa.2
--R
--R
@@ -564,13 +570,15 @@ cc22:=bb2-aa.2
--E
@
-\section{\cite{1}:14.89~~~~~$\displaystyle\int{\sqrt{ax+b}~dx}$}
+\section{\cite{1}:14.89~~~~~$\displaystyle
+\int{\sqrt{ax+b}~dx}$}
$$\int{\sqrt{ax+b}}=
-\frac{2\sqrt{(ax+b)^3}}{3a}$$
+\frac{2\sqrt{(ax+b)^3}}{3a}
+$$
<<*>>=
)clear all
---S 39 of 92
+--S 39
aa:=integrate(sqrt(a*x+b),x)
--R
--R
@@ -582,7 +590,7 @@ aa:=integrate(sqrt(a*x+b),x)
--E
@
<<*>>=
---S 40 of 92
+--S 40
bb:=(2*sqrt((a*x+b)^3))/(3*a)
--R
--R
@@ -595,7 +603,7 @@ bb:=(2*sqrt((a*x+b)^3))/(3*a)
--E
@
<<*>>=
---S 41 of 92
+--S 41
cc:=aa-bb
--R
--R +----------------------------+
@@ -608,7 +616,7 @@ cc:=aa-bb
@
Since this didn't simplify we could check each answer using the derivative
<<*>>=
---S 42 of 92
+--S 42
target:=sqrt(a*x+b)
--R
--R +-------+
@@ -618,7 +626,7 @@ target:=sqrt(a*x+b)
@
We take the derivative of Axiom's answer
<<*>>=
---S 43 of 92
+--S 43
t1:=D(aa,x)
--R
--R a x + b
@@ -630,7 +638,7 @@ t1:=D(aa,x)
@
And we subtract the target from the derivative of Axiom's answer
<<*>>=
---S 44 of 92
+--S 44
t1-target
--R
--R (6) 0
@@ -639,7 +647,7 @@ t1-target
@
So they are equal. Now we do the same with Spiegel's answer
<<*>>=
---S 45 of 92
+--S 45
t2:=D(bb,x)
--R
--R 2 2 2
@@ -653,7 +661,7 @@ t2:=D(bb,x)
@
The numerator is
<<*>>=
---S 46 of 92
+--S 46
nn:=(a*x+b)^2
--R
--R 2 2 2
@@ -662,7 +670,7 @@ nn:=(a*x+b)^2
--E
@
<<*>>=
---S 47 of 92
+--S 47
mm:=(a*x+b)^3
--R
--R 3 3 2 2 2 3
@@ -672,7 +680,7 @@ mm:=(a*x+b)^3
@
which expands to Spiegel's version.
<<*>>=
---S 48 of 92
+--S 48 14:89 Schaums and Axiom differ by a constant
result=nn/sqrt(mm)
--R
--R 2 2 2
@@ -686,13 +694,15 @@ result=nn/sqrt(mm)
@
and this reduces to $\sqrt{ax+b}$
-\section{\cite{1}:14.90~~~~~$\displaystyle\int{x\sqrt{ax+b}~dx}$}
+\section{\cite{1}:14.90~~~~~$\displaystyle
+\int{x\sqrt{ax+b}~dx}$}
$$\int{x\sqrt{ax+b}}=
-\frac{2(3ax-2b)}{15a^2}~\sqrt{(ax+b)^3}$$
+\frac{2(3ax-2b)}{15a^2}~\sqrt{(ax+b)^3}
+$$
<<*>>=
)clear all
---S 49 of 92
+--S 49
aa:=integrate(x*sqrt(a*x+b),x)
--R
--R
@@ -703,9 +713,8 @@ aa:=integrate(x*sqrt(a*x+b),x)
--R 15a
--R Type: Union(Expression
Integer,...)
--E
-@
-<<*>>=
---S 50 of 92
+
+--S 50
bb:=(2*(3*a*x-2*b))/(15*a^2)*sqrt((a*x+b)^3)
--R
--R
@@ -717,9 +726,8 @@ bb:=(2*(3*a*x-2*b))/(15*a^2)*sqrt((a*x+b)^3)
--R 15a
--R Type: Expression
Integer
--E
-@
-<<*>>=
---S 51 of 92
+
+--S 51
cc:=aa-bb
--R
--R (3)
@@ -737,15 +745,14 @@ cc:=aa-bb
@
If we had the terms
<<*>>=
---S 52 of 92
+--S 52
t1:=(3*a*x-2*b)
--R
--R (4) 3a x - 2b
--R Type: Polynomial
Integer
--E
-@
-<<*>>=
---S 53 of 92
+
+--S 53
t2:=(a*x+b)
--R
--R (5) a x + b
@@ -754,7 +761,7 @@ t2:=(a*x+b)
@
We can construct the Axiom result
<<*>>=
---S 54 of 92
+--S 54
2*t1*t2*sqrt(t2)/(15*a^2)
--R
--R 2 2 2 +-------+
@@ -767,7 +774,7 @@ We can construct the Axiom result
@
and we can construct the Spiegel result
<<*>>=
---S 55 of 92
+--S 55
2*t1*sqrt(t2^3)/(15*a^2)
--R
--R +----------------------------+
@@ -781,7 +788,7 @@ and we can construct the Spiegel result
@
the difference of these two depends on
<<*>>=
---S 56 of 92
+--S 56 14:90 Axiom cannot simplify this expression
t2*sqrt(t2)-sqrt(t2^3)
--R
--R +----------------------------+
@@ -791,14 +798,16 @@ t2*sqrt(t2)-sqrt(t2^3)
--E
@
-\section{\cite{1}:14.91~~~~~$\displaystyle\int{x^2\sqrt{ax+b}~dx}$}
+\section{\cite{1}:14.91~~~~~$\displaystyle
+\int{x^2\sqrt{ax+b}~dx}$}
$$\int{x^2\sqrt{ax+b}}=
-\frac{2(15a^2x^2-12abx+8b^2)}{105a^2}~\sqrt{(a+bx)^3}$$
+\frac{2(15a^2x^2-12abx+8b^2)}{105a^2}~\sqrt{(a+bx)^3}
+$$
Note: the sqrt term is almost certainly $\sqrt{(ax+b)}$
<<*>>=
)clear all
---S 57 of 92
+--S 57
aa:=integrate(x^2*sqrt(a*x+b),x)
--R
--R
@@ -809,9 +818,8 @@ aa:=integrate(x^2*sqrt(a*x+b),x)
--R 105a
--R Type: Union(Expression
Integer,...)
--E
-@
-<<*>>=
---S 58 of 92
+
+--S 58
bb:=(2*(15*a^2*x^2-12*a*b*x+8*b^2))/(105*a^2)*sqrt((a*x+b)^3)
--R
--R
@@ -823,9 +831,8 @@
bb:=(2*(15*a^2*x^2-12*a*b*x+8*b^2))/(105*a^2)*sqrt((a*x+b)^3)
--R 105a
--R Type: Expression
Integer
--E
-@
-<<*>>=
---S 59 of 92
+
+--S 59 14:91 Axiom cannot simplify this expression
cc:=aa-bb
--R
--R
@@ -843,13 +850,15 @@ cc:=aa-bb
--E
@
-\section{\cite{1}:14.92~~~~~$\displaystyle\int{\frac{\sqrt{ax+b}}{x}~dx}$}
+\section{\cite{1}:14.92~~~~~$\displaystyle
+\int{\frac{\sqrt{ax+b}}{x}~dx}$}
$$\int{\frac{\sqrt{ax+b}}{x}}=
-2\sqrt{ax+b}+b~\int{\frac{1}{x\sqrt{ax+b}}}$$
+2\sqrt{ax+b}+b~\int{\frac{1}{x\sqrt{ax+b}}}
+$$
<<*>>=
)clear all
---S 60 of 92
+--S 60
aa:=integrate(sqrt(a*x+b)/x,x)
--R
--R
@@ -865,9 +874,8 @@ aa:=integrate(sqrt(a*x+b)/x,x)
--R \|- b
--R Type: Union(List Expression
Integer,...)
--E
-@
-<<*>>=
---S 61 of 92
+
+--S 61
dd:=integrate(1/(x*sqrt(a*x+b)),x)
--R
--R
@@ -880,9 +888,8 @@ dd:=integrate(1/(x*sqrt(a*x+b)),x)
--R \|b \|- b
--R Type: Union(List Expression
Integer,...)
--E
-@
-<<*>>=
---S 62 of 92
+
+--S 62
bb1:=2*sqrt(a*x+b)+b*dd.1
--R
--R
@@ -895,9 +902,8 @@ bb1:=2*sqrt(a*x+b)+b*dd.1
--R \|b
--R Type: Expression
Integer
--E
-@
-<<*>>=
---S 63 of 92
+
+--S 63
bb2:=2*sqrt(a*x+b)+b*dd.2
--R
--R
@@ -910,9 +916,8 @@ bb2:=2*sqrt(a*x+b)+b*dd.2
--R \|- b
--R Type: Expression
Integer
--E
-@
-<<*>>=
---S 64 of 92
+
+--S 64
cc11:=bb1-aa.1
--R
--R
@@ -926,9 +931,8 @@ cc11:=bb1-aa.1
--R \|b
--R Type: Expression
Integer
--E
-@
-<<*>>=
---S 65 of 92
+
+--S 65
cc12:=bb1-aa.2
--R
--R
@@ -942,9 +946,8 @@ cc12:=bb1-aa.2
--R \|b
--R Type: Expression
Integer
--E
-@
-<<*>>=
---S 66 of 92
+
+--S 66
cc21:=bb2-aa.1
--R
--R
@@ -958,9 +961,8 @@ cc21:=bb2-aa.1
--R \|- b
--R Type: Expression
Integer
--E
-@
-<<*>>=
---S 67 of 92
+
+--S 67 14:92 Axiom cannot simplify this expression
cc22:=bb2-aa.2
--R
--R
@@ -976,13 +978,15 @@ cc22:=bb2-aa.2
--E
@
-\section{\cite{1}:14.93~~~~~$\displaystyle\int{\frac{\sqrt{ax+b}}{x^2}~dx}$}
+\section{\cite{1}:14.93~~~~~$\displaystyle
+\int{\frac{\sqrt{ax+b}}{x^2}~dx}$}
$$\int{\frac{\sqrt{ax+b}}{x^2}}=
--\frac{\sqrt{ax+b}}{x}+\frac{a}{2}~\int{\frac{1}{x\sqrt{ax+b}}}$$
+-\frac{\sqrt{ax+b}}{x}+\frac{a}{2}~\int{\frac{1}{x\sqrt{ax+b}}}
+$$
<<*>>=
)clear all
---S 68 of 92
+--S 68
aa:=integrate(sqrt(a*x+b)/x^2,x)
--R
--R
@@ -1003,9 +1007,8 @@ aa:=integrate(sqrt(a*x+b)/x^2,x)
--R x\|- b
--R Type: Union(List Expression
Integer,...)
--E
-@
-<<*>>=
---S 69 of 92
+
+--S 69
dd:=integrate(1/(x*sqrt(a*x+b)),x)
--R
--R
@@ -1018,9 +1021,8 @@ dd:=integrate(1/(x*sqrt(a*x+b)),x)
--R \|b \|- b
--R Type: Union(List Expression
Integer,...)
--E
-@
-<<*>>=
---S 70 of 92
+
+--S 70
bb1:=-sqrt(a*x+b)/x+a/2*dd.1
--R
--R
@@ -1033,9 +1035,8 @@ bb1:=-sqrt(a*x+b)/x+a/2*dd.1
--R 2x\|b
--R Type: Expression
Integer
--E
-@
-<<*>>=
---S 71 of 92
+
+--S 71
bb2:=-sqrt(a*x+b)/x+a/2*dd.2
--R
--R
@@ -1048,18 +1049,16 @@ bb2:=-sqrt(a*x+b)/x+a/2*dd.2
--R x\|- b
--R Type: Expression
Integer
--E
-@
-<<*>>=
---S 72 of 92
+
+--S 72
cc11:=bb1-aa.1
--R
--R
--R (5) 0
--R Type: Expression
Integer
--E
-@
-<<*>>=
---S 73 of 92
+
+--S 73
cc21:=bb-aa.1
--R
--R
@@ -1073,9 +1072,8 @@ cc21:=bb-aa.1
--R 2x\|b
--R Type: Expression
Integer
--E
-@
-<<*>>=
---S 74 of 92
+
+--S 74
cc12:=bb1-aa.2
--R
--R
@@ -1089,9 +1087,8 @@ cc12:=bb1-aa.2
--R 2\|- b \|b
--R Type: Expression
Integer
--E
-@
-<<*>>=
---S 75 of 92
+
+--S 75 14:93 Schaums and Axiom agree
cc22:=bb2-aa.2
--R
--R
@@ -1100,14 +1097,16 @@ cc22:=bb2-aa.2
--E
@
-\section{\cite{1}:14.94~~~~~$\displaystyle\int{\frac{x^m}{\sqrt{ax+b}}~dx}$}
+\section{\cite{1}:14.94~~~~~$\displaystyle
+\int{\frac{x^m}{\sqrt{ax+b}}~dx}$}
$$\int{\frac{x^m}{\sqrt{ax+b}}}=
\frac{2x^m\sqrt{ax+b}}{(2m+1)a}-\frac{2mb}{(2m+1)a}
-~\int{\frac{x^{m-1}}{\sqrt{ax+b}}}$$
+~\int{\frac{x^{m-1}}{\sqrt{ax+b}}}
+$$
<<*>>=
)clear all
---S 76 of 92
+--S 76 14:94 Axiom cannot do this integral
aa:=integrate(x^m/sqrt(a*x+b),x)
--R
--R
@@ -1120,14 +1119,16 @@ aa:=integrate(x^m/sqrt(a*x+b),x)
--E
@
-\section{\cite{1}:14.95~~~~~$\displaystyle\int{\frac{dx}{x^m\sqrt{ax+b}}}$}
+\section{\cite{1}:14.95~~~~~$\displaystyle
+\int{\frac{dx}{x^m\sqrt{ax+b}}}$}
$$\int{\frac{1}{x^m\sqrt{ax+b}}}=
-\frac{\sqrt{ax+b}}{(m-1)bx^{m-1}}-\frac{(2m-3)a}{(2m-2)b}
-~\int{\frac{1}{x^{m-1}\sqrt{ax+b}}}$$
+~\int{\frac{1}{x^{m-1}\sqrt{ax+b}}}
+$$
<<*>>=
)clear all
---S 77 of 92
+--S 77 14:95 Axiom cannot do this integral
aa:=integrate(1/(x^m*sqrt(a*x+b)),x)
--R
--R
@@ -1140,14 +1141,16 @@ aa:=integrate(1/(x^m*sqrt(a*x+b)),x)
--E
@
-\section{\cite{1}:14.96~~~~~$\displaystyle\int{x^m\sqrt{ax+b}~dx}$}
+\section{\cite{1}:14.96~~~~~$\displaystyle
+\int{x^m\sqrt{ax+b}~dx}$}
$$\int{x^m\sqrt{ax+b}}=
\frac{2x^m}{(2m+3)a}(ax+b)^{3/2}
--\frac{2mb}{(2m+3)a}~\int{x^{m-1}\sqrt{ax+b}}$$
+-\frac{2mb}{(2m+3)a}~\int{x^{m-1}\sqrt{ax+b}}
+$$
<<*>>=
)clear all
---S 78 of 92
+--S 78 14:96 Axiom cannot do this integral
aa:=integrate(x^m*sqrt(a*x+b),x)
--R
--R
@@ -1159,14 +1162,16 @@ aa:=integrate(x^m*sqrt(a*x+b),x)
--E
@
-\section{\cite{1}:14.97~~~~~$\displaystyle\int{\frac{\sqrt{ax+b}}{x^m}~dx}$}
+\section{\cite{1}:14.97~~~~~$\displaystyle
+\int{\frac{\sqrt{ax+b}}{x^m}~dx}$}
$$\int{\frac{\sqrt{ax+b}}{x^m}}=
-\frac{\sqrt{ax+b}}{(m-1)x^{m-1}}
-+\frac{a}{2(m-1)}~\int{\frac{1}{x^{m-1}\sqrt{ax+b}}}$$
++\frac{a}{2(m-1)}~\int{\frac{1}{x^{m-1}\sqrt{ax+b}}}
+$$
<<*>>=
)clear all
---S 79 of 92
+--S 79 14:97 Axiom cannot do this integral
aa:=integrate(sqrt(a*x+b)/x^m,x)
--R
--R
@@ -1179,15 +1184,17 @@ aa:=integrate(sqrt(a*x+b)/x^m,x)
--E
@
-\section{\cite{1}:14.98~~~~~$\displaystyle\int{\frac{\sqrt{ax+b}}{x^m}~dx}$}
+\section{\cite{1}:14.98~~~~~$\displaystyle
+\int{\frac{\sqrt{ax+b}}{x^m}~dx}$}
$$\int{\frac{\sqrt{ax+b}}{x^m}}=
\frac{-(ax+b)^{3/2}}{(m-1)bx^{m-1}}
--\frac{(2m-5)a}{(2m-2)b}~\int{\frac{\sqrt{ax+b}}{x^{m-1}}}$$
+-\frac{(2m-5)a}{(2m-2)b}~\int{\frac{\sqrt{ax+b}}{x^{m-1}}}
+$$
Note: 14.98 is the same as 14.97
<<*>>=
)clear all
---S 80 of 92
+--S 80 14:98 Axiom cannot do this integral
aa:=integrate(sqrt(a*x+b)/x^m,x)
--R
--R
@@ -1200,13 +1207,15 @@ aa:=integrate(sqrt(a*x+b)/x^m,x)
--E
@
-\section{\cite{1}:14.99~~~~~$\displaystyle\int{(ax+b)^{m/2}~dx}$}
+\section{\cite{1}:14.99~~~~~$\displaystyle
+\int{(ax+b)^{m/2}~dx}$}
$$\int{(ax+b)^{m/2}}=
-\frac{2(ax+b)^{(m+2)/2}}{a(m+2)}$$
+\frac{2(ax+b)^{(m+2)/2}}{a(m+2)}
+$$
<<*>>=
)clear all
---S 81 of 92
+--S 81
aa:=integrate((a*x+b)^(m/2),x)
--R
--R
@@ -1218,9 +1227,8 @@ aa:=integrate((a*x+b)^(m/2),x)
--R a m + 2a
--R Type: Union(Expression
Integer,...)
--E
-@
-<<*>>=
---S 82 of 92
+
+--S 82
bb:=(2*(a*x+b)^((m+2)/2))/(a*(m+2))
--R
--R
@@ -1232,9 +1240,8 @@ bb:=(2*(a*x+b)^((m+2)/2))/(a*(m+2))
--R a m + 2a
--R Type: Expression
Integer
--E
-@
-<<*>>=
---S 83 of 92
+
+--S 83 14:99 Axiom cannot simplify this expression
cc:=aa-bb
--R
--R
@@ -1248,14 +1255,16 @@ cc:=aa-bb
--E
@
-\section{\cite{1}:14.100~~~~~$\displaystyle\int{x(ax+b)^{m/2}~dx}$}
+\section{\cite{1}:14.100~~~~~$\displaystyle
+\int{x(ax+b)^{m/2}~dx}$}
$$\int{x(ax+b)^{m/2}}=
\frac{2(ax+b)^{(m+4)/2}}{a^2(m+4)}
--\frac{2b(ax+b)^{(m+2)/2}}{a^2(m+2)}$$
+-\frac{2b(ax+b)^{(m+2)/2}}{a^2(m+2)}
+$$
<<*>>=
)clear all
---S 84 of 92
+--S 84
aa:=integrate(x*(a*x+b)^(m/2),x)
--R
--R
@@ -1268,9 +1277,8 @@ aa:=integrate(x*(a*x+b)^(m/2),x)
--R a m + 6a m + 8a
--R Type: Union(Expression
Integer,...)
--E
-@
-<<*>>=
---S 85 of 92
+
+--S 85
bb:=(2*(a*x+b)^((m+4)/2))/(a^2*(m+4))-(2*b*(a*x+b)^((m+2)/2))/(a^2*(m+2))
--R
--R
@@ -1283,9 +1291,8 @@
bb:=(2*(a*x+b)^((m+4)/2))/(a^2*(m+4))-(2*b*(a*x+b)^((m+2)/2))/(a^2*(m+2))
--R a m + 6a m + 8a
--R Type: Expression
Integer
--E
-@
-<<*>>=
---S 86 of 92
+
+--S 86 14:100 Axiom cannot simplify this expression
cc:=aa-bb
--R
--R
@@ -1306,15 +1313,17 @@ cc:=aa-bb
--E
@
-\section{\cite{1}:14.101~~~~~$\displaystyle\int{x^2(ax+b)^{m/2}~dx}$}
+\section{\cite{1}:14.101~~~~~$\displaystyle
+\int{x^2(ax+b)^{m/2}~dx}$}
$$\int{x^2(ax+b)^{m/2}}=
\frac{2(ax+b)^{(m+6)/2}}{a^3(m+6)}
-\frac{4b(ax+b)^{(m+4)/2}}{a^3(m+4)}
-+\frac{2b^2(ax+b)^{(m+2)/2}}{a^3(m+2)}$$
++\frac{2b^2(ax+b)^{(m+2)/2}}{a^3(m+2)}
+$$
<<*>>=
)clear all
---S 87 of 92
+--S 87
aa:=integrate(x^2*(a*x+b)^(m/2),x)
--R
--R
@@ -1331,9 +1340,8 @@ aa:=integrate(x^2*(a*x+b)^(m/2),x)
--R a m + 12a m + 44a m + 48a
--R Type: Union(Expression
Integer,...)
--E
-@
-<<*>>=
---S 88 of 92
+
+--S 88
bb:=(2*(a*x+b)^((m+6)/2))/(a^3*(m+6))-_
(4*b*(a*x+b)^((m+4)/2))/(a^3*(m+4))+_
(2*b^2*(a*x+b)^((m+2)/2))/(a^3*(m+2))
@@ -1354,9 +1362,8 @@ bb:=(2*(a*x+b)^((m+6)/2))/(a^3*(m+6))-_
--R a m + 12a m + 44a m + 48a
--R Type: Expression
Integer
--E
-@
-<<*>>=
---S 89 of 92
+
+--S 89 14:101 Axiom cannot simplify this expression
cc:=aa-bb
--R
--R
@@ -1385,14 +1392,16 @@ cc:=aa-bb
--E
@
-\section{\cite{1}:14.102~~~~~$\displaystyle\int{\frac{(ax+b)^{m/2}}{x}~dx}$}
+\section{\cite{1}:14.102~~~~~$\displaystyle
+\int{\frac{(ax+b)^{m/2}}{x}~dx}$}
$$\int{\frac{(ax+b)^{m/2}}{x}}=
\frac{2(ax+b)^{m/2}}{m}
-+b~\int{\frac{(ax+b)^{(m-2)/2}}{x}}$$
++b~\int{\frac{(ax+b)^{(m-2)/2}}{x}}
+$$
<<*>>=
)clear all
---S 90 of 92
+--S 90 14:102 Axiom cannot do this integral
aa:=integrate((a*x+b)^(m/2)/x,x)
--R
--R
@@ -1409,11 +1418,12 @@ aa:=integrate((a*x+b)^(m/2)/x,x)
\int{\frac{(ax+b)^{m/2}}{x^2}~dx}$}
$$\int{\frac{(ax+b)^{m/2}}{x^2}}=
-\frac{(ax+b)^{(m+2)/2}}{bx}
-+\frac{ma}{2b}~\int{\frac{(ax+b)^{m/2}}{x}}$$
++\frac{ma}{2b}~\int{\frac{(ax+b)^{m/2}}{x}}
+$$
<<*>>=
)clear all
---S 91 of 92
+--S 91 14:103 Axiom cannot do this integral
aa:=integrate((a*x+b)^(m/2)/x^2,x)
--R
--R
@@ -1431,11 +1441,12 @@ aa:=integrate((a*x+b)^(m/2)/x^2,x)
\int{\frac{dx}{x(ax+b)^{m/2}}}$}
$$\int{\frac{1}{x(ax+b)^{m/2}}}=
\frac{2}{(m-2)b(ax+b)^{(m-2)/2}}
-+\frac{1}{b}~\int{\frac{1}{x(ax+b)^{(m-2)/2}}}$$
++\frac{1}{b}~\int{\frac{1}{x(ax+b)^{(m-2)/2}}}
+$$
<<*>>=
)clear all
---S 92 of 92
+--S 92 14:104 Axiom cannot do this integral
aa:=integrate(1/(x*(a*x+b)^(m/2)),x)
--R
--R
@@ -1448,9 +1459,7 @@ aa:=integrate(1/(x*(a*x+b)^(m/2)),x)
--I %L (b + %L a)
--R Type: Union(Expression
Integer,...)
--E
-@
-<<*>>=
)spool
)lisp (bye)
@
diff --git a/src/input/schaum3.input.pamphlet b/src/input/schaum3.input.pamphlet
index e1e030d..bed98c8 100644
--- a/src/input/schaum3.input.pamphlet
+++ b/src/input/schaum3.input.pamphlet
@@ -16,7 +16,7 @@ $$\int{\frac{1}{(ax+b)(px+q)}}=
)set message auto off
)clear all
---S 1 of 11
+--S 1
aa:=integrate(1/((a*x+b)*(p*x+q)),x)
--R
--R
@@ -25,9 +25,8 @@ aa:=integrate(1/((a*x+b)*(p*x+q)),x)
--R a q - b p
--R Type: Union(Expression
Integer,...)
--E
-@
-<<*>>=
---S 2 of 11
+
+--S 2
bb:=1/(b*p-a*q)*log((p*x+q)/(a*x+b))
--R
--R
@@ -38,9 +37,8 @@ bb:=1/(b*p-a*q)*log((p*x+q)/(a*x+b))
--R a q - b p
--R Type: Expression
Integer
--E
-@
-<<*>>=
---S 3 of 11
+
+--S 3
cc:=aa-bb
--R
--R
@@ -51,6 +49,41 @@ cc:=aa-bb
--R a q - b p
--R Type: Expression
Integer
--E
+
+--S 4
+logdiv:=rule(log(a)-log(b) == log(a/b))
+--R
+--R a
+--I (4) - log(b) + log(a) + %I == log(-) + %I
+--R b
+--R Type: RewriteRule(Integer,Integer,Expression
Integer)
+--E
+
+--S 5
+dd:=logdiv cc
+--R
+--R 1
+--R log(a x + b) + log(-------)
+--R a x + b
+--R (5) ---------------------------
+--R a q - b p
+--R Type: Expression
Integer
+--E
+
+--S 6
+logmul:=rule(log(a)+log(b) == log(a*b))
+--R
+--I (6) log(b) + log(a) + %J == log(a b) + %J
+--R Type: RewriteRule(Integer,Integer,Expression
Integer)
+--E
+
+--S 7 14:105 Schaums and Axiom agree
+ee:=logmul dd
+--R
+--R (7) 0
+--R Type: Expression
Integer
+--E
+
@
\section{\cite{1}:14.106~~~~~$\displaystyle\int{\frac{x~dx}{(ax+b)(px+q)}}$}
@@ -59,7 +92,7 @@ $$\int{\frac{x}{(ax+b)(px+q)}}=
<<*>>=
)clear all
---S 4 of 11
+--S 8
aa:=integrate(x/((a*x+b)*(p*x+q)),x)
--R
--R
@@ -69,9 +102,8 @@ aa:=integrate(x/((a*x+b)*(p*x+q)),x)
--R a p q - a b p
--R Type: Union(Expression
Integer,...)
--E
-@
-<<*>>=
---S 5 of 11
+
+--S 9
bb:=1/(b*p-a*q)*(b/a*log(a*x+b)-q/p*log(p*x+q))
--R
--R
@@ -81,9 +113,8 @@ bb:=1/(b*p-a*q)*(b/a*log(a*x+b)-q/p*log(p*x+q))
--R a p q - a b p
--R Type: Expression
Integer
--E
-@
-<<*>>=
---S 6 of 11
+
+--S 10 14:106 Schaums and Axiom agree
cc:=aa-bb
--R
--R
@@ -100,7 +131,7 @@ $$\int{\frac{1}{(ax+b)^2(px+q)}}=
<<*>>=
)clear all
---S 7 of 11
+--S 11
aa:=integrate(1/((a*x+b)^2*(p*x+q)),x)
--R
--R
@@ -110,9 +141,8 @@ aa:=integrate(1/((a*x+b)^2*(p*x+q)),x)
--R (a q - 2a b p q + a b p )x + a b q - 2a b p q + b p
--R Type: Union(Expression
Integer,...)
--E
-@
-<<*>>=
---S 8 of 11
+
+--S 12
bb:=1/(b*p-a*q)*(1/(a*x+b)+p/(b*p-a*q)*log((p*x+q)/(a*x+b)))
--R
--R
@@ -124,9 +154,8 @@ bb:=1/(b*p-a*q)*(1/(a*x+b)+p/(b*p-a*q)*log((p*x+q)/(a*x+b)))
--R (a q - 2a b p q + a b p )x + a b q - 2a b p q + b p
--R Type: Expression
Integer
--E
-@
-<<*>>=
---S 9 of 11
+
+--S 13
cc:=aa-bb
--R
--R
@@ -138,6 +167,23 @@ cc:=aa-bb
--R a q - 2a b p q + b p
--R Type: Expression
Integer
--E
+
+--S 14
+divlog:=rule(log(a/b) == log(a) - log(b))
+--R
+--R a
+--R (4) log(-) == - log(b) + log(a)
+--R b
+--R Type: RewriteRule(Integer,Integer,Expression
Integer)
+--E
+
+--S 15 14:107 Schaums and Axiom agree
+dd:=divlog cc
+--R
+--R (5) 0
+--R Type: Expression
Integer
+--E
+
@
\section{\cite{1}:14.108~~~~~$\displaystyle\int{\frac{x~dx}{(ax+b)^2(px+q)}}$}
@@ -149,7 +195,7 @@ $$\int{\frac{x}{(ax+b)^2(px+q)}}=
<<*>>=
)clear all
---S 10 of 11
+--S 16
aa:=integrate(x/((a*x+b)^2*(p*x+q)),x)
--R
--R
@@ -161,9 +207,8 @@ aa:=integrate(x/((a*x+b)^2*(p*x+q)),x)
--R (a q - 2a b p q + a b p )x + a b q - 2a b p q + a b p
--R Type: Union(Expression
Integer,...)
--E
-@
-<<*>>=
---S 11 of 11
+
+--S 17
bb:=1/(b*p-a*q)*(q/(b*p-a*q)*log((a*x+b)/(p*x+q))-b/(a*(a*x+b)))
--R
--R
@@ -175,8 +220,8 @@
bb:=1/(b*p-a*q)*(q/(b*p-a*q)*log((a*x+b)/(p*x+q))-b/(a*(a*x+b)))
--R (a q - 2a b p q + a b p )x + a b q - 2a b p q + a b p
--R Type: Expression
Integer
--E
-@
-<<*>>=
+
+--S 18
cc:=aa-bb
--R
--R
@@ -188,6 +233,22 @@ cc:=aa-bb
--R a q - 2a b p q + b p
--R Type: Expression
Integer
--E
+
+--S 19
+divlog:=rule(log(a/b) == log(a) - log(b))
+--R
+--R a
+--R (4) log(-) == - log(b) + log(a)
+--R b
+--R Type: RewriteRule(Integer,Integer,Expression
Integer)
+--E
+
+--S 20 14:108 Schaums and Axiom agree
+dd:=divlog cc
+--R
+--R (5) 0
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.109~~~~~$\displaystyle
@@ -198,7 +259,7 @@ $$\frac{b^2}{(bp-aq)a^2(ax+b)}+\frac{1}{(bp-aq)^2}
<<*>>=
)clear all
---S
+--S 21
aa:=integrate(x^2/((a*x+b)^2*(p*x+q)),x)
--R
--R
@@ -213,9 +274,8 @@ aa:=integrate(x^2/((a*x+b)^2*(p*x+q)),x)
--R (a p q - 2a b p q + a b p )x + a b p q - 2a b p q + a b p
--R Type: Union(Expression
Integer,...)
--E
-@
-<<*>>=
---S
+
+--S 22
bb:=b^2/((b*p-a*q)*a^2*(a*x+b))+_
1/(b*p-a*q)^2*(q^2/p*log(p*x+q)+((b*(b*p-2*a*q))/a^2)*log(a*x+b))
--R
@@ -231,9 +291,8 @@ bb:=b^2/((b*p-a*q)*a^2*(a*x+b))+_
--R (a p q - 2a b p q + a b p )x + a b p q - 2a b p q + a b p
--R Type: Expression
Integer
--E
-@
-<<*>>=
---S
+
+--S 23 14:109 Schaums and Axiom agree
cc:=aa-bb
--R
--R
@@ -250,7 +309,7 @@ a(m+n-2)~\int{\frac{1}{(ax+b)^m(px+q)^{n-1}}}\right\}$$
<<*>>=
)clear all
---S
+--S 24 14:110 Axiom cannot do this integral
aa:=integrate(1/((a*x+b)^m*(p*x+q)^n),x)
--R
--R
@@ -261,81 +320,14 @@ aa:=integrate(1/((a*x+b)^m*(p*x+q)^n),x)
--I (b + %L a) (q + %L p)
--R Type: Union(Expression
Integer,...)
--E
-@
-<<*>>=
---S
-dd:=integrate(1/((a*x+b)^m*(p*x+q)^(n-1)),x)
---R
---R
---R x
---R ++ 1
---I (2) | -------------------------- d%L
---R ++ m n - 1
---I (b + %L a) (q + %L p)
---R Type: Union(Expression
Integer,...)
---E
-@
-<<*>>=
---S
-bb:=-1/((n-1)*(b*p-a*q))*(1/((a*x+b)^(m-1)*(p*x+q)^(n-1))+a*(m+n-2)*dd)
---R
---R
---R (3)
---R m - 1 n - 1
---R (a n + a m - 2a)(a x + b) (p x + q)
---R *
---R x
---R ++ 1
---I | -------------------------- d%L
---R ++ m n - 1
---I (b + %L a) (q + %L p)
---R +
---R 1
---R /
---R m - 1 n - 1
---R ((a n - a)q + (- b n + b)p)(a x + b) (p x + q)
---R Type: Expression
Integer
---E
@
-<<*>>=
---S
-cc:=aa-bb
---R
---R
---R (4)
---R m - 1 n - 1
---R (- a n - a m + 2a)(a x + b) (p x + q)
---R *
---R x
---R ++ 1
---I | -------------------------- d%L
---R ++ m n - 1
---I (b + %L a) (q + %L p)
---R +
---R m - 1 n - 1
---R ((a n - a)q + (- b n + b)p)(a x + b) (p x + q)
---R *
---R x
---R ++ 1
---I | ---------------------- d%L
---R ++ m n
---I (b + %L a) (q + %L p)
---R +
---R - 1
---R /
---R m - 1 n - 1
---R ((a n - a)q + (- b n + b)p)(a x + b) (p x + q)
---R Type: Expression
Integer
---E
-@
-
\section{\cite{1}:14.111~~~~~$\displaystyle\int{\frac{ax+b}{px+q}~dx}$}
$$\int{\frac{ax+b}{px+q}}=\frac{ax}{p}+\frac{bp-aq}{p^2}~\ln(px+q)$$
<<*>>=
)clear all
---S
+--S 25
aa:=integrate((a*x+b)/(p*x+q),x)
--R
--R
@@ -345,9 +337,8 @@ aa:=integrate((a*x+b)/(p*x+q),x)
--R p
--R Type: Union(Expression
Integer,...)
--E
-@
-<<*>>=
---S
+
+--S 26
bb:=(a*x)/p+(b*p-a*q)/p^2*log(p*x+q)
--R
--R
@@ -357,9 +348,8 @@ bb:=(a*x)/p+(b*p-a*q)/p^2*log(p*x+q)
--R p
--R Type: Expression
Integer
--E
-@
-<<*>>=
---S
+
+--S 27 14:111 Schaums and Axiom agree
cc:=aa-bb
--R
--R
@@ -383,7 +373,7 @@ $$\int{\frac{(ax+b)^m}{(px+q)^n}}=\left\{
<<*>>=
)clear all
---S
+--S 28 14:112 Axiom cannot do this integral
aa:=integrate((a*x+b)^m/(p*x+q)^n,x)
--R
--R
@@ -393,7 +383,6 @@ aa:=integrate((a*x+b)^m/(p*x+q)^n,x)
--R ++ n
--I (q + %L p)
--R Type: Union(Expression
Integer,...)
---R
--E
<<*>>=
)spool
diff --git a/src/input/schaum4.input.pamphlet b/src/input/schaum4.input.pamphlet
index efd9edc..0edbbf9 100644
--- a/src/input/schaum4.input.pamphlet
+++ b/src/input/schaum4.input.pamphlet
@@ -16,7 +16,7 @@ $$\int{\frac{px+q}{\sqrt{ax+b}}}=
)set message auto off
)clear all
---S 1 of 7
+--S 1
aa:=integrate((p*x+q)/sqrt(a*x+b),x)
--R
--R
@@ -27,6 +27,24 @@ aa:=integrate((p*x+q)/sqrt(a*x+b),x)
--R 3a
--R Type: Union(Expression
Integer,...)
--E
+
+--S 2
+bb:=(2*(a*p*x+3*a*q-2*b*p))/(3*a^2)*sqrt(a*x+b)
+--R
+--R +-------+
+--R (2a p x + 6a q - 4b p)\|a x + b
+--R (2) --------------------------------
+--R 2
+--R 3a
+--R Type: Expression
Integer
+--E
+
+--S 3 14:113 Schaums and Axiom agree
+cc:=aa-bb
+--R
+--R (3) 0
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.114~~~~~$\displaystyle
@@ -43,7 +61,7 @@ $$
<<*>>=
)clear all
---S 2 of 7
+--S 4
aa:=integrate(1/((p*x+q)*sqrt(a*x+b)),x)
--R
--R
@@ -68,6 +86,157 @@ aa:=integrate(1/((p*x+q)*sqrt(a*x+b)),x)
--R \|a p q - b p
--R Type: Union(List Expression
Integer,...)
--E
+
+--S 5
+aa1:=aa.1
+--R
+--R (2)
+--R +--------------+
+--R 2 +-------+ | 2
+--R (2a p q - 2b p )\|a x + b + (a p x - a q + 2b p)\|- a p q + b p
+--R log(------------------------------------------------------------------)
+--R p x + q
+--R -----------------------------------------------------------------------
+--R +--------------+
+--R | 2
+--R \|- a p q + b p
+--R Type: Expression
Integer
+--E
+
+--S 6
+aa2:=aa.2
+--R
+--R +------------+
+--R | 2 +-------+
+--R \|a p q - b p \|a x + b
+--R 2atan(-------------------------)
+--R a q - b p
+--R (3) --------------------------------
+--R +------------+
+--R | 2
+--R \|a p q - b p
+--R Type: Expression
Integer
+--E
+
+--S 7
+bb1:=1/sqrt(b*p-a*q)*log((sqrt(p*(a*x+b))-sqrt(b*p-a*q))/(sqrt(p*(a*x+b))+sqrt(b*p-a*q)))
+--R
+--R +-----------+ +-----------+
+--R \|a p x + b p - \|- a q + b p
+--R log(-------------------------------)
+--R +-----------+ +-----------+
+--R \|a p x + b p + \|- a q + b p
+--R (4) ------------------------------------
+--R +-----------+
+--R \|- a q + b p
+--R Type: Expression
Integer
+--E
+
+--S 8
+bb2:=2/(sqrt(a*q-b*p)*sqrt(p))*atan(sqrt((p*(a*x+b))/(a*q-b*p)))
+--R
+--R +-----------+
+--R |a p x + b p
+--R 2atan( |----------- )
+--R \| a q - b p
+--R (5) ---------------------
+--R +-+ +---------+
+--R \|p \|a q - b p
+--R Type: Expression
Integer
+--E
+
+--S 9
+cc1:=aa1-bb1
+--R
+--R (6)
+--R +-----------+
+--R \|- a q + b p
+--R *
+--R
+--------------+
+--R 2 +-------+ | 2
+--R (2a p q - 2b p )\|a x + b + (a p x - a q + 2b p)\|- a p q + b p
+--R
log(------------------------------------------------------------------)
+--R p x + q
+--R +
+--R +--------------+ +-----------+ +-----------+
+--R | 2 \|a p x + b p - \|- a q + b p
+--R - \|- a p q + b p log(-------------------------------)
+--R +-----------+ +-----------+
+--R \|a p x + b p + \|- a q + b p
+--R /
+--R +--------------+
+--R | 2 +-----------+
+--R \|- a p q + b p \|- a q + b p
+--R Type: Expression
Integer
+--E
+
+--S 10
+cc2:=aa1-bb2
+--R
+--R (7)
+--R +-+ +---------+
+--R \|p \|a q - b p
+--R *
+--R
+--------------+
+--R 2 +-------+ | 2
+--R (2a p q - 2b p )\|a x + b + (a p x - a q + 2b p)\|- a p q + b p
+--R
log(------------------------------------------------------------------)
+--R p x + q
+--R +
+--R +--------------+ +-----------+
+--R | 2 |a p x + b p
+--R - 2\|- a p q + b p atan( |----------- )
+--R \| a q - b p
+--R /
+--R +--------------+
+--R | 2 +-+ +---------+
+--R \|- a p q + b p \|p \|a q - b p
+--R Type: Expression
Integer
+--E
+
+--S 11
+cc3:=aa2-bb1
+--R
+--R (8)
+--R +------------+ +-----------+ +-----------+
+--R | 2 \|a p x + b p - \|- a q + b p
+--R - \|a p q - b p log(-------------------------------)
+--R +-----------+ +-----------+
+--R \|a p x + b p + \|- a q + b p
+--R +
+--R +------------+
+--R | 2 +-------+
+--R +-----------+ \|a p q - b p \|a x + b
+--R 2\|- a q + b p atan(-------------------------)
+--R a q - b p
+--R /
+--R +------------+
+--R +-----------+ | 2
+--R \|- a q + b p \|a p q - b p
+--R Type: Expression
Integer
+--E
+
+--S 12 14:114 Axiom cannot simplify these answers
+cc4:=aa2-bb2
+--R
+--R (9)
+--R +------------+
+--R | 2 +-------+
+--R +-+ +---------+ \|a p q - b p \|a x + b
+--R 2\|p \|a q - b p atan(-------------------------)
+--R a q - b p
+--R +
+--R +------------+ +-----------+
+--R | 2 |a p x + b p
+--R - 2\|a p q - b p atan( |----------- )
+--R \| a q - b p
+--R /
+--R +------------+
+--R +-+ +---------+ | 2
+--R \|p \|a q - b p \|a p q - b p
+--R Type: Expression
Integer
+--E
+
@
\section{\cite{1}:14.115~~~~~$\displaystyle\int{\frac{\sqrt{ax+b}}{px+q}}~dx$}
@@ -83,7 +252,7 @@ $$\int{\frac{\sqrt{ax+b}}{px+q}}=
<<*>>=
)clear all
---S 3 of 7
+--S 13
aa:=integrate(sqrt(a*x+b)/(p*x+q),x)
--R
--R
@@ -112,6 +281,150 @@ aa:=integrate(sqrt(a*x+b)/(p*x+q),x)
--R p
--R Type: Union(List Expression
Integer,...)
--E
+
+--S 14
+aa1:=aa.1
+--R
+--R (2)
+--R +-----------+
+--R |- a q + b p +-------+
+--R +-----------+ - 2p |----------- \|a x + b + a p x - a q + 2b p
+--R |- a q + b p \| p
+--R |----------- log(-------------------------------------------------)
+--R \| p p x + q
+--R +
+--R +-------+
+--R 2\|a x + b
+--R /
+--R p
+--R Type: Expression
Integer
+--E
+
+--S 15
+aa2:=aa.2
+--R
+--R +---------+ +-------+
+--R |a q - b p \|a x + b +-------+
+--R - 2 |--------- atan(------------ + 2\|a x + b
+--R \| p +---------+
+--R |a q - b p
+--R |---------
+--R \| p
+--R (3) -----------------------------------------------
+--R p
+--R Type: Expression
Integer
+--E
+
+--S 16
+bb1:=(2*sqrt(a*x+b))/p+sqrt(b*p-a*q)/(p*sqrt(p))*log((sqrt(p*(a*x+b))-sqrt(b*p-a*q))/(sqrt(p*(a*x+b))+sqrt(b*p-a*q)))
+--R
+--R +-----------+ +-----------+
+--R +-----------+ \|a p x + b p - \|- a q + b p +-+ +-------+
+--R \|- a q + b p log(-------------------------------) + 2\|p \|a x + b
+--R +-----------+ +-----------+
+--R \|a p x + b p + \|- a q + b p
+--R (4) --------------------------------------------------------------------
+--R +-+
+--R p\|p
+--R Type: Expression
Integer
+--E
+
+--S 17
+bb2:=(2*sqrt(a*x+b))/p-(2*sqrt(a*q-b*p))/(p*sqrt(p))*atan(sqrt((p*(a*x+b))/(a*q-b*p)))
+--R
+--R +-----------+
+--R +---------+ |a p x + b p +-+ +-------+
+--R - 2\|a q - b p atan( |----------- ) + 2\|p \|a x + b
+--R \| a q - b p
+--R (5) -----------------------------------------------------
+--R +-+
+--R p\|p
+--R Type: Expression
Integer
+--E
+
+--S 18
+cc1:=aa1-bb1
+--R
+--R (6)
+--R +-----------+ +-----------+
+--R +-----------+ \|a p x + b p - \|- a q + b p
+--R - \|- a q + b p log(-------------------------------)
+--R +-----------+ +-----------+
+--R \|a p x + b p + \|- a q + b p
+--R +
+--R +-----------+
+--R |- a q + b p +-------+
+--R +-----------+ - 2p |----------- \|a x + b + a p x - a q +
2b p
+--R |- a q + b p +-+ \| p
+--R |----------- \|p
log(-------------------------------------------------)
+--R \| p p x + q
+--R /
+--R +-+
+--R p\|p
+--R Type: Expression
Integer
+--E
+
+--S 19
+cc2:=aa1-bb2
+--R
+--R (7)
+--R +-----------+
+--R |- a q + b p +-------+
+--R +-----------+ - 2p |----------- \|a x + b + a p x - a q +
2b p
+--R |- a q + b p +-+ \| p
+--R |----------- \|p
log(-------------------------------------------------)
+--R \| p p x + q
+--R +
+--R +-----------+
+--R +---------+ |a p x + b p
+--R 2\|a q - b p atan( |----------- )
+--R \| a q - b p
+--R /
+--R +-+
+--R p\|p
+--R Type: Expression
Integer
+--E
+
+--S 20
+cc3:=aa2-bb1
+--R
+--R (8)
+--R +-----------+ +-----------+
+--R +-----------+ \|a p x + b p - \|- a q + b p
+--R - \|- a q + b p log(-------------------------------)
+--R +-----------+ +-----------+
+--R \|a p x + b p + \|- a q + b p
+--R +
+--R +---------+ +-------+
+--R +-+ |a q - b p \|a x + b
+--R - 2\|p |--------- atan(------------)
+--R \| p +---------+
+--R |a q - b p
+--R |---------
+--R \| p
+--R /
+--R +-+
+--R p\|p
+--R Type: Expression
Integer
+--E
+
+--S 21 14:115 Axiom cannot simplify these answers
+cc4:=aa2-bb2
+--R
+--R (9)
+--R +---------+ +-------+ +-----------+
+--R +-+ |a q - b p \|a x + b +---------+ |a p x + b p
+--R - 2\|p |--------- atan(------------) + 2\|a q - b p atan( |----------- )
+--R \| p +---------+ \| a q - b p
+--R |a q - b p
+--R |---------
+--R \| p
+--R -------------------------------------------------------------------------
+--R +-+
+--R p\|p
+--R Type: Expression
Integer
+--E
+
@
\section{\cite{1}:14.116~~~~~$\displaystyle\int{(px+b)^n\sqrt{ax+b}}~dx$}
@@ -122,7 +435,7 @@ $$\int{(px+b)^n\sqrt{ax+b}}=
<<*>>=
)clear all
---S 4 of 7
+--S 22 14:116 Axiom cannot compute this integral
aa:=integrate((p*x+q)^n*sqrt(a*x+b),x)
--R
--R
@@ -144,7 +457,7 @@ $$\int{\frac{1}{(px+b)^n\sqrt{ax+b}}}=
<<*>>=
)clear all
---S 5 of 7
+--S 23 14:117 Axiom cannot compute this integral
aa:=integrate(1/((p*x+q)^n*sqrt(a*x+b)),x)
--R
--R
@@ -166,7 +479,7 @@ $$\int{\frac{(px+q)^n}{\sqrt{ax+b}}}=
<<*>>=
)clear all
---S 6 of 7
+--S 24 14:118 Axiom cannot compute this integral
aa:=integrate((p*x+q)^n/sqrt(a*x+b),x)
--R
--R
@@ -187,7 +500,7 @@ $$\int{\frac{\sqrt{ax+b}}{(px+q)^n}}=
<<*>>=
)clear all
---S 7 of 7
+--S 25 14:119 Axiom cannot compute this integral
aa:=integrate(sqrt(a*x+b)/(p*x+q)^n,x)
--R
--R
diff --git a/src/input/schaum5.input.pamphlet b/src/input/schaum5.input.pamphlet
index f8bccc8..50ea618 100644
--- a/src/input/schaum5.input.pamphlet
+++ b/src/input/schaum5.input.pamphlet
@@ -22,7 +22,7 @@ $$\int{\frac{1}{\sqrt{(ax+b)(px+q)}}}=
)set message auto off
)clear all
---S 1 of 5
+--S 1
aa:=integrate(1/sqrt((a*x+b)*(p*x+q)),x)
--R
--R
@@ -53,6 +53,157 @@ aa:=integrate(1/sqrt((a*x+b)*(p*x+q)),x)
--R \|- a p
--R Type: Union(List Expression
Integer,...)
--E
+
+--S 2
+aa1:=aa.1
+--R
+--R (2)
+--R log
+--R +---------------------------+
+--R +---+ +---+ | 2
+--R (2\|a p \|b q - 2a p x)\|a p x + (a q + b p)x + b q
+--R +
+--R +---+ 2 +---+
+--R 2a p x\|b q + (- 2a p x + (- a q - b p)x - 2b q)\|a p
+--R /
+--R +---------------------------+
+--R +---+ | 2
+--R 2\|b q \|a p x + (a q + b p)x + b q + (- a q - b p)x - 2b q
+--R /
+--R +---+
+--R \|a p
+--R Type: Expression
Integer
+--E
+
+--S 3
+aa2:=aa.2
+--R
+--R +---------------------------+
+--R +-----+ | 2 +-----+ +---+
+--R \|- a p \|a p x + (a q + b p)x + b q - \|- a p \|b q
+--R 2atan(-------------------------------------------------------)
+--R a p x
+--R (3) --------------------------------------------------------------
+--R +-----+
+--R \|- a p
+--R Type: Expression
Integer
+--E
+
+--S 4
+bb1:=2/sqrt(a*p)*log(sqrt(a*(p*x+q))+sqrt(p*(a*x+b)))
+--R
+--R +-----------+ +-----------+
+--R 2log(\|a p x + a q + \|a p x + b p )
+--R (4) -------------------------------------
+--R +---+
+--R \|a p
+--R Type: Expression
Integer
+--E
+
+--S 5
+bb2:=2/sqrt(-a*p)*atan(sqrt((-p*(a*x+b))/(a*(p*x+q))))
+--R
+--R +-------------+
+--R |- a p x - b p
+--R 2atan( |------------- )
+--R \| a p x + a q
+--R (5) -----------------------
+--R +-----+
+--R \|- a p
+--R Type: Expression
Integer
+--E
+
+--S 6
+cc1:=aa1-bb1
+--R
+--R (6)
+--R +-----------+ +-----------+
+--R - 2log(\|a p x + a q + \|a p x + b p )
+--R +
+--R log
+--R +---------------------------+
+--R +---+ +---+ | 2
+--R (2\|a p \|b q - 2a p x)\|a p x + (a q + b p)x + b q
+--R +
+--R +---+ 2 +---+
+--R 2a p x\|b q + (- 2a p x + (- a q - b p)x - 2b q)\|a p
+--R /
+--R +---------------------------+
+--R +---+ | 2
+--R 2\|b q \|a p x + (a q + b p)x + b q + (- a q - b p)x - 2b q
+--R /
+--R +---+
+--R \|a p
+--R Type: Expression
Integer
+--E
+
+--S 7
+cc2:=aa1-bb2
+--R
+--R (7)
+--R +-----+
+--R \|- a p
+--R *
+--R log
+--R +---------------------------+
+--R +---+ +---+ | 2
+--R (2\|a p \|b q - 2a p x)\|a p x + (a q + b p)x + b q
+--R +
+--R +---+ 2 +---+
+--R 2a p x\|b q + (- 2a p x + (- a q - b p)x - 2b q)\|a p
+--R /
+--R +---------------------------+
+--R +---+ | 2
+--R 2\|b q \|a p x + (a q + b p)x + b q + (- a q - b p)x - 2b q
+--R +
+--R +-------------+
+--R +---+ |- a p x - b p
+--R - 2\|a p atan( |------------- )
+--R \| a p x + a q
+--R /
+--R +-----+ +---+
+--R \|- a p \|a p
+--R Type: Expression
Integer
+--E
+
+--S 8
+cc3:=aa2-bb1
+--R
+--R (8)
+--R +-----+ +-----------+ +-----------+
+--R - 2\|- a p log(\|a p x + a q + \|a p x + b p )
+--R +
+--R +---------------------------+
+--R +-----+ | 2 +-----+ +---+
+--R +---+ \|- a p \|a p x + (a q + b p)x + b q - \|- a p \|b q
+--R 2\|a p atan(-------------------------------------------------------)
+--R a p x
+--R /
+--R +-----+ +---+
+--R \|- a p \|a p
+--R Type: Expression
Integer
+--E
+
+--S 9 14:120 Axiom cannot simplify these answers
+cc4:=aa2-bb2
+--R
+--R (9)
+--R +---------------------------+
+--R +-----+ | 2 +-----+ +---+
+--R \|- a p \|a p x + (a q + b p)x + b q - \|- a p \|b q
+--R 2atan(-------------------------------------------------------)
+--R a p x
+--R +
+--R +-------------+
+--R |- a p x - b p
+--R - 2atan( |------------- )
+--R \| a p x + a q
+--R /
+--R +-----+
+--R \|- a p
+--R Type: Expression
Integer
+--E
+
@
\section{\cite{1}:14.121~~~~~$\displaystyle
@@ -64,7 +215,7 @@ $$
<<*>>=
)clear all
---S 2 of 5
+--S 10
aa:=integrate(x/sqrt((a*x+b)*(p*x+q)),x)
--R
--R
@@ -133,6 +284,187 @@ aa:=integrate(x/sqrt((a*x+b)*(p*x+q)),x)
--R ]
--R Type: Union(List Expression
Integer,...)
--E
+
+--S 11
+bb1:=integrate(1/(sqrt(a*x+b)*(p*x+q)),x)
+--R
+--R (2)
+--R +--------------+
+--R 2 +-------+ | 2
+--R (2a p q - 2b p )\|a x + b + (a p x - a q + 2b p)\|- a p q + b p
+--R log(------------------------------------------------------------------)
+--R p x + q
+--R [-----------------------------------------------------------------------,
+--R +--------------+
+--R | 2
+--R \|- a p q + b p
+--R +------------+
+--R | 2 +-------+
+--R \|a p q - b p \|a x + b
+--R 2atan(-------------------------)
+--R a q - b p
+--R --------------------------------]
+--R +------------+
+--R | 2
+--R \|a p q - b p
+--R Type: Union(List Expression
Integer,...)
+--E
+
+--S 12
+bb2:=sqrt((a*x+b)*(p*x+q))/(a*p)-(b*p+a*q)/(2*a*p)
+--R
+--R +---------------------------+
+--R | 2
+--R 2\|a p x + (a q + b p)x + b q - a q - b p
+--R (3) -------------------------------------------
+--R 2a p
+--R Type: Expression
Integer
+--E
+
+--S 13
+bb:=bb2*bb1
+--R
+--R (4)
+--R [
+--R +---------------------------+
+--R | 2
+--R (2\|a p x + (a q + b p)x + b q - a q - b p)
+--R *
+--R
+--------------+
+--R 2 +-------+ | 2
+--R (2a p q - 2b p )\|a x + b + (a p x - a q + 2b p)\|- a p q + b p
+--R
log(------------------------------------------------------------------)
+--R p x + q
+--R /
+--R +--------------+
+--R | 2
+--R 2a p\|- a p q + b p
+--R ,
+--R +------------+
+--R +---------------------------+ | 2
+-------+
+--R | 2 \|a p q - b p \|a x
+ b
+--R (2\|a p x + (a q + b p)x + b q - a q - b
p)atan(-------------------------)
+--R a q - b p
+--R
----------------------------------------------------------------------------
+--R +------------+
+--R | 2
+--R a p\|a p q - b p
+--R ]
+--R Type: Vector Expression
Integer
+--E
+
+--S 14 14:121 Axiom cannot simplify this answer
+cc:=aa-bb
+--R
+--R (5)
+--R [
+--R +---+ +---+ +---+
+--R ((2a q + 2b p)\|a p \|b q + ((2a q + 2b p)x + 4b q)\|a p )
+--R *
+--R +---------------------------+
+--R | 2
+--R \|a p x + (a q + b p)x + b q
+--R +
+--R 2 +---+ +---+
+--R (- 4a p x + (- 4a q - 4b p)x - 4b q)\|a p \|b q
+--R +
+--R 2 2 2 2 2 2 +---+
+--R ((- a q - 2a b p q - b p )x - 2a b q - 2b p q)\|a p
+--R *
+--R
+--------------+
+--R 2 +-------+ |
2
+--R (2a p q - 2b p )\|a x + b + (a p x - a q + 2b p)\|- a p q + b
p
+--R
log(------------------------------------------------------------------)
+--R p x + q
+--R +
+--R +--------------+
+---------------------------+
+--R | 2 +---+ | 2
+--R (2a q + 2b p)\|- a p q + b p \|b q \|a p x + (a q + b p)x +
b q
+--R +
+--R
+--------------+
+--R 2 2 2 2 2 2 |
2
+--R ((- a q - 2a b p q - b p )x - 2a b q - 2b p q)\|- a p q + b p
+--R *
+--R log
+--R +---------------------------+
+--R +---+ +---+ | 2
+--R (2\|a p \|b q + 2a p x)\|a p x + (a q + b p)x + b q
+--R +
+--R +---+ 2 +---+
+--R - 2a p x\|b q + (- 2a p x + (- a q - b p)x - 2b q)\|a p
+--R /
+--R +---------------------------+
+--R +---+ | 2
+--R 2\|b q \|a p x + (a q + b p)x + b q + (- a q - b p)x - 2b
q
+--R +
+--R +--------------+
+---------------------------+
+--R | 2 +---+ | 2
+--R (- 2a q - 2b p)x\|- a p q + b p \|a p \|a p x + (a q + b p)x + b
q
+--R +
+--R +--------------+
+--R 2 | 2 +---+ +---+
+--R (4a p x + (2a q + 2b p)x)\|- a p q + b p \|a p \|b q
+--R /
+--R +--------------+ +---------------------------+
+--R | 2 +---+ +---+ | 2
+--R 4a p\|- a p q + b p \|a p \|b q \|a p x + (a q + b p)x + b q
+--R +
+--R +--------------+
+--R 2 2 | 2 +---+
+--R ((- 2a p q - 2a b p )x - 4a b p q)\|- a p q + b p \|a p
+--R ,
+--R
+--R +------------+
+---------------------------+
+--R +---+ | 2 | 2
+--R (- 2a q - 2b p)\|b q \|a p q - b p \|a p x + (a q + b p)x +
b q
+--R +
+--R +------------+
+--R 2 2 2 2 2 2 | 2
+--R ((a q + 2a b p q + b p )x + 2a b q + 2b p q)\|a p q - b p
+--R *
+--R +---------------------------+
+--R +-----+ | 2 +-----+ +---+
+--R \|- a p \|a p x + (a q + b p)x + b q - \|- a p \|b q
+--R atan(-------------------------------------------------------)
+--R a p x
+--R +
+--R +-----+ +---+
+-----+
+--R ((2a q + 2b p)\|- a p \|b q + ((2a q + 2b p)x + 4b q)\|- a
p )
+--R *
+--R +---------------------------+
+--R | 2
+--R \|a p x + (a q + b p)x + b q
+--R +
+--R 2 +-----+ +---+
+--R (- 4a p x + (- 4a q - 4b p)x - 4b q)\|- a p \|b q
+--R +
+--R 2 2 2 2 2 2 +-----+
+--R ((- a q - 2a b p q - b p )x - 2a b q - 2b p q)\|- a p
+--R *
+--R +------------+
+--R | 2 +-------+
+--R \|a p q - b p \|a x + b
+--R atan(-------------------------)
+--R a q - b p
+--R +
+--R +------------+ +---------------------------+
+--R +-----+ | 2 | 2
+--R (- a q - b p)x\|- a p \|a p q - b p \|a p x + (a q + b p)x + b q
+--R +
+--R +------------+
+--R 2 +-----+ +---+ | 2
+--R (2a p x + (a q + b p)x)\|- a p \|b q \|a p q - b p
+--R /
+--R +------------+ +---------------------------+
+--R +-----+ +---+ | 2 | 2
+--R 2a p\|- a p \|b q \|a p q - b p \|a p x + (a q + b p)x + b q
+--R +
+--R +------------+
+--R 2 2 +-----+ | 2
+--R ((- a p q - a b p )x - 2a b p q)\|- a p \|a p q - b p
+--R ]
+--R Type: Vector Expression
Integer
+--E
@
\section{\cite{1}:14.122~~~~~$\displaystyle\int{\sqrt{(ax+b)(px+q)}}~dx$}
@@ -143,7 +475,7 @@ $$
<<*>>=
)clear all
---S 3 of 5
+--S 15
aa:=integrate(sqrt((a*x+b)*(p*x+q)),x)
--R
--R
@@ -289,7 +621,639 @@ aa:=integrate(sqrt((a*x+b)*(p*x+q)),x)
--R \|- a p
--R ]
--R Type: Union(List Expression
Integer,...)
---E
+--E
+@
+Since there are two parts to the aa variable we split them:
+<<*>>=
+--S 16
+aa1:=aa.1
+--R
+--R (2)
+--R 3 3 2 2 2 2 3 3 2 3 2
2
+--R (4a q - 4a b p q - 4a b p q + 4b p )x + 8a b q - 16a b p q
+--R +
+--R 3 2
+--R 8b p q
+--R *
+--R +---------------------------+
+--R +---+ | 2
+--R \|b q \|a p x + (a q + b p)x + b q
+--R +
+--R 4 4 3 3 2 2 2 2 3 3 4 4 2
+--R (- a q - 4a b p q + 10a b p q - 4a b p q - b p )x
+--R +
+--R 3 4 2 2 3 3 2 2 4 3 2 2 4 3
3
+--R (- 8a b q + 8a b p q + 8a b p q - 8b p q)x - 8a b q + 16a b
p q
+--R +
+--R 4 2 2
+--R - 8b p q
+--R *
+--R log
+--R +---------------------------+
+--R +---+ +---+ | 2
+--R (2\|a p \|b q + 2a p x)\|a p x + (a q + b p)x + b q
+--R +
+--R +---+ 2 +---+
+--R - 2a p x\|b q + (- 2a p x + (- a q - b p)x - 2b q)\|a p
+--R /
+--R +---------------------------+
+--R +---+ | 2
+--R 2\|b q \|a p x + (a q + b p)x + b q + (- a q - b p)x - 2b q
+--R +
+--R 3 2 2 2 2 3 3
+--R (- 4a p q - 24a b p q - 4a b p )x
+--R +
+--R 3 3 2 2 2 2 3 3 2
+--R (- 2a q - 46a b p q - 46a b p q - 2b p )x
+--R +
+--R 2 3 2 2 3 2
+--R (- 8a b q - 48a b p q - 8b p q)x
+--R *
+--R +---------------------------+
+--R +---+ | 2
+--R \|a p \|a p x + (a q + b p)x + b q
+--R +
+--R 3 2 2 3 4 3 2 2 2 2 3 3
+--R (16a p q + 16a b p )x + (24a p q + 80a b p q + 24a b p )x
+--R +
+--R 3 3 2 2 2 2 3 3 2
+--R (6a q + 74a b p q + 74a b p q + 6b p )x
+--R +
+--R 2 3 2 2 3 2
+--R (8a b q + 48a b p q + 8b p q)x
+--R *
+--R +---+ +---+
+--R \|a p \|b q
+--R /
+--R 2 2 +---+ +---+
+--R ((32a p q + 32a b p )x + 64a b p q)\|a p \|b q
+--R *
+--R +---------------------------+
+--R | 2
+--R \|a p x + (a q + b p)x + b q
+--R +
+--R 3 2 2 2 2 3 2 2 2 2 2
+--R (- 8a p q - 48a b p q - 8a b p )x + (- 64a b p q - 64a b p q)x
+--R +
+--R 2 2
+--R - 64a b p q
+--R *
+--R +---+
+--R \|a p
+--R Type: Expression
Integer
+--E
+
+--S 17
+aa2:=aa.2
+--R
+--R (3)
+--R 3 3 2 2 2 2 3 3 2 3 2
2
+--R (- 4a q + 4a b p q + 4a b p q - 4b p )x - 8a b q + 16a b
p q
+--R +
+--R 3 2
+--R - 8b p q
+--R *
+--R +---------------------------+
+--R +---+ | 2
+--R \|b q \|a p x + (a q + b p)x + b q
+--R +
+--R 4 4 3 3 2 2 2 2 3 3 4 4 2
+--R (a q + 4a b p q - 10a b p q + 4a b p q + b p )x
+--R +
+--R 3 4 2 2 3 3 2 2 4 3 2 2 4 3
3
+--R (8a b q - 8a b p q - 8a b p q + 8b p q)x + 8a b q - 16a b p q
+--R +
+--R 4 2 2
+--R 8b p q
+--R *
+--R +---------------------------+
+--R +-----+ | 2 +-----+ +---+
+--R \|- a p \|a p x + (a q + b p)x + b q - \|- a p \|b q
+--R atan(-------------------------------------------------------)
+--R a p x
+--R +
+--R 3 2 2 2 2 3 3
+--R (- 2a p q - 12a b p q - 2a b p )x
+--R +
+--R 3 3 2 2 2 2 3 3 2
+--R (- a q - 23a b p q - 23a b p q - b p )x
+--R +
+--R 2 3 2 2 3 2
+--R (- 4a b q - 24a b p q - 4b p q)x
+--R *
+--R +---------------------------+
+--R +-----+ | 2
+--R \|- a p \|a p x + (a q + b p)x + b q
+--R +
+--R 3 2 2 3 4 3 2 2 2 2 3 3
+--R (8a p q + 8a b p )x + (12a p q + 40a b p q + 12a b p )x
+--R +
+--R 3 3 2 2 2 2 3 3 2
+--R (3a q + 37a b p q + 37a b p q + 3b p )x
+--R +
+--R 2 3 2 2 3 2
+--R (4a b q + 24a b p q + 4b p q)x
+--R *
+--R +-----+ +---+
+--R \|- a p \|b q
+--R /
+--R 2 2 +-----+ +---+
+--R ((16a p q + 16a b p )x + 32a b p q)\|- a p \|b q
+--R *
+--R +---------------------------+
+--R | 2
+--R \|a p x + (a q + b p)x + b q
+--R +
+--R 3 2 2 2 2 3 2 2 2 2 2
+--R (- 4a p q - 24a b p q - 4a b p )x + (- 32a b p q - 32a b p q)x
+--R +
+--R 2 2
+--R - 32a b p q
+--R *
+--R +-----+
+--R \|- a p
+--R Type: Expression
Integer
+--E
+@
+We break the books answer into 3 parts, the first term, the coefficient
+of the second term, and the integral.
+<<*>>=
+--S 18
+bba:=((2*a*p*x+b*p+a*q)/(4*a*p))*sqrt((a*x+b)*(p*x+q))
+--R
+--R +---------------------------+
+--R | 2
+--R (2a p x + a q + b p)\|a p x + (a q + b p)x + b q
+--R (4) --------------------------------------------------
+--R 4a p
+--R Type: Expression
Integer
+--E
+
+--S 19
+bbb:=-(b*p-a*q)^2/(8*a*p)
+--R
+--R 2 2 2 2
+--R - a q + 2a b p q - b p
+--R (5) ------------------------
+--R 8a p
+--R Type: Fraction Polynomial
Integer
+--E
+
+--S 20
+bbc:=integrate(1/sqrt((a*x+b)*(p*x+q)),x)
+--R
+--R (6)
+--R [
+--R log
+--R +---------------------------+
+--R +---+ +---+ | 2
+--R (2\|a p \|b q - 2a p x)\|a p x + (a q + b p)x + b q
+--R +
+--R +---+ 2 +---+
+--R 2a p x\|b q + (- 2a p x + (- a q - b p)x - 2b q)\|a p
+--R /
+--R +---------------------------+
+--R +---+ | 2
+--R 2\|b q \|a p x + (a q + b p)x + b q + (- a q - b p)x - 2b q
+--R /
+--R +---+
+--R \|a p
+--R ,
+--R +---------------------------+
+--R +-----+ | 2 +-----+ +---+
+--R \|- a p \|a p x + (a q + b p)x + b q - \|- a p \|b q
+--R 2atan(-------------------------------------------------------)
+--R a p x
+--R --------------------------------------------------------------]
+--R +-----+
+--R \|- a p
+--R Type: Union(List Expression
Integer,...)
+--E
+@
+Since the integral has two parts, we break them apart
+<<*>>=
+--S 21
+bbc1:=bbc.1
+--R
+--R (7)
+--R log
+--R +---------------------------+
+--R +---+ +---+ | 2
+--R (2\|a p \|b q - 2a p x)\|a p x + (a q + b p)x + b q
+--R +
+--R +---+ 2 +---+
+--R 2a p x\|b q + (- 2a p x + (- a q - b p)x - 2b q)\|a p
+--R /
+--R +---------------------------+
+--R +---+ | 2
+--R 2\|b q \|a p x + (a q + b p)x + b q + (- a q - b p)x - 2b q
+--R /
+--R +---+
+--R \|a p
+--R Type: Expression
Integer
+--E
+
+--S 22
+bbc2:=bbc.2
+--R
+--R +---------------------------+
+--R +-----+ | 2 +-----+ +---+
+--R \|- a p \|a p x + (a q + b p)x + b q - \|- a p \|b q
+--R 2atan(-------------------------------------------------------)
+--R a p x
+--R (8) --------------------------------------------------------------
+--R +-----+
+--R \|- a p
+--R Type: Expression
Integer
+--E
+@
+And now we construct the two bb answers based on the integral parts
+<<*>>=
+--S 23
+bb1:=bba+bbb*bbc1
+--R
+--R (9)
+--R 2 2 2 2
+--R (- a q + 2a b p q - b p )
+--R *
+--R log
+--R +---------------------------+
+--R +---+ +---+ | 2
+--R (2\|a p \|b q - 2a p x)\|a p x + (a q + b p)x + b q
+--R +
+--R +---+ 2 +---+
+--R 2a p x\|b q + (- 2a p x + (- a q - b p)x - 2b q)\|a p
+--R /
+--R +---------------------------+
+--R +---+ | 2
+--R 2\|b q \|a p x + (a q + b p)x + b q + (- a q - b p)x - 2b q
+--R +
+--R +---------------------------+
+--R +---+ | 2
+--R (4a p x + 2a q + 2b p)\|a p \|a p x + (a q + b p)x + b q
+--R /
+--R +---+
+--R 8a p\|a p
+--R Type: Expression
Integer
+--E
+
+--S 24
+bb2:=bba+bbb*bbc2
+--R
+--R (10)
+--R 2 2 2 2
+--R (- a q + 2a b p q - b p )
+--R *
+--R +---------------------------+
+--R +-----+ | 2 +-----+ +---+
+--R \|- a p \|a p x + (a q + b p)x + b q - \|- a p \|b q
+--R atan(-------------------------------------------------------)
+--R a p x
+--R +
+--R +---------------------------+
+--R +-----+ | 2
+--R (2a p x + a q + b p)\|- a p \|a p x + (a q + b p)x + b q
+--R /
+--R +-----+
+--R 4a p\|- a p
+--R Type: Expression
Integer
+--E
+@
+So there are 4 possible combinations that might yield an answer.
+We construct all four.
+<<*>>=
+--S 25
+cc1:=aa1-bb1
+--R
+--R (11)
+--R 3 3 2 2 2 2 3 3 2 3 2
2
+--R (4a q - 4a b p q - 4a b p q + 4b p )x + 8a b q - 16a b p q
+--R +
+--R 3 2
+--R 8b p q
+--R *
+--R +---------------------------+
+--R +---+ | 2
+--R \|b q \|a p x + (a q + b p)x + b q
+--R +
+--R 4 4 3 3 2 2 2 2 3 3 4 4 2
+--R (- a q - 4a b p q + 10a b p q - 4a b p q - b p )x
+--R +
+--R 3 4 2 2 3 3 2 2 4 3 2 2 4 3
3
+--R (- 8a b q + 8a b p q + 8a b p q - 8b p q)x - 8a b q + 16a b
p q
+--R +
+--R 4 2 2
+--R - 8b p q
+--R *
+--R log
+--R +---------------------------+
+--R +---+ +---+ | 2
+--R (2\|a p \|b q + 2a p x)\|a p x + (a q + b p)x + b q
+--R +
+--R +---+ 2 +---+
+--R - 2a p x\|b q + (- 2a p x + (- a q - b p)x - 2b q)\|a p
+--R /
+--R +---------------------------+
+--R +---+ | 2
+--R 2\|b q \|a p x + (a q + b p)x + b q + (- a q - b p)x - 2b q
+--R +
+--R 3 3 2 2 2 2 3 3 2 3 2
2
+--R (4a q - 4a b p q - 4a b p q + 4b p )x + 8a b q - 16a b p q
+--R +
+--R 3 2
+--R 8b p q
+--R *
+--R +---------------------------+
+--R +---+ | 2
+--R \|b q \|a p x + (a q + b p)x + b q
+--R +
+--R 4 4 3 3 2 2 2 2 3 3 4 4 2
+--R (- a q - 4a b p q + 10a b p q - 4a b p q - b p )x
+--R +
+--R 3 4 2 2 3 3 2 2 4 3 2 2 4 3
3
+--R (- 8a b q + 8a b p q + 8a b p q - 8b p q)x - 8a b q + 16a b
p q
+--R +
+--R 4 2 2
+--R - 8b p q
+--R *
+--R log
+--R +---------------------------+
+--R +---+ +---+ | 2
+--R (2\|a p \|b q - 2a p x)\|a p x + (a q + b p)x + b q
+--R +
+--R +---+ 2 +---+
+--R 2a p x\|b q + (- 2a p x + (- a q - b p)x - 2b q)\|a p
+--R /
+--R +---------------------------+
+--R +---+ | 2
+--R 2\|b q \|a p x + (a q + b p)x + b q + (- a q - b p)x - 2b q
+--R +
+--R 2 3 2 2 3 2 2 3 3 2 +---+
+--R ((8a b q + 16a b p q + 8b p q)x + 16a b q + 16b p q )\|a p
+--R *
+--R +---------------------------+
+--R | 2
+--R \|a p x + (a q + b p)x + b q
+--R +
+--R 3 3 2 2 2 2 3 3 2
+--R (- 2a q - 14a b p q - 14a b p q - 2b p )x
+--R +
+--R 2 3 2 2 3 2 2 3 3 2
+--R (- 16a b q - 32a b p q - 16b p q)x - 16a b q - 16b p q
+--R *
+--R +---+ +---+
+--R \|a p \|b q
+--R /
+--R 2 2 +---+ +---+
+--R ((32a p q + 32a b p )x + 64a b p q)\|a p \|b q
+--R *
+--R +---------------------------+
+--R | 2
+--R \|a p x + (a q + b p)x + b q
+--R +
+--R 3 2 2 2 2 3 2 2 2 2 2
+--R (- 8a p q - 48a b p q - 8a b p )x + (- 64a b p q - 64a b p q)x
+--R +
+--R 2 2
+--R - 64a b p q
+--R *
+--R +---+
+--R \|a p
+--R Type: Expression
Integer
+--E
+
+--S 26
+cc2:=aa1-bb2
+--R
+--R (12)
+--R 3 3 2 2 2 2 3 3 2 3 2
2
+--R (4a q - 4a b p q - 4a b p q + 4b p )x + 8a b q - 16a b p q
+--R +
+--R 3 2
+--R 8b p q
+--R *
+--R +---------------------------+
+--R +-----+ +---+ | 2
+--R \|- a p \|b q \|a p x + (a q + b p)x + b q
+--R +
+--R 4 4 3 3 2 2 2 2 3 3 4 4 2
+--R (- a q - 4a b p q + 10a b p q - 4a b p q - b p )x
+--R +
+--R 3 4 2 2 3 3 2 2 4 3 2 2 4
+--R (- 8a b q + 8a b p q + 8a b p q - 8b p q)x - 8a b q
+--R +
+--R 3 3 4 2 2
+--R 16a b p q - 8b p q
+--R *
+--R +-----+
+--R \|- a p
+--R *
+--R log
+--R +---------------------------+
+--R +---+ +---+ | 2
+--R (2\|a p \|b q + 2a p x)\|a p x + (a q + b p)x + b q
+--R +
+--R +---+ 2 +---+
+--R - 2a p x\|b q + (- 2a p x + (- a q - b p)x - 2b q)\|a p
+--R /
+--R +---------------------------+
+--R +---+ | 2
+--R 2\|b q \|a p x + (a q + b p)x + b q + (- a q - b p)x - 2b q
+--R +
+--R 3 3 2 2 2 2 3 3 2 3 2
2
+--R (8a q - 8a b p q - 8a b p q + 8b p )x + 16a b q - 32a b p
q
+--R +
+--R 3 2
+--R 16b p q
+--R *
+--R +---------------------------+
+--R +---+ +---+ | 2
+--R \|a p \|b q \|a p x + (a q + b p)x + b q
+--R +
+--R 4 4 3 3 2 2 2 2 3 3 4 4 2
+--R (- 2a q - 8a b p q + 20a b p q - 8a b p q - 2b p )x
+--R +
+--R 3 4 2 2 3 3 2 2 4 3 2 2 4
+--R (- 16a b q + 16a b p q + 16a b p q - 16b p q)x - 16a b q
+--R +
+--R 3 3 4 2 2
+--R 32a b p q - 16b p q
+--R *
+--R +---+
+--R \|a p
+--R *
+--R +---------------------------+
+--R +-----+ | 2 +-----+ +---+
+--R \|- a p \|a p x + (a q + b p)x + b q - \|- a p \|b q
+--R atan(-------------------------------------------------------)
+--R a p x
+--R +
+--R 2 3 2 2 3 2 2 3 3 2 +-----+
+---+
+--R ((8a b q + 16a b p q + 8b p q)x + 16a b q + 16b p q )\|- a p
\|a p
+--R *
+--R +---------------------------+
+--R | 2
+--R \|a p x + (a q + b p)x + b q
+--R +
+--R 3 3 2 2 2 2 3 3 2
+--R (- 2a q - 14a b p q - 14a b p q - 2b p )x
+--R +
+--R 2 3 2 2 3 2 2 3 3 2
+--R (- 16a b q - 32a b p q - 16b p q)x - 16a b q - 16b p q
+--R *
+--R +-----+ +---+ +---+
+--R \|- a p \|a p \|b q
+--R /
+--R 2 2 +-----+ +---+ +---+
+--R ((32a p q + 32a b p )x + 64a b p q)\|- a p \|a p \|b q
+--R *
+--R +---------------------------+
+--R | 2
+--R \|a p x + (a q + b p)x + b q
+--R +
+--R 3 2 2 2 2 3 2 2 2 2 2
+--R (- 8a p q - 48a b p q - 8a b p )x + (- 64a b p q - 64a b p q)x
+--R +
+--R 2 2
+--R - 64a b p q
+--R *
+--R +-----+ +---+
+--R \|- a p \|a p
+--R Type: Expression
Integer
+--E
+
+--S 27
+cc3:=aa1-bb1
+--R
+--R (13)
+--R 3 3 2 2 2 2 3 3 2 3 2
2
+--R (4a q - 4a b p q - 4a b p q + 4b p )x + 8a b q - 16a b p q
+--R +
+--R 3 2
+--R 8b p q
+--R *
+--R +---------------------------+
+--R +---+ | 2
+--R \|b q \|a p x + (a q + b p)x + b q
+--R +
+--R 4 4 3 3 2 2 2 2 3 3 4 4 2
+--R (- a q - 4a b p q + 10a b p q - 4a b p q - b p )x
+--R +
+--R 3 4 2 2 3 3 2 2 4 3 2 2 4 3
3
+--R (- 8a b q + 8a b p q + 8a b p q - 8b p q)x - 8a b q + 16a b
p q
+--R +
+--R 4 2 2
+--R - 8b p q
+--R *
+--R log
+--R +---------------------------+
+--R +---+ +---+ | 2
+--R (2\|a p \|b q + 2a p x)\|a p x + (a q + b p)x + b q
+--R +
+--R +---+ 2 +---+
+--R - 2a p x\|b q + (- 2a p x + (- a q - b p)x - 2b q)\|a p
+--R /
+--R +---------------------------+
+--R +---+ | 2
+--R 2\|b q \|a p x + (a q + b p)x + b q + (- a q - b p)x - 2b q
+--R +
+--R 3 3 2 2 2 2 3 3 2 3 2
2
+--R (4a q - 4a b p q - 4a b p q + 4b p )x + 8a b q - 16a b p q
+--R +
+--R 3 2
+--R 8b p q
+--R *
+--R +---------------------------+
+--R +---+ | 2
+--R \|b q \|a p x + (a q + b p)x + b q
+--R +
+--R 4 4 3 3 2 2 2 2 3 3 4 4 2
+--R (- a q - 4a b p q + 10a b p q - 4a b p q - b p )x
+--R +
+--R 3 4 2 2 3 3 2 2 4 3 2 2 4 3
3
+--R (- 8a b q + 8a b p q + 8a b p q - 8b p q)x - 8a b q + 16a b
p q
+--R +
+--R 4 2 2
+--R - 8b p q
+--R *
+--R log
+--R +---------------------------+
+--R +---+ +---+ | 2
+--R (2\|a p \|b q - 2a p x)\|a p x + (a q + b p)x + b q
+--R +
+--R +---+ 2 +---+
+--R 2a p x\|b q + (- 2a p x + (- a q - b p)x - 2b q)\|a p
+--R /
+--R +---------------------------+
+--R +---+ | 2
+--R 2\|b q \|a p x + (a q + b p)x + b q + (- a q - b p)x - 2b q
+--R +
+--R 2 3 2 2 3 2 2 3 3 2 +---+
+--R ((8a b q + 16a b p q + 8b p q)x + 16a b q + 16b p q )\|a p
+--R *
+--R +---------------------------+
+--R | 2
+--R \|a p x + (a q + b p)x + b q
+--R +
+--R 3 3 2 2 2 2 3 3 2
+--R (- 2a q - 14a b p q - 14a b p q - 2b p )x
+--R +
+--R 2 3 2 2 3 2 2 3 3 2
+--R (- 16a b q - 32a b p q - 16b p q)x - 16a b q - 16b p q
+--R *
+--R +---+ +---+
+--R \|a p \|b q
+--R /
+--R 2 2 +---+ +---+
+--R ((32a p q + 32a b p )x + 64a b p q)\|a p \|b q
+--R *
+--R +---------------------------+
+--R | 2
+--R \|a p x + (a q + b p)x + b q
+--R +
+--R 3 2 2 2 2 3 2 2 2 2 2
+--R (- 8a p q - 48a b p q - 8a b p )x + (- 64a b p q - 64a b p q)x
+--R +
+--R 2 2
+--R - 64a b p q
+--R *
+--R +---+
+--R \|a p
+--R Type: Expression
Integer
+--E
+
+--S 28 14:122 Axiom cannot simplify this answer
+cc4:=aa2-bb2
+--R
+--R (14)
+--R 2 3 2 2 3 2 2 3 3 2
+--R ((4a b q + 8a b p q + 4b p q)x + 8a b q + 8b p q )
+--R *
+--R +---------------------------+
+--R | 2
+--R \|a p x + (a q + b p)x + b q
+--R +
+--R 3 3 2 2 2 2 3 3 2
+--R (- a q - 7a b p q - 7a b p q - b p )x
+--R +
+--R 2 3 2 2 3 2 2 3 3 2
+--R (- 8a b q - 16a b p q - 8b p q)x - 8a b q - 8b p q
+--R *
+--R +---+
+--R \|b q
+--R /
+--R
+---------------------------+
+--R 2 2 +---+ | 2
+--R ((16a p q + 16a b p )x + 32a b p q)\|b q \|a p x + (a q + b p)x + b
q
+--R +
+--R 3 2 2 2 2 3 2 2 2 2 2
+--R (- 4a p q - 24a b p q - 4a b p )x + (- 32a b p q - 32a b p q)x
+--R +
+--R 2 2
+--R - 32a b p q
+--R Type: Expression
Integer
+--E
+
@
\section{\cite{1}:14.123~~~~~$\displaystyle\int{\sqrt{\frac{px+q}{ax+b}}}~dx$}
@@ -300,7 +1264,7 @@ $$
<<*>>=
)clear all
---S 4 of 5
+--S 29
aa:=integrate(sqrt((p*x+q)/(a*x+b)),x)
--R
--R
@@ -332,6 +1296,310 @@ aa:=integrate(sqrt((p*x+q)/(a*x+b)),x)
--R a\|- a p
--R Type: Union(List Expression
Integer,...)
--E
+
+--S 30
+aa1:=aa.1
+--R
+--R (2)
+--R
+-------+
+--R +---+ 2 |p x +
q
+--R (a q - b p)log((2a p x + a q + b p)\|a p + (2a p x + 2a b p)
|------- )
+--R \|a x +
b
+--R +
+--R +-------+
+--R |p x + q +---+
+--R (2a x + 2b) |------- \|a p
+--R \|a x + b
+--R /
+--R +---+
+--R 2a\|a p
+--R Type: Expression
Integer
+--E
+
+--S 31
+aa2:=aa.2
+--R
+--R +-------+
+--R +-----+ |p x + q
+--R \|- a p |------- +-------+
+--R \|a x + b +-----+ |p x + q
+--R (a q - b p)atan(------------------) + (a x + b)\|- a p |-------
+--R p \|a x + b
+--R (3) -----------------------------------------------------------------
+--R +-----+
+--R a\|- a p
+--R Type: Expression
Integer
+--E
+
+--S 32
+bba:=sqrt((a*x+b)*(p*x+q))/a
+--R
+--R +---------------------------+
+--R | 2
+--R \|a p x + (a q + b p)x + b q
+--R (4) ------------------------------
+--R a
+--R Type: Expression
Integer
+--E
+
+--S 33
+bbb:=(a*q-b*p)/(2*a)
+--R
+--R a q - b p
+--R (5) ---------
+--R 2a
+--R Type: Fraction Polynomial
Integer
+--E
+
+--S 34
+bbc:=integrate(1/(sqrt((a*x+b)*(p*x+q))),x)
+--R
+--R (6)
+--R [
+--R log
+--R +---------------------------+
+--R +---+ +---+ | 2
+--R (2\|a p \|b q - 2a p x)\|a p x + (a q + b p)x + b q
+--R +
+--R +---+ 2 +---+
+--R 2a p x\|b q + (- 2a p x + (- a q - b p)x - 2b q)\|a p
+--R /
+--R +---------------------------+
+--R +---+ | 2
+--R 2\|b q \|a p x + (a q + b p)x + b q + (- a q - b p)x - 2b q
+--R /
+--R +---+
+--R \|a p
+--R ,
+--R +---------------------------+
+--R +-----+ | 2 +-----+ +---+
+--R \|- a p \|a p x + (a q + b p)x + b q - \|- a p \|b q
+--R 2atan(-------------------------------------------------------)
+--R a p x
+--R --------------------------------------------------------------]
+--R +-----+
+--R \|- a p
+--R Type: Union(List Expression
Integer,...)
+--E
+
+--S 35
+bbc1:=bbc.1
+--R
+--R (7)
+--R log
+--R +---------------------------+
+--R +---+ +---+ | 2
+--R (2\|a p \|b q - 2a p x)\|a p x + (a q + b p)x + b q
+--R +
+--R +---+ 2 +---+
+--R 2a p x\|b q + (- 2a p x + (- a q - b p)x - 2b q)\|a p
+--R /
+--R +---------------------------+
+--R +---+ | 2
+--R 2\|b q \|a p x + (a q + b p)x + b q + (- a q - b p)x - 2b q
+--R /
+--R +---+
+--R \|a p
+--R Type: Expression
Integer
+--E
+
+--S 36
+bbc2:=bbc.2
+--R
+--R +---------------------------+
+--R +-----+ | 2 +-----+ +---+
+--R \|- a p \|a p x + (a q + b p)x + b q - \|- a p \|b q
+--R 2atan(-------------------------------------------------------)
+--R a p x
+--R (8) --------------------------------------------------------------
+--R +-----+
+--R \|- a p
+--R Type: Expression
Integer
+--E
+
+--S 37
+bb1:=bba+bbb*bbc1
+--R
+--R (9)
+--R (a q - b p)
+--R *
+--R log
+--R +---------------------------+
+--R +---+ +---+ | 2
+--R (2\|a p \|b q - 2a p x)\|a p x + (a q + b p)x + b q
+--R +
+--R +---+ 2 +---+
+--R 2a p x\|b q + (- 2a p x + (- a q - b p)x - 2b q)\|a p
+--R /
+--R +---------------------------+
+--R +---+ | 2
+--R 2\|b q \|a p x + (a q + b p)x + b q + (- a q - b p)x - 2b q
+--R +
+--R +---------------------------+
+--R +---+ | 2
+--R 2\|a p \|a p x + (a q + b p)x + b q
+--R /
+--R +---+
+--R 2a\|a p
+--R Type: Expression
Integer
+--E
+
+--S 38
+bb2:=bba+bbb*bbc2
+--R
+--R (10)
+--R +---------------------------+
+--R +-----+ | 2 +-----+
+---+
+--R \|- a p \|a p x + (a q + b p)x + b q - \|- a p \|b
q
+--R (a q - b
p)atan(-------------------------------------------------------)
+--R a p x
+--R +
+--R +---------------------------+
+--R +-----+ | 2
+--R \|- a p \|a p x + (a q + b p)x + b q
+--R /
+--R +-----+
+--R a\|- a p
+--R Type: Expression
Integer
+--E
+
+--S 39
+cc1:=aa1-bb1
+--R
+--R (11)
+--R
+-------+
+--R +---+ 2 |p x +
q
+--R (a q - b p)log((2a p x + a q + b p)\|a p + (2a p x + 2a b p)
|------- )
+--R \|a x +
b
+--R +
+--R (- a q + b p)
+--R *
+--R log
+--R +---------------------------+
+--R +---+ +---+ | 2
+--R (2\|a p \|b q - 2a p x)\|a p x + (a q + b p)x + b q
+--R +
+--R +---+ 2 +---+
+--R 2a p x\|b q + (- 2a p x + (- a q - b p)x - 2b q)\|a p
+--R /
+--R +---------------------------+
+--R +---+ | 2
+--R 2\|b q \|a p x + (a q + b p)x + b q + (- a q - b p)x - 2b q
+--R +
+--R +---------------------------+ +-------+
+--R +---+ | 2 |p x + q +---+
+--R - 2\|a p \|a p x + (a q + b p)x + b q + (2a x + 2b) |------- \|a p
+--R \|a x + b
+--R /
+--R +---+
+--R 2a\|a p
+--R Type: Expression
Integer
+--E
+
+--S 40
+cc2:=aa1-bb2
+--R
+--R (12)
+--R +-----+
+--R (a q - b p)\|- a p
+--R *
+--R +-------+
+--R +---+ 2 |p x + q
+--R log((2a p x + a q + b p)\|a p + (2a p x + 2a b p) |------- )
+--R \|a x + b
+--R +
+--R +---+
+--R (- 2a q + 2b p)\|a p
+--R *
+--R +---------------------------+
+--R +-----+ | 2 +-----+ +---+
+--R \|- a p \|a p x + (a q + b p)x + b q - \|- a p \|b q
+--R atan(-------------------------------------------------------)
+--R a p x
+--R +
+--R +---------------------------+
+--R +-----+ +---+ | 2
+--R - 2\|- a p \|a p \|a p x + (a q + b p)x + b q
+--R +
+--R +-------+
+--R +-----+ |p x + q +---+
+--R (2a x + 2b)\|- a p |------- \|a p
+--R \|a x + b
+--R /
+--R +-----+ +---+
+--R 2a\|- a p \|a p
+--R Type: Expression
Integer
+--E
+
+--S 41
+cc3:=aa2-bb1
+--R
+--R (13)
+--R +-----+
+--R (- a q + b p)\|- a p
+--R *
+--R log
+--R +---------------------------+
+--R +---+ +---+ | 2
+--R (2\|a p \|b q - 2a p x)\|a p x + (a q + b p)x + b q
+--R +
+--R +---+ 2 +---+
+--R 2a p x\|b q + (- 2a p x + (- a q - b p)x - 2b q)\|a p
+--R /
+--R +---------------------------+
+--R +---+ | 2
+--R 2\|b q \|a p x + (a q + b p)x + b q + (- a q - b p)x - 2b q
+--R +
+--R +-------+
+--R +-----+ |p x + q
+--R \|- a p |-------
+--R +---+ \|a x + b
+--R (2a q - 2b p)\|a p atan(------------------)
+--R p
+--R +
+--R +---------------------------+
+--R +-----+ +---+ | 2
+--R - 2\|- a p \|a p \|a p x + (a q + b p)x + b q
+--R +
+--R +-------+
+--R +-----+ |p x + q +---+
+--R (2a x + 2b)\|- a p |------- \|a p
+--R \|a x + b
+--R /
+--R +-----+ +---+
+--R 2a\|- a p \|a p
+--R Type: Expression
Integer
+--E
+
+--S 42 14:88 Axiom cannot simplify these results
+cc4:=aa2-bb2
+--R
+--R (14)
+--R (- a q + b p)
+--R *
+--R +---------------------------+
+--R +-----+ | 2 +-----+ +---+
+--R \|- a p \|a p x + (a q + b p)x + b q - \|- a p \|b q
+--R atan(-------------------------------------------------------)
+--R a p x
+--R +
+--R +-------+
+--R +-----+ |p x + q
+--R \|- a p |-------
+--R \|a x + b
+--R (a q - b p)atan(------------------)
+--R p
+--R +
+--R +---------------------------+
+-------+
+--R +-----+ | 2 +-----+ |p x + q
+--R - \|- a p \|a p x + (a q + b p)x + b q + (a x + b)\|- a p |-------
+--R \|a x + b
+--R /
+--R +-----+
+--R a\|- a p
+--R Type: Expression
Integer
+--E
+
@
\section{\cite{1}:14.124~~~~~$\displaystyle
@@ -342,7 +1610,7 @@ $$
<<*>>=
)clear all
---S 5 of 5
+--S 43
aa:=integrate(1/((p*x+q)*sqrt((a*x+b)*(p*x+q))),x)
--R
--R
@@ -354,6 +1622,38 @@ aa:=integrate(1/((p*x+q)*sqrt((a*x+b)*(p*x+q))),x)
--R Type: Union(Expression
Integer,...)
--E
+--S 44
+bb:=(2*sqrt(a*x+b))/((a*q-b*p)*sqrt(p*x+q))
+--R
+--R +-------+
+--R 2\|a x + b
+--R (2) ---------------------
+--R +-------+
+--R (a q - b p)\|p x + q
+--R Type: Expression
Integer
+--E
+
+--S 45 14:124 Axiom cannot simplify this result
+cc:=aa-bb
+--R
+--R (3)
+--R +---------------------------+
+--R +-------+ | 2
+-------+
+--R - 2q\|a x + b \|a p x + (a q + b p)x + b q + (2a q - 2b p)x\|p x +
q
+--R +
+--R +---+ +-------+
+--R (2p x + 2q)\|b q \|a x + b
+--R /
+--R +---------------------------+
+--R 2 +-------+ | 2
+--R (a q - b p q)\|p x + q \|a p x + (a q + b p)x + b q
+--R +
+--R 2 2 +---+ +-------+
+--R ((- a p q + b p )x - a q + b p q)\|b q \|p x + q
+--R Type: Expression
Integer
+--E
+
+
)spool
)lisp (bye)
@
diff --git a/src/input/schaum6.input.pamphlet b/src/input/schaum6.input.pamphlet
index 9a08dc6..9b6f4f0 100644
--- a/src/input/schaum6.input.pamphlet
+++ b/src/input/schaum6.input.pamphlet
@@ -15,7 +15,7 @@ $$\int{\frac{1}{x^2+a^2}}=\frac{1}{a}\tan^{-1}\frac{x}{a}$$
)set message auto off
)clear all
---S 1 of 19
+--S 1
aa:=integrate(1/(x^2+a^2),x)
--R
--R
@@ -26,6 +26,25 @@ aa:=integrate(1/(x^2+a^2),x)
--R a
--R Type: Union(Expression
Integer,...)
--E
+
+--S 2
+bb:=(1/a)*atan(x/a)
+--R
+--R x
+--R atan(-)
+--R a
+--R (2) -------
+--R a
+--R Type: Expression
Integer
+--E
+
+--S 3 14:125 Schaums and Axiom agree
+cc:=aa-bb
+--R
+--R (3) 0
+--R Type: Expression
Integer
+--E
+
@
\section{\cite{1}:14.126~~~~~$\displaystyle\int{\frac{x~dx}{x^2+a^2}}$}
@@ -33,7 +52,7 @@ $$\int{\frac{x}{x^2+a^2}}=\frac{1}{2}\ln(x^2+a^2)$$
<<*>>=
)clear all
---S 2 of 19
+--S 4
aa:=integrate(x/(x^2+a^2),x)
--R
--R
@@ -43,6 +62,23 @@ aa:=integrate(x/(x^2+a^2),x)
--R 2
--R Type: Union(Expression
Integer,...)
--E
+
+--S 5
+bb:=(1/2)*log(x^2+a^2)
+--R
+--R 2 2
+--R log(x + a )
+--R (2) ------------
+--R 2
+--R Type: Expression
Integer
+--E
+
+--S 6 14:126 Schaums and Axiom agree
+cc:=aa-bb
+--R
+--R (3) 0
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.127~~~~~$\displaystyle\int{\frac{x^2~dx}{x^2+a^2}}$}
@@ -50,7 +86,7 @@ $$\int{\frac{x^2}{x^2+a^2}}=x-a\tan^{-1}\frac{x}{a}$$
<<*>>=
)clear all
---S 3 of 19
+--S 7
aa:=integrate(x^2/(x^2+a^2),x)
--R
--R
@@ -59,6 +95,23 @@ aa:=integrate(x^2/(x^2+a^2),x)
--R a
--R Type: Union(Expression
Integer,...)
--E
+
+--S 8
+bb:=x-a*atan(x/a)
+--R
+--R x
+--R (2) - a atan(-) + x
+--R a
+--R Type: Expression
Integer
+--E
+
+--S 9 14:127 Schaums and Axiom agree
+cc:=aa-bb
+--R
+--R (3) 0
+--R Type: Expression
Integer
+--E
+
@
\section{\cite{1}:14.128~~~~~$\displaystyle\int{\frac{x^3~dx}{x^2+a^2}}$}
@@ -67,7 +120,7 @@
$$\int{\frac{x^3}{x^2+a^2}}=\frac{x^2}{2}-\frac{a^2}{2}\ln(x^2+a^2)$$
<<*>>=
)clear all
---S 4 of 19
+--S 10
aa:=integrate(x^3/(x^2+a^2),x)
--R
--R
@@ -77,6 +130,23 @@ aa:=integrate(x^3/(x^2+a^2),x)
--R 2
--R Type: Union(Expression
Integer,...)
--E
+
+--S 11
+bb:=x^2/2-a^2/2*log(x^2+a^2)
+--R
+--R 2 2 2 2
+--R - a log(x + a ) + x
+--R (2) ---------------------
+--R 2
+--R Type: Expression
Integer
+--E
+
+--S 12 14:128 Schaums and Axiom agree
+cc:=aa-bb
+--R
+--R (3) 0
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.129~~~~~$\displaystyle\int{\frac{dx}{x(x^2+a^2)}}$}
@@ -86,7 +156,7 @@ $$
<<*>>=
)clear all
---S 5 of 19
+--S 13
aa:=integrate(1/(x*(x^2+a^2)),x)
--R
--R
@@ -97,6 +167,70 @@ aa:=integrate(1/(x*(x^2+a^2)),x)
--R 2a
--R Type: Union(Expression
Integer,...)
--E
+
+--S 14
+bb:=1/(2*a^2)*log(x^2/(x^2+a^2))
+--R
+--R 2
+--R x
+--R log(-------)
+--R 2 2
+--R x + a
+--R (2) ------------
+--R 2
+--R 2a
+--R Type: Expression
Integer
+--E
+
+--S 15
+cc:=aa-bb
+--R
+--R 2
+--R 2 2 x
+--R - log(x + a ) + 2log(x) - log(-------)
+--R 2 2
+--R x + a
+--R (3) ---------------------------------------
+--R 2
+--R 2a
+--R Type: Expression
Integer
+--E
+
+--S 16
+divlog:=rule(log(a/b) == log(a) - log(b))
+--R
+--R a
+--R (4) log(-) == - log(b) + log(a)
+--R b
+--R Type: RewriteRule(Integer,Integer,Expression
Integer)
+--E
+
+--S 17
+dd:=divlog cc
+--R
+--R 2
+--R - log(x ) + 2log(x)
+--R (5) -------------------
+--R 2
+--R 2a
+--R Type: Expression
Integer
+--E
+
+--S 18
+logpow:=rule(log(a^n) == n*log(a))
+--R
+--R n
+--R (6) log(a ) == n log(a)
+--R Type: RewriteRule(Integer,Integer,Expression
Integer)
+--E
+
+--S 19 14:129 Schaums and Axiom agree
+ee:=logpow dd
+--R
+--R (7) 0
+--R Type: Expression
Integer
+--E
+
@
\section{\cite{1}:14.130~~~~~$\displaystyle\int{\frac{dx}{x^2(x^2+a^2)}}$}
@@ -106,7 +240,7 @@ $$
<<*>>=
)clear all
---S 6 of 19
+--S 20
aa:=integrate(1/(x^2*(x^2+a^2)),x)
--R
--R
@@ -118,6 +252,26 @@ aa:=integrate(1/(x^2*(x^2+a^2)),x)
--R a x
--R Type: Union(Expression
Integer,...)
--E
+
+--S 21
+bb:=-1/(a^2*x)-1/a^3*atan(x/a)
+--R
+--R x
+--R - x atan(-) - a
+--R a
+--R (2) ---------------
+--R 3
+--R a x
+--R Type: Expression
Integer
+--E
+
+--S 22 14:130 Schaums and Axiom agree
+cc:=aa-bb
+--R
+--R (3) 0
+--R Type: Expression
Integer
+--E
+
@
\section{\cite{1}:14.131~~~~~$\displaystyle\int{\frac{dx}{x^3(x^2+a^2)}}$}
@@ -127,7 +281,7 @@ $$
<<*>>=
)clear all
---S 7 of 19
+--S 23
aa:=integrate(1/(x^3*(x^2+a^2)),x)
--R
--R
@@ -138,6 +292,70 @@ aa:=integrate(1/(x^3*(x^2+a^2)),x)
--R 2a x
--R Type: Union(Expression
Integer,...)
--E
+
+--S 24
+bb:=-1/(2*a^2*x^2)-1/(2*a^4)*log(x^2/(x^2+a^2))
+--R
+--R 2
+--R 2 x 2
+--R - x log(-------) - a
+--R 2 2
+--R x + a
+--R (2) ---------------------
+--R 4 2
+--R 2a x
+--R Type: Expression
Integer
+--E
+
+--S 25
+cc:=aa-bb
+--R
+--R 2
+--R 2 2 x
+--R log(x + a ) - 2log(x) + log(-------)
+--R 2 2
+--R x + a
+--R (3) -------------------------------------
+--R 4
+--R 2a
+--R Type: Expression
Integer
+--E
+
+--S 26
+divlog:=rule(log(a/b) == log(a) - log(b))
+--R
+--R a
+--R (4) log(-) == - log(b) + log(a)
+--R b
+--R Type: RewriteRule(Integer,Integer,Expression
Integer)
+--E
+
+--S 27
+dd:=divlog cc
+--R
+--R 2
+--R log(x ) - 2log(x)
+--R (5) -----------------
+--R 4
+--R 2a
+--R Type: Expression
Integer
+--E
+
+--S 28
+logpow:=rule(log(a^n) == n*log(a))
+--R
+--R n
+--R (6) log(a ) == n log(a)
+--R Type: RewriteRule(Integer,Integer,Expression
Integer)
+--E
+
+--S 29 14:131 Schaums and Axiom agree
+ee:=logpow dd
+--R
+--R (7) 0
+--R Type: Expression
Integer
+--E
+
@
\section{\cite{1}:14.132~~~~~$\displaystyle\int{\frac{dx}{(x^2+a^2)^2}}$}
@@ -147,7 +365,7 @@ $$
<<*>>=
)clear all
---S 8 of 19
+--S 30
aa:=integrate(1/((x^2+a^2)^2),x)
--R
--R
@@ -159,6 +377,26 @@ aa:=integrate(1/((x^2+a^2)^2),x)
--R 2a x + 2a
--R Type: Union(Expression
Integer,...)
--E
+
+--S 31
+bb:=x/(2*a^2*(x^2+a^2))+1/(2*a^3)*atan(x/a)
+--R
+--R 2 2 x
+--R (x + a )atan(-) + a x
+--R a
+--R (2) ----------------------
+--R 3 2 5
+--R 2a x + 2a
+--R Type: Expression
Integer
+--E
+
+--S 32 14:132 Schaums and Axiom agree
+cc:=aa-bb
+--R
+--R (3) 0
+--R Type: Expression
Integer
+--E
+
@
\section{\cite{1}:14.133~~~~~$\displaystyle\int{\frac{x~dx}{(x^2+a^2)^2}}$}
@@ -168,7 +406,7 @@ $$
<<*>>=
)clear all
---S 9 of 19
+--S 33
aa:=integrate(x/((x^2+a^2)^2),x)
--R
--R
@@ -178,6 +416,23 @@ aa:=integrate(x/((x^2+a^2)^2),x)
--R 2x + 2a
--R Type: Union(Expression
Integer,...)
--E
+
+--S 34
+bb:=-1/(2*(x^2+a^2))
+--R
+--R 1
+--R (2) - ---------
+--R 2 2
+--R 2x + 2a
+--R Type: Fraction Polynomial
Integer
+--E
+
+--S 35 14:133 Schaums and Axiom agree
+cc:=aa-bb
+--R
+--R (3) 0
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.134~~~~~$\displaystyle\int{\frac{x^2dx}{(x^2+a^2)^2}}$}
@@ -187,7 +442,7 @@ $$
<<*>>=
)clear all
---S 10 of 19
+--S 36
aa:=integrate(x^2/((x^2+a^2)^2),x)
--R
--R
@@ -199,6 +454,25 @@ aa:=integrate(x^2/((x^2+a^2)^2),x)
--R 2a x + 2a
--R Type: Union(Expression
Integer,...)
--E
+
+--S 37
+bb:=-x/(2*(x^2+a^2))+1/(2*a)*atan(x/a)
+--R
+--R 2 2 x
+--R (x + a )atan(-) - a x
+--R a
+--R (2) ----------------------
+--R 2 3
+--R 2a x + 2a
+--R Type: Expression
Integer
+--E
+
+--S 38 14:134 Schaums and Axiom differ by a constant
+cc:=aa-bb
+--R
+--R (3) 0
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.135~~~~~$\displaystyle\int{\frac{x^3dx}{(x^2+a^2)^2}}$}
@@ -208,7 +482,7 @@ $$
<<*>>=
)clear all
---S 11 of 19
+--S 39
aa:=integrate(x^3/((x^2+a^2)^2),x)
--R
--R
@@ -219,6 +493,24 @@ aa:=integrate(x^3/((x^2+a^2)^2),x)
--R 2x + 2a
--R Type: Union(Expression
Integer,...)
--E
+
+--S 40
+bb:=a^2/(2*(x^2+a^2))+1/2*log(x^2+a^2)
+--R
+--R 2 2 2 2 2
+--R (x + a )log(x + a ) + a
+--R (2) --------------------------
+--R 2 2
+--R 2x + 2a
+--R Type: Expression
Integer
+--E
+
+--S 41 14:135 Schaums and Axiom agree
+cc:=aa-bb
+--R
+--R (3) 0
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.136~~~~~$\displaystyle\int{\frac{dx}{x(x^2+a^2)^2}}$}
@@ -228,7 +520,7 @@ $$
<<*>>=
)clear all
---S 12 of 19
+--S 42
aa:=integrate(1/(x*(x^2+a^2)^2),x)
--R
--R
@@ -239,6 +531,69 @@ aa:=integrate(1/(x*(x^2+a^2)^2),x)
--R 2a x + 2a
--R Type: Union(Expression
Integer,...)
--E
+
+--S 43
+bb:=1/(2*a^2*(x^2+a^2))+1/(2*a^4)*log(x^2/(x^2+a^2))
+--R
+--R 2
+--R 2 2 x 2
+--R (x + a )log(-------) + a
+--R 2 2
+--R x + a
+--R (2) --------------------------
+--R 4 2 6
+--R 2a x + 2a
+--R Type: Expression
Integer
+--E
+
+--S 44
+cc:=aa-bb
+--R
+--R 2
+--R 2 2 x
+--R - log(x + a ) + 2log(x) - log(-------)
+--R 2 2
+--R x + a
+--R (3) ---------------------------------------
+--R 4
+--R 2a
+--R Type: Expression
Integer
+--E
+
+--S 45
+divlog:=rule(log(a/b) == log(a) - log(b))
+--R
+--R a
+--R (4) log(-) == - log(b) + log(a)
+--R b
+--R Type: RewriteRule(Integer,Integer,Expression
Integer)
+--E
+
+--S 46
+dd:=divlog cc
+--R
+--R 2
+--R - log(x ) + 2log(x)
+--R (5) -------------------
+--R 4
+--R 2a
+--R Type: Expression
Integer
+--E
+
+--S 47
+logpow:=rule(log(a^n) == n*log(a))
+--R
+--R n
+--R (6) log(a ) == n log(a)
+--R Type: RewriteRule(Integer,Integer,Expression
Integer)
+--E
+
+--S 48 14:136 Schaums and Axiom agree
+ee:=logpow dd
+--R
+--R (7) 0
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.137~~~~~$\displaystyle\int{\frac{dx}{x^2(x^2+a^2)^2}}$}
@@ -248,18 +603,36 @@ $$
<<*>>=
)clear all
---S 13 of 19
-aa:=integrate(1/((x^2+a^2)^2),x)
---R
+--S 49
+aa:=integrate(1/(x^2*(x^2+a^2)^2),x)
--R
---R 2 2 x
---R (x + a )atan(-) + a x
---R a
---R (1) ----------------------
---R 3 2 5
---R 2a x + 2a
+--R 3 2 x 2 3
+--R (- 3x - 3a x)atan(-) - 3a x - 2a
+--R a
+--R (1) -----------------------------------
+--R 5 3 7
+--R 2a x + 2a x
--R Type: Union(Expression
Integer,...)
---E
+--E
+
+--S 50
+bb:=-1/(a^4*x)-x/(2*a^4*(x^2+a^2))-3/(2*a^5)*atan(x/a)
+--R
+--R 3 2 x 2 3
+--R (- 3x - 3a x)atan(-) - 3a x - 2a
+--R a
+--R (2) -----------------------------------
+--R 5 3 7
+--R 2a x + 2a x
+--R Type: Expression
Integer
+--E
+
+--S 51 14:137 Schaums and Axiom agree
+cc:=aa-bb
+--R
+--R (3) 0
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.138~~~~~$\displaystyle\int{\frac{dx}{x^3(x^2+a^2)^2}}$}
@@ -270,7 +643,7 @@ $$
<<*>>=
)clear all
---S 14 of 19
+--S 52
aa:=integrate(1/(x^3*(x^2+a^2)^2),x)
--R
--R
@@ -281,6 +654,70 @@ aa:=integrate(1/(x^3*(x^2+a^2)^2),x)
--R 2a x + 2a x
--R Type: Union(Expression
Integer,...)
--E
+
+--S 53
+bb:=-1/(2*a^4*x^2)-1/(2*a^4*(x^2+a^2))-1/a^6*log(x^2/(x^2+a^2))
+--R
+--R 2
+--R 4 2 2 x 2 2 4
+--R (- 2x - 2a x )log(-------) - 2a x - a
+--R 2 2
+--R x + a
+--R (2) ----------------------------------------
+--R 6 4 8 2
+--R 2a x + 2a x
+--R Type: Expression
Integer
+--E
+
+--S 54
+cc:=aa-bb
+--R
+--R 2
+--R 2 2 x
+--R log(x + a ) - 2log(x) + log(-------)
+--R 2 2
+--R x + a
+--R (3) -------------------------------------
+--R 6
+--R a
+--R Type: Expression
Integer
+--E
+
+--S 55
+divlog:=rule(log(a/b) == log(a) - log(b))
+--R
+--R a
+--R (4) log(-) == - log(b) + log(a)
+--R b
+--R Type: RewriteRule(Integer,Integer,Expression
Integer)
+--E
+
+--S 56
+dd:=divlog cc
+--R
+--R 2
+--R log(x ) - 2log(x)
+--R (5) -----------------
+--R 6
+--R a
+--R Type: Expression
Integer
+--E
+
+--S 57
+logpow:=rule(log(a^n) == n*log(a))
+--R
+--R n
+--R (6) log(a ) == n log(a)
+--R Type: RewriteRule(Integer,Integer,Expression
Integer)
+--E
+
+--S 58 14:138 Schaums and Axiom agree
+ee:=logpow dd
+--R
+--R (7) 0
+--R Type: Expression
Integer
+--E
+
@
\section{\cite{1}:14.139~~~~~$\displaystyle\int{\frac{dx}{(x^2+a^2)^n}}$}
@@ -291,7 +728,7 @@ $$
<<*>>=
)clear all
---S 15 of 19
+--S 59 14:139 Axiom cannot do this integral
aa:=integrate(1/((x^2+a^2)^n),x)
--R
--R
@@ -311,7 +748,7 @@ $$
<<*>>=
)clear all
---S 16 of 19
+--S 60
aa:=integrate(x/((x^2+a^2)^n),x)
--R
--R
@@ -323,6 +760,48 @@ aa:=integrate(x/((x^2+a^2)^n),x)
--R (2n - 2)%e
--R Type: Union(Expression
Integer,...)
--E
+
+--S 61
+bb:=-1/(2*(n-1)*(x^2+a^2)^(n-1))
+--R
+--R 1
+--R (2) - ----------------------
+--R 2 2 n - 1
+--R (2n - 2)(x + a )
+--R Type: Expression
Integer
+--E
+
+--S 62
+cc:=aa-bb
+--R
+--R 2 2
+--R n log(x + a ) 2 2 2 2 n - 1
+--R %e + (- x - a )(x + a )
+--R (3) --------------------------------------------
+--R 2 2
+--R 2 2 n - 1 n log(x + a )
+--R (2n - 2)(x + a ) %e
+--R Type: Expression
Integer
+--E
+
+--S 63
+explog:=rule(%e^(n*log(x)) == x^n)
+--R
+--R n log(x) n
+--R (4) %e == x
+--R Type: RewriteRule(Integer,Integer,Expression
Integer)
+--E
+
+--S 64 14:140 Schaums and Axiom cannot simplify this expression
+dd:=explog cc
+--R
+--R 2 2 n 2 2 2 2 n - 1
+--R (x + a ) + (- x - a )(x + a )
+--R (5) --------------------------------------
+--R 2 2 n - 1 2 2 n
+--R (2n - 2)(x + a ) (x + a )
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.141~~~~~$\displaystyle\int{\frac{dx}{x(x^2+a^2)^n}}$}
@@ -333,7 +812,7 @@ $$
<<*>>=
)clear all
---S 17 of 19
+--S 65 14:141 Axiom cannot do this integral
aa:=integrate(1/(x*(x^2+a^2)^n),x)
--R
--R
@@ -354,7 +833,7 @@ $$
<<*>>=
)clear all
---S 18 of 19
+--S 66 14:142 Axiom cannot do this integral
aa:=integrate(x^m/((x^2+a^2)^n),x)
--R
--R
@@ -375,7 +854,7 @@ $$
<<*>>=
)clear all
---S 19 of 19
+--S 67 14:143 Axiom cannot do this integral
aa:=integrate(1/(x^m*(x^2+a^2)^n),x)
--R
--R
diff --git a/src/input/schaum7.input.pamphlet b/src/input/schaum7.input.pamphlet
index 9226da0..3062d2e 100644
--- a/src/input/schaum7.input.pamphlet
+++ b/src/input/schaum7.input.pamphlet
@@ -16,7 +16,7 @@ $$\int{\frac{1}{x^2-a^2}}=-\frac{1}{a}\coth^{-1}\frac{x}{a}$$
)set message auto off
)clear all
---S 1 of 19
+--S 1
aa:=integrate(1/(x^2-a^2),x)
--R
--R
@@ -25,6 +25,45 @@ aa:=integrate(1/(x^2-a^2),x)
--R 2a
--R Type: Union(Expression
Integer,...)
--E
+
+--S 2
+bb:=1/(2*a)*log((x-a)/(x+a))
+--R
+--R x - a
+--R log(-----)
+--R x + a
+--R (2) ----------
+--R 2a
+--R Type: Expression
Integer
+--E
+
+--S 3
+cc:=aa-bb
+--R
+--R x - a
+--R - log(x + a) + log(x - a) - log(-----)
+--R x + a
+--R (3) --------------------------------------
+--R 2a
+--R Type: Expression
Integer
+--E
+
+--S 4
+divlog:=rule(log(a/b) == log(a) - log(b))
+--R
+--R a
+--R (4) log(-) == - log(b) + log(a)
+--R b
+--R Type: RewriteRule(Integer,Integer,Expression
Integer)
+--E
+
+--S 5 14:144 Schaums and Axiom agree
+dd:=divlog cc
+--R
+--R (5) 0
+--R Type: Expression
Integer
+--E
+
@
\section{\cite{1}:14.145~~~~~$\displaystyle\int{\frac{x~dx}{x^2-a^2}}$}
@@ -32,7 +71,7 @@ $$\int{\frac{x}{x^2-a^2}}=\frac{1}{2}\ln(x^2-a^2)$$
<<*>>=
)clear all
---S 2 of 19
+--S 6
aa:=integrate(x/(x^2-a^2),x)
--R
--R
@@ -42,6 +81,24 @@ aa:=integrate(x/(x^2-a^2),x)
--R 2
--R Type: Union(Expression
Integer,...)
--E
+
+--S 7
+bb:=1/2*log(x^2-a^2)
+--R
+--R 2 2
+--R log(x - a )
+--R (2) ------------
+--R 2
+--R Type: Expression
Integer
+--E
+
+--S 8 14:145 Schaums and Axiom agree
+cc:=aa-bb
+--R
+--R (3) 0
+--R Type: Expression
Integer
+--E
+
@
\section{\cite{1}:14.146~~~~~$\displaystyle\int{\frac{x^2~dx}{x^2-a^2}}$}
@@ -49,7 +106,7 @@
$$\int{\frac{x^2}{x^2-a^2}}=x+\frac{a}{2}\ln\left(\frac{x-a}{x+a}\right)$$
<<*>>=
)clear all
---S 3 of 19
+--S 9
aa:=integrate(x^2/(x^2-a^2),x)
--R
--R
@@ -58,6 +115,45 @@ aa:=integrate(x^2/(x^2-a^2),x)
--R 2
--R Type: Union(Expression
Integer,...)
--E
+
+--S 10
+bb:=x+a/2*log((x-a)/(x+a))
+--R
+--R x - a
+--R a log(-----) + 2x
+--R x + a
+--R (2) -----------------
+--R 2
+--R Type: Expression
Integer
+--E
+
+--S 11
+cc:=aa-bb
+--R
+--R x - a
+--R - a log(x + a) + a log(x - a) - a log(-----)
+--R x + a
+--R (3) --------------------------------------------
+--R 2
+--R Type: Expression
Integer
+--E
+
+--S 12
+divlog:=rule(log(a/b) == log(a) - log(b))
+--R
+--R a
+--R (4) log(-) == - log(b) + log(a)
+--R b
+--R Type: RewriteRule(Integer,Integer,Expression
Integer)
+--E
+
+--S 13 14:146 Schaums and Axiom agree
+dd:=divlog cc
+--R
+--R (5) 0
+--R Type: Expression
Integer
+--E
+
@
\section{\cite{1}:14.147~~~~~$\displaystyle\int{\frac{x^3~dx}{x^2-a^2}}$}
@@ -66,7 +162,7 @@
$$\int{\frac{x^3}{x^2-a^2}}=\frac{x^2}{2}+\frac{a^2}{2}\ln(x^2-a^2)$$
<<*>>=
)clear all
---S 4 of 19
+--S 14
aa:=integrate(x^3/(x^2-a^2),x)
--R
--R
@@ -76,6 +172,24 @@ aa:=integrate(x^3/(x^2-a^2),x)
--R 2
--R Type: Union(Expression
Integer,...)
--E
+
+--S 15
+bb:=x^2/2+a^2/2*log(x^2-a^2)
+--R
+--R 2 2 2 2
+--R a log(x - a ) + x
+--R (2) -------------------
+--R 2
+--R Type: Expression
Integer
+--E
+
+--S 16 14:147 Schaums and Axiom agree
+cc:=aa-bb
+--R
+--R (3) 0
+--R Type: Expression
Integer
+--E
+
@
\section{\cite{1}:14.148~~~~~$\displaystyle\int{\frac{dx}{x(x^2-a^2)}}$}
@@ -85,7 +199,7 @@ $$
<<*>>=
)clear all
---S 5 of 19
+--S 17
aa:=integrate(1/(x*(x^2-a^2)),x)
--R
--R
@@ -96,6 +210,70 @@ aa:=integrate(1/(x*(x^2-a^2)),x)
--R 2a
--R Type: Union(Expression
Integer,...)
--E
+
+--S 18
+bb:=1/(2*a^2)*log((x^2-a^2)/x^2)
+--R
+--R 2 2
+--R x - a
+--R log(-------)
+--R 2
+--R x
+--R (2) ------------
+--R 2
+--R 2a
+--R Type: Expression
Integer
+--E
+
+--S 19
+cc:=aa-bb
+--R
+--R 2 2
+--R 2 2 x - a
+--R log(x - a ) - 2log(x) - log(-------)
+--R 2
+--R x
+--R (3) -------------------------------------
+--R 2
+--R 2a
+--R Type: Expression
Integer
+--E
+
+--S 20
+divlog:=rule(log(a/b) == log(a) - log(b))
+--R
+--R a
+--R (4) log(-) == - log(b) + log(a)
+--R b
+--R Type: RewriteRule(Integer,Integer,Expression
Integer)
+--E
+
+--S 21
+dd:=divlog cc
+--R
+--R 2
+--R log(x ) - 2log(x)
+--R (5) -----------------
+--R 2
+--R 2a
+--R Type: Expression
Integer
+--E
+
+--S 22
+logpow:=rule(log(a^n) == n*log(a))
+--R
+--R n
+--R (6) log(a ) == n log(a)
+--R Type: RewriteRule(Integer,Integer,Expression
Integer)
+--E
+
+--S 23 14:148 Schaums and Axiom agree
+ee:=logpow dd
+--R
+--R (7) 0
+--R Type: Expression
Integer
+--E
+
@
\section{\cite{1}:14.149~~~~~$\displaystyle\int{\frac{dx}{x^2(x^2-a^2)}}$}
@@ -105,7 +283,7 @@ $$
<<*>>=
)clear all
---S 6 of 19
+--S 24
aa:=integrate(1/(x^2*(x^2-a^2)),x)
--R
--R
@@ -115,6 +293,47 @@ aa:=integrate(1/(x^2*(x^2-a^2)),x)
--R 2a x
--R Type: Union(Expression
Integer,...)
--E
+
+--S 25
+bb:=1/(a^2*x)+1/(2*a^3)*log((x-a)/(x+a))
+--R
+--R x - a
+--R x log(-----) + 2a
+--R x + a
+--R (2) -----------------
+--R 3
+--R 2a x
+--R Type: Expression
Integer
+--E
+
+--S 26
+cc:=aa-bb
+--R
+--R x - a
+--R - log(x + a) + log(x - a) - log(-----)
+--R x + a
+--R (3) --------------------------------------
+--R 3
+--R 2a
+--R Type: Expression
Integer
+--E
+
+--S 27
+divlog:=rule(log(a/b) == log(a) - log(b))
+--R
+--R a
+--R (4) log(-) == - log(b) + log(a)
+--R b
+--R Type: RewriteRule(Integer,Integer,Expression
Integer)
+--E
+
+--S 28 14:149 Schaums and Axiom agree
+dd:=divlog cc
+--R
+--R (5) 0
+--R Type: Expression
Integer
+--E
+
@
\section{\cite{1}:14.150~~~~~$\displaystyle\int{\frac{dx}{x^3(x^2-a^2)}}$}
@@ -124,7 +343,7 @@ $$
<<*>>=
)clear all
---S 7 of 19
+--S 29
aa:=integrate(1/(x^3*(x^2-a^2)),x)
--R
--R
@@ -135,6 +354,73 @@ aa:=integrate(1/(x^3*(x^2-a^2)),x)
--R 2a x
--R Type: Union(Expression
Integer,...)
--E
+
+--S 30
+bb:=1/(2*a^2*x*2)-1/(2*a^4)*log(x^2/(x^2-a^2))
+--R
+--R 2
+--R x 2
+--R - 2x log(-------) + a
+--R 2 2
+--R x - a
+--R (2) ----------------------
+--R 4
+--R 4a x
+--R Type: Expression
Integer
+--E
+
+--S 31
+cc:=aa-bb
+--R
+--R 2
+--R 2 2 2 2 2 x 2 2
+--R 2x log(x - a ) - 4x log(x) + 2x log(-------) - a x + 2a
+--R 2 2
+--R x - a
+--R (3) ---------------------------------------------------------
+--R 4 2
+--R 4a x
+--R Type: Expression
Integer
+--E
+
+--S 32
+divlog:=rule(log(a/b) == log(a) - log(b))
+--R
+--R a
+--R (4) log(-) == - log(b) + log(a)
+--R b
+--R Type: RewriteRule(Integer,Integer,Expression
Integer)
+--E
+
+--S 33
+dd:=divlog cc
+--R
+--R 2 2 2 2 2
+--R 2x log(x ) - 4x log(x) - a x + 2a
+--R (5) ----------------------------------
+--R 4 2
+--R 4a x
+--R Type: Expression
Integer
+--E
+
+--S 34
+logpow:=rule(log(a^n) == n*log(a))
+--R
+--R n
+--R (6) log(a ) == n log(a)
+--R Type: RewriteRule(Integer,Integer,Expression
Integer)
+--E
+
+--S 35 14:150 Schaums and Axiom DISAGREE BY A NON-CONSTANT
+ee:=logpow dd
+--R
+--R - x + 2
+--R (7) -------
+--R 2 2
+--R 4a x
+--R Type: Expression
Integer
+--E
+
@
\section{\cite{1}:14.151~~~~~$\displaystyle\int{\frac{dx}{(x^2-a^2)^2}}$}
@@ -144,7 +430,7 @@ $$
<<*>>=
)clear all
---S 8 of 19
+--S 36
aa:=integrate(1/((x^2-a^2)^2),x)
--R
--R
@@ -155,6 +441,47 @@ aa:=integrate(1/((x^2-a^2)^2),x)
--R 4a x - 4a
--R Type: Union(Expression
Integer,...)
--E
+
+--S 37
+bb:=-x/(2*a^2*(x^2-a^2))-1/(4*a^3)*log((x-a)/(x+a))
+--R
+--R 2 2 x - a
+--R (- x + a )log(-----) - 2a x
+--R x + a
+--R (2) ----------------------------
+--R 3 2 5
+--R 4a x - 4a
+--R Type: Expression
Integer
+--E
+
+--S 38
+cc:=aa-bb
+--R
+--R x - a
+--R log(x + a) - log(x - a) + log(-----)
+--R x + a
+--R (3) ------------------------------------
+--R 3
+--R 4a
+--R Type: Expression
Integer
+--E
+
+--S 39
+divlog:=rule(log(a/b) == log(a) - log(b))
+--R
+--R a
+--R (4) log(-) == - log(b) + log(a)
+--R b
+--R Type: RewriteRule(Integer,Integer,Expression
Integer)
+--E
+
+--S 40 14:151 Schaums and Axiom agree
+dd:=divlog cc
+--R
+--R (5) 0
+--R Type: Expression
Integer
+--E
+
@
\section{\cite{1}:14.152~~~~~$\displaystyle\int{\frac{x~dx}{(x^2-a^2)^2}}$}
@@ -164,7 +491,7 @@ $$
<<*>>=
)clear all
---S 9 of 19
+--S 41
aa:=integrate(x/((x^2-a^2)^2),x)
--R
--R
@@ -174,6 +501,24 @@ aa:=integrate(x/((x^2-a^2)^2),x)
--R 2x - 2a
--R Type: Union(Expression
Integer,...)
--E
+
+--S 42
+bb:=-1/(2*(x^2-a^2))
+--R
+--R 1
+--R (2) - ---------
+--R 2 2
+--R 2x - 2a
+--R Type: Fraction Polynomial
Integer
+--E
+
+--S 43 14:152 Schaums and Axiom agree
+cc:=aa-bb
+--R
+--R (3) 0
+--R Type: Expression
Integer
+--E
+
@
\section{\cite{1}:14.153~~~~~$\displaystyle\int{\frac{x^2dx}{(x^2-a^2)^2}}$}
@@ -183,7 +528,7 @@ $$
<<*>>=
)clear all
---S 10 of 19
+--S 44
aa:=integrate(x^2/((x^2-a^2)^2),x)
--R
--R
@@ -194,6 +539,46 @@ aa:=integrate(x^2/((x^2-a^2)^2),x)
--R 4a x - 4a
--R Type: Union(Expression
Integer,...)
--E
+
+--S 45
+bb:=-x/(2*(x^2-a^2))+1/(4*a)*log((x-a)/(x+a))
+--R
+--R 2 2 x - a
+--R (x - a )log(-----) - 2a x
+--R x + a
+--R (2) --------------------------
+--R 2 3
+--R 4a x - 4a
+--R Type: Expression
Integer
+--E
+
+--S 46
+cc:=aa-bb
+--R
+--R x - a
+--R - log(x + a) + log(x - a) - log(-----)
+--R x + a
+--R (3) --------------------------------------
+--R 4a
+--R Type: Expression
Integer
+--E
+
+--S 47
+divlog:=rule(log(a/b) == log(a) - log(b))
+--R
+--R a
+--R (4) log(-) == - log(b) + log(a)
+--R b
+--R Type: RewriteRule(Integer,Integer,Expression
Integer)
+--E
+
+--S 48 14:153 Schaums and Axiom agree
+dd:=divlog cc
+--R
+--R (5) 0
+--R Type: Expression
Integer
+--E
+
@
\section{\cite{1}:14.154~~~~~$\displaystyle\int{\frac{x^3dx}{(x^2-a^2)^2}}$}
@@ -203,7 +588,7 @@ $$
<<*>>=
)clear all
---S 11 of 19
+--S 49
aa:=integrate(x^3/((x^2-a^2)^2),x)
--R
--R
@@ -214,6 +599,25 @@ aa:=integrate(x^3/((x^2-a^2)^2),x)
--R 2x - 2a
--R Type: Union(Expression
Integer,...)
--E
+
+--S 50
+bb:=-a^2/(2*(x^2-a^2))+1/2*log(x^2-a^2)
+--R
+--R 2 2 2 2 2
+--R (x - a )log(x - a ) - a
+--R (2) --------------------------
+--R 2 2
+--R 2x - 2a
+--R Type: Expression
Integer
+--E
+
+--S 51 14:154 Schaums and Axiom agree
+cc:=aa-bb
+--R
+--R (3) 0
+--R Type: Expression
Integer
+--E
+
@
\section{\cite{1}:14.155~~~~~$\displaystyle\int{\frac{dx}{x(x^2-a^2)^2}}$}
@@ -223,7 +627,7 @@ $$
<<*>>=
)clear all
---S 12 of 19
+--S 52
aa:=integrate(1/(x*(x^2-a^2)^2),x)
--R
--R
@@ -234,6 +638,70 @@ aa:=integrate(1/(x*(x^2-a^2)^2),x)
--R 2a x - 2a
--R Type: Union(Expression
Integer,...)
--E
+
+--S 53
+bb:=-1/(2*a^2*(x^2-a^2))+1/(2*a^4)*log(x^2/(x^2-a^2))
+--R
+--R 2
+--R 2 2 x 2
+--R (x - a )log(-------) - a
+--R 2 2
+--R x - a
+--R (2) --------------------------
+--R 4 2 6
+--R 2a x - 2a
+--R Type: Expression
Integer
+--E
+
+--S 54
+cc:=aa-bb
+--R
+--R 2
+--R 2 2 x
+--R - log(x - a ) + 2log(x) - log(-------)
+--R 2 2
+--R x - a
+--R (3) ---------------------------------------
+--R 4
+--R 2a
+--R Type: Expression
Integer
+--E
+
+--S 55
+divlog:=rule(log(a/b) == log(a) - log(b))
+--R
+--R a
+--R (4) log(-) == - log(b) + log(a)
+--R b
+--R Type: RewriteRule(Integer,Integer,Expression
Integer)
+--E
+
+--S 56
+dd:=divlog cc
+--R
+--R 2
+--R - log(x ) + 2log(x)
+--R (5) -------------------
+--R 4
+--R 2a
+--R Type: Expression
Integer
+--E
+
+--S 57
+logpow:=rule(log(a^n) == n*log(a))
+--R
+--R n
+--R (6) log(a ) == n log(a)
+--R Type: RewriteRule(Integer,Integer,Expression
Integer)
+--E
+
+--S 58 14:155 Schaums and Axiom agree
+ee:=logpow dd
+--R
+--R (7) 0
+--R Type: Expression
Integer
+--E
+
@
\section{\cite{1}:14.156~~~~~$\displaystyle\int{\frac{dx}{x^2(x^2-a^2)^2}}$}
@@ -244,17 +712,57 @@ $$
<<*>>=
)clear all
---S 13 of 19
-aa:=integrate(1/((x^2-a^2)^2),x)
---R
+--S 59
+aa:=integrate(1/(x^2*(x^2-a^2)^2),x)
--R
---R 2 2 2 2
---R (x - a )log(x + a) + (- x + a )log(x - a) - 2a x
---R (1) --------------------------------------------------
---R 3 2 5
---R 4a x - 4a
+--R 3 2 3 2 2 3
+--R (3x - 3a x)log(x + a) + (- 3x + 3a x)log(x - a) - 6a x + 4a
+--R (1) ---------------------------------------------------------------
+--R 5 3 7
+--R 4a x - 4a x
--R Type: Union(Expression
Integer,...)
--E
+
+--S 60
+bb:=-1/(a^4*x)-x/(2*a^4*(x^2-a^2))-3/(4*a^5)*log((x-a)/(x+a))
+--R
+--R 3 2 x - a 2 3
+--R (- 3x + 3a x)log(-----) - 6a x + 4a
+--R x + a
+--R (2) --------------------------------------
+--R 5 3 7
+--R 4a x - 4a x
+--R Type: Expression
Integer
+--E
+
+--S 61
+cc:=aa-bb
+--R
+--R x - a
+--R 3log(x + a) - 3log(x - a) + 3log(-----)
+--R x + a
+--R (3) ---------------------------------------
+--R 5
+--R 4a
+--R Type: Expression
Integer
+--E
+
+--S 62
+divlog:=rule(log(a/b) == log(a) - log(b))
+--R
+--R a
+--R (4) log(-) == - log(b) + log(a)
+--R b
+--R Type: RewriteRule(Integer,Integer,Expression
Integer)
+--E
+
+--S 63 14:156 Schaums and Axiom agree
+dd:=divlog cc
+--R
+--R (5) 0
+--R Type: Expression
Integer
+--E
+
@
\section{\cite{1}:14.157~~~~~$\displaystyle\int{\frac{dx}{x^3(x^2-a^2)^2}}$}
@@ -265,7 +773,7 @@ $$
<<*>>=
)clear all
---S 14 of 19
+--S 64
aa:=integrate(1/(x^3*(x^2-a^2)^2),x)
--R
--R
@@ -276,6 +784,70 @@ aa:=integrate(1/(x^3*(x^2-a^2)^2),x)
--R 2a x - 2a x
--R Type: Union(Expression
Integer,...)
--E
+
+--S 65
+bb:=-1/(2*a^4*x^2)-1/(2*a^4*(x^2-a^2))+1/a^6*log(x^2/(x^2-a^2))
+--R
+--R 2
+--R 4 2 2 x 2 2 4
+--R (2x - 2a x )log(-------) - 2a x + a
+--R 2 2
+--R x - a
+--R (2) --------------------------------------
+--R 6 4 8 2
+--R 2a x - 2a x
+--R Type: Expression
Integer
+--E
+
+--S 66
+cc:=aa-bb
+--R
+--R 2
+--R 2 2 x
+--R - log(x - a ) + 2log(x) - log(-------)
+--R 2 2
+--R x - a
+--R (3) ---------------------------------------
+--R 6
+--R a
+--R Type: Expression
Integer
+--E
+
+--S 67
+divlog:=rule(log(a/b) == log(a) - log(b))
+--R
+--R a
+--R (4) log(-) == - log(b) + log(a)
+--R b
+--R Type: RewriteRule(Integer,Integer,Expression
Integer)
+--E
+
+--S 68
+dd:=divlog cc
+--R
+--R 2
+--R - log(x ) + 2log(x)
+--R (5) -------------------
+--R 6
+--R a
+--R Type: Expression
Integer
+--E
+
+--S 69
+logpow:=rule(log(a^n) == n*log(a))
+--R
+--R n
+--R (6) log(a ) == n log(a)
+--R Type: RewriteRule(Integer,Integer,Expression
Integer)
+--E
+
+--S 70 14:157 Schaums and Axiom agree
+ee:=logpow dd
+--R
+--R (7) 0
+--R Type: Expression
Integer
+--E
+
@
\section{\cite{1}:14.158~~~~~$\displaystyle\int{\frac{dx}{(x^2-a^2)^n}}$}
@@ -286,7 +858,7 @@ $$
<<*>>=
)clear all
---S 15 of 19
+--S 71 14:158 Axiom cannot do this integral
aa:=integrate(1/((x^2-a^2)^n),x)
--R
--R
@@ -306,7 +878,7 @@ $$
<<*>>=
)clear all
---S 16 of 19
+--S 72
aa:=integrate(x/((x^2-a^2)^n),x)
--R
--R
@@ -318,6 +890,49 @@ aa:=integrate(x/((x^2-a^2)^n),x)
--R (2n - 2)%e
--R Type: Union(Expression
Integer,...)
--E
+
+--S 73
+bb:=-1/(2*(n-1)*(x^2-a^2)^(n-1))
+--R
+--R 1
+--R (2) - ----------------------
+--R 2 2 n - 1
+--R (2n - 2)(x - a )
+--R Type: Expression
Integer
+--E
+
+--S 74
+cc:=aa-bb
+--R
+--R 2 2
+--R n log(x - a ) 2 2 2 2 n - 1
+--R %e + (- x + a )(x - a )
+--R (3) --------------------------------------------
+--R 2 2
+--R 2 2 n - 1 n log(x - a )
+--R (2n - 2)(x - a ) %e
+--R Type: Expression
Integer
+--E
+
+--S 75
+explog:=rule(%e^(n*log(x)) == x^n)
+--R
+--R n log(x) n
+--R (4) %e == x
+--R Type: RewriteRule(Integer,Integer,Expression
Integer)
+--E
+
+--S 76 14:159 Axiom cannot simplify this expression
+dd:=explog cc
+--R
+--R 2 2 n 2 2 2 2 n - 1
+--R (x - a ) + (- x + a )(x - a )
+--R (5) --------------------------------------
+--R 2 2 n - 1 2 2 n
+--R (2n - 2)(x - a ) (x - a )
+--R Type: Expression
Integer
+--E
+
@
\section{\cite{1}:14.160~~~~~$\displaystyle\int{\frac{dx}{x(x^2-a^2)^n}}$}
@@ -328,7 +943,7 @@ $$
<<*>>=
)clear all
---S 17 of 19
+--S 77 14:160 Axiom cannot compute this integral
aa:=integrate(1/(x*(x^2-a^2)^n),x)
--R
--R
@@ -349,7 +964,7 @@ $$
<<*>>=
)clear all
---S 18 of 19
+--S 78 14:161 Axiom cannot compute this integral
aa:=integrate(x^m/((x^2-a^2)^n),x)
--R
--R
@@ -370,7 +985,7 @@ $$
<<*>>=
)clear all
---S 19 of 19
+--S 79 14:162 Axiom cannot compute this integral
aa:=integrate(1/(x^m*(x^2-a^2)^n),x)
--R
--R
- [Axiom-developer] 20080416.01.tpd.patch (CATS Schaums-Axiom equivalence testing (2-7)),
daly <=