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[Axiom-developer] 20080419.01.tpd.patch
|
From: |
daly |
|
Subject: |
[Axiom-developer] 20080419.01.tpd.patch |
|
Date: |
Sun, 20 Apr 2008 02:01:32 -0500 |
schaum12.input.pamphlet
14:265 Axiom cannot simplify these expressions
14:266 Axiom cannot simplify these expressions
14:267 Schaums and Axiom differ by a constant
14:268 Axiom cannot compute this integral
14:269 Schaums and Axiom differ by a constant
14:270 Schaums and Axiom differ by a constant
14:271 Axiom cannot compute this integral
14:272 Axiom cannot simplify these expressions
14:273 Schaums and Axiom differ by a constant
14:274 Schaums and Axiom agree
14:275 Axiom cannot compute this integral
14:276 Axiom cannot compute this integral
14:277 Schaums and Axiom differ by a constant
14:278 Schaums and Axiom differ by a constant
14:279 Axiom cannot compute this integral
=========================================================================
diff --git a/changelog b/changelog
index c10550c..d61af40 100644
--- a/changelog
+++ b/changelog
@@ -1,3 +1,4 @@
+20080419 tpd src/input/schaum12.input show Schaums-Axiom equivalence
20080418 tpd src/input/schaum11.input show Schaums-Axiom equivalence
20080418 tpd src/input/schaum10.input fix Schaums-Axiom equivalence
20080418 tpd src/input/schaum9.input fix Schaums-Axiom equivalence
diff --git a/src/input/schaum12.input.pamphlet
b/src/input/schaum12.input.pamphlet
index 644ebed..1c0d0b2 100644
--- a/src/input/schaum12.input.pamphlet
+++ b/src/input/schaum12.input.pamphlet
@@ -24,7 +24,7 @@ $$
)set message auto off
)clear all
---S 1 of 15
+--S 1
aa:=integrate(1/sqrt(a*x^2+b*x+c),x)
--R
--R
@@ -55,6 +55,152 @@ aa:=integrate(1/sqrt(a*x^2+b*x+c),x)
--R \|- a
--R Type: Union(List Expression
Integer,...)
--E
+
+--S 2
+bb1:=2/sqrt(4*a*c-b^2)*atan((2*a*x+b)/sqrt(4*a*c-b^2))
+--R
+--R
+--R 2a x + b
+--R 2atan(------------)
+--R +---------+
+--R | 2
+--R \|4a c - b
+--R (2) -------------------
+--R +---------+
+--R | 2
+--R \|4a c - b
+--R Type: Expression
Integer
+--E
+
+--S 3
+bb2:=1/sqrt(b^2-4*a*c)*log((2*a*x+b-sqrt(b^2-4*a*c))/(2*a*x+b+sqrt(b^2-4*a*c)))
+--R
+--R
+--R +-----------+
+--R | 2
+--R - \|- 4a c + b + 2a x + b
+--R log(---------------------------)
+--R +-----------+
+--R | 2
+--R \|- 4a c + b + 2a x + b
+--R (3) --------------------------------
+--R +-----------+
+--R | 2
+--R \|- 4a c + b
+--R Type: Expression
Integer
+--E
+
+--S 4
+cc1:=aa.1-bb1
+--R
+--R (4)
+--R +---------+
+--R | 2
+--R \|4a c - b
+--R *
+--R log
+--R +--------------+
+--R +-+ +-+ | 2 +-+
+--R (2\|a \|c - 2a x)\|a x + b x + c + 2a x\|c
+--R +
+--R 2 +-+
+--R (- 2a x - b x - 2c)\|a
+--R /
+--R +--------------+
+--R +-+ | 2
+--R 2\|c \|a x + b x + c - b x - 2c
+--R +
+--R +-+ 2a x + b
+--R - 2\|a atan(------------)
+--R +---------+
+--R | 2
+--R \|4a c - b
+--R /
+--R +---------+
+--R +-+ | 2
+--R \|a \|4a c - b
+--R Type: Expression
Integer
+--E
+
+--S 5
+cc2:=aa.1-bb2
+--R
+--R (5)
+--R +-----------+
+--R | 2
+--R \|- 4a c + b
+--R *
+--R log
+--R +--------------+
+--R +-+ +-+ | 2 +-+
+--R (2\|a \|c - 2a x)\|a x + b x + c + 2a x\|c
+--R +
+--R 2 +-+
+--R (- 2a x - b x - 2c)\|a
+--R /
+--R +--------------+
+--R +-+ | 2
+--R 2\|c \|a x + b x + c - b x - 2c
+--R +
+--R +-----------+
+--R | 2
+--R +-+ - \|- 4a c + b + 2a x + b
+--R - \|a log(---------------------------)
+--R +-----------+
+--R | 2
+--R \|- 4a c + b + 2a x + b
+--R /
+--R +-----------+
+--R | 2 +-+
+--R \|- 4a c + b \|a
+--R Type: Expression
Integer
+--E
+
+--S 6
+cc3:=aa.2-bb1
+--R
+--R (6)
+--R +--------------+
+--R +---------+ +---+ | 2 +---+ +-+
+--R | 2 \|- a \|a x + b x + c - \|- a \|c
+--R 2\|4a c - b atan(------------------------------------)
+--R a x
+--R +
+--R +---+ 2a x + b
+--R - 2\|- a atan(------------)
+--R +---------+
+--R | 2
+--R \|4a c - b
+--R /
+--R +---------+
+--R +---+ | 2
+--R \|- a \|4a c - b
+--R Type: Expression
Integer
+--E
+
+--S 7 14:265 Axiom cannot simplify these expressions
+cc4:=aa.2-bb2
+--R
+--R (7)
+--R +-----------+
+--R | 2
+--R +---+ - \|- 4a c + b + 2a x + b
+--R - \|- a log(---------------------------)
+--R +-----------+
+--R | 2
+--R \|- 4a c + b + 2a x + b
+--R +
+--R +--------------+
+--R +-----------+ +---+ | 2 +---+ +-+
+--R | 2 \|- a \|a x + b x + c - \|- a \|c
+--R 2\|- 4a c + b atan(------------------------------------)
+--R a x
+--R /
+--R +-----------+
+--R | 2 +---+
+--R \|- 4a c + b \|- a
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.266~~~~~$\displaystyle
@@ -65,7 +211,7 @@ $$
<<*>>=
)clear all
---S 2 of 15
+--S 8
aa:=integrate(x/sqrt(a*x^2+b*x+c),x)
--R
--R
@@ -116,6 +262,316 @@ aa:=integrate(x/sqrt(a*x^2+b*x+c),x)
--R ]
--R Type: Union(List Expression
Integer,...)
--E
+
+--S 9
+t1:=integrate(1/(a*x^2+b*x+c),x)
+--R
+--R
+--R (2)
+--R [
+--R log
+--R +-----------+
+--R 2 2 2 | 2 2 2
+--R (2a x + 2a b x - 2a c + b )\|- 4a c + b + (8a c - 2a b )x
+--R +
+--R 3
+--R 4a b c - b
+--R /
+--R 2
+--R a x + b x + c
+--R /
+--R +-----------+
+--R | 2
+--R \|- 4a c + b
+--R ,
+--R +---------+
+--R | 2
+--R (2a x + b)\|4a c - b
+--R 2atan(----------------------)
+--R 2
+--R 4a c - b
+--R -----------------------------]
+--R +---------+
+--R | 2
+--R \|4a c - b
+--R Type: Union(List Expression
Integer,...)
+--E
+
+--S 10
+bb1:=1/(2*a)*log(a*x^2+b*x+c)-b/(2*a)*t1.1
+--R
+--R
+--R (3)
+--R -
+--R b
+--R *
+--R log
+--R +-----------+
+--R 2 2 2 | 2 2
2
+--R (2a x + 2a b x - 2a c + b )\|- 4a c + b + (8a c - 2a
b )x
+--R +
+--R 3
+--R 4a b c - b
+--R /
+--R 2
+--R a x + b x + c
+--R +
+--R +-----------+
+--R 2 | 2
+--R log(a x + b x + c)\|- 4a c + b
+--R /
+--R +-----------+
+--R | 2
+--R 2a\|- 4a c + b
+--R Type: Expression
Integer
+--E
+
+--S 11
+bb2:=1/(2*a)*log(a*x^2+b*x+c)-b/(2*a)*t1.2
+--R
+--R
+--R +---------+
+--R | 2 +---------+
+--R (2a x + b)\|4a c - b 2 | 2
+--R - 2b atan(----------------------) + log(a x + b x + c)\|4a c - b
+--R 2
+--R 4a c - b
+--R (4) -------------------------------------------------------------------
+--R +---------+
+--R | 2
+--R 2a\|4a c - b
+--R Type: Expression
Integer
+--E
+
+--S 12
+cc1:=aa.1-bb1
+--R
+--R (5)
+--R +-----------+ +--------------+
+-----------+
+--R | 2 +-+ | 2 2 |
2
+--R (2b\|- 4a c + b \|c \|a x + b x + c + (- b x - 2b c)\|- 4a c +
b )
+--R *
+--R log
+--R +--------------+
+--R +-+ +-+ | 2 +-+
+--R (2\|a \|c + 2a x)\|a x + b x + c - 2a x\|c
+--R +
+--R 2 +-+
+--R (- 2a x - b x - 2c)\|a
+--R /
+--R +--------------+
+--R +-+ | 2
+--R 2\|c \|a x + b x + c - b x - 2c
+--R +
+--R +--------------+
+--R +-+ +-+ | 2 2 +-+
+--R (2b\|a \|c \|a x + b x + c + (- b x - 2b c)\|a )
+--R *
+--R log
+--R +-----------+
+--R 2 2 2 | 2 2 2
+--R (2a x + 2a b x - 2a c + b )\|- 4a c + b + (8a c - 2a b )x
+--R +
+--R 3
+--R 4a b c - b
+--R /
+--R 2
+--R a x + b x + c
+--R +
+--R +-----------+ +-----------+
+--R 2 | 2 +-+ +-+ | 2
+-+
+--R (- 2log(a x + b x + c)\|- 4a c + b \|a \|c - 2b x\|- 4a c + b
\|a )
+--R *
+--R +--------------+
+--R | 2
+--R \|a x + b x + c
+--R +
+--R +-----------+
+--R 2 | 2 +-+ +-+
+--R (4a x + 2b x)\|- 4a c + b \|a \|c
+--R +
+--R +-----------+
+--R 2 | 2 +-+
+--R (b x + 2c)log(a x + b x + c)\|- 4a c + b \|a
+--R /
+--R +-----------+ +--------------+
+--R | 2 +-+ +-+ | 2
+--R 4a\|- 4a c + b \|a \|c \|a x + b x + c
+--R +
+--R +-----------+
+--R | 2 +-+
+--R (- 2a b x - 4a c)\|- 4a c + b \|a
+--R Type: Expression
Integer
+--E
+
+--S 13
+cc2:=aa.2-bb1
+--R
+--R (6)
+--R +--------------+
+--R +---+ +-+ | 2 2 +---+
+--R (2b\|- a \|c \|a x + b x + c + (- b x - 2b c)\|- a )
+--R *
+--R log
+--R +-----------+
+--R 2 2 2 | 2 2 2
+--R (2a x + 2a b x - 2a c + b )\|- 4a c + b + (8a c - 2a b )x
+--R +
+--R 3
+--R 4a b c - b
+--R /
+--R 2
+--R a x + b x + c
+--R +
+--R +-----------+ +--------------+
+-----------+
+--R | 2 +-+ | 2 2 |
2
+--R (- 4b\|- 4a c + b \|c \|a x + b x + c + (2b x + 4b c)\|- 4a c +
b )
+--R *
+--R +--------------+
+--R +---+ | 2 +---+ +-+
+--R \|- a \|a x + b x + c - \|- a \|c
+--R atan(------------------------------------)
+--R a x
+--R +
+--R +-----------+
+--R 2 | 2 +---+ +-+
+--R - 2log(a x + b x + c)\|- 4a c + b \|- a \|c
+--R +
+--R +-----------+
+--R | 2 +---+
+--R - 2b x\|- 4a c + b \|- a
+--R *
+--R +--------------+
+--R | 2
+--R \|a x + b x + c
+--R +
+--R +-----------+
+--R 2 | 2 +---+ +-+
+--R (4a x + 2b x)\|- 4a c + b \|- a \|c
+--R +
+--R +-----------+
+--R 2 | 2 +---+
+--R (b x + 2c)log(a x + b x + c)\|- 4a c + b \|- a
+--R /
+--R +-----------+ +--------------+
+--R | 2 +---+ +-+ | 2
+--R 4a\|- 4a c + b \|- a \|c \|a x + b x + c
+--R +
+--R +-----------+
+--R | 2 +---+
+--R (- 2a b x - 4a c)\|- 4a c + b \|- a
+--R Type: Expression
Integer
+--E
+
+--S 14
+cc3:=aa.2-bb1
+--R
+--R (7)
+--R +--------------+
+--R +---+ +-+ | 2 2 +---+
+--R (2b\|- a \|c \|a x + b x + c + (- b x - 2b c)\|- a )
+--R *
+--R log
+--R +-----------+
+--R 2 2 2 | 2 2 2
+--R (2a x + 2a b x - 2a c + b )\|- 4a c + b + (8a c - 2a b )x
+--R +
+--R 3
+--R 4a b c - b
+--R /
+--R 2
+--R a x + b x + c
+--R +
+--R +-----------+ +--------------+
+-----------+
+--R | 2 +-+ | 2 2 |
2
+--R (- 4b\|- 4a c + b \|c \|a x + b x + c + (2b x + 4b c)\|- 4a c +
b )
+--R *
+--R +--------------+
+--R +---+ | 2 +---+ +-+
+--R \|- a \|a x + b x + c - \|- a \|c
+--R atan(------------------------------------)
+--R a x
+--R +
+--R +-----------+
+--R 2 | 2 +---+ +-+
+--R - 2log(a x + b x + c)\|- 4a c + b \|- a \|c
+--R +
+--R +-----------+
+--R | 2 +---+
+--R - 2b x\|- 4a c + b \|- a
+--R *
+--R +--------------+
+--R | 2
+--R \|a x + b x + c
+--R +
+--R +-----------+
+--R 2 | 2 +---+ +-+
+--R (4a x + 2b x)\|- 4a c + b \|- a \|c
+--R +
+--R +-----------+
+--R 2 | 2 +---+
+--R (b x + 2c)log(a x + b x + c)\|- 4a c + b \|- a
+--R /
+--R +-----------+ +--------------+
+--R | 2 +---+ +-+ | 2
+--R 4a\|- 4a c + b \|- a \|c \|a x + b x + c
+--R +
+--R +-----------+
+--R | 2 +---+
+--R (- 2a b x - 4a c)\|- 4a c + b \|- a
+--R Type: Expression
Integer
+--E
+
+--S 15 14:266 Axiom cannot simplify these expressions
+cc4:=aa.2-bb2
+--R
+--R (8)
+--R +---------+ +--------------+ +---------+
+--R +-+ | 2 | 2 2 | 2
+--R (- 4b\|c \|4a c - b \|a x + b x + c + (2b x + 4b c)\|4a c - b )
+--R *
+--R +--------------+
+--R +---+ | 2 +---+ +-+
+--R \|- a \|a x + b x + c - \|- a \|c
+--R atan(------------------------------------)
+--R a x
+--R +
+--R +--------------+
+--R +---+ +-+ | 2 2 +---+
+--R (4b\|- a \|c \|a x + b x + c + (- 2b x - 4b c)\|- a )
+--R *
+--R +---------+
+--R | 2
+--R (2a x + b)\|4a c - b
+--R atan(----------------------)
+--R 2
+--R 4a c - b
+--R +
+--R +---------+
+--R 2 +---+ +-+ +---+ | 2
+--R (- 2log(a x + b x + c)\|- a \|c - 2b x\|- a )\|4a c - b
+--R *
+--R +--------------+
+--R | 2
+--R \|a x + b x + c
+--R +
+--R 2 +---+ +-+ 2 +---+
+--R ((4a x + 2b x)\|- a \|c + (b x + 2c)log(a x + b x + c)\|- a )
+--R *
+--R +---------+
+--R | 2
+--R \|4a c - b
+--R /
+--R +---------+ +--------------+
+--R +---+ +-+ | 2 | 2
+--R 4a\|- a \|c \|4a c - b \|a x + b x + c
+--R +
+--R +---------+
+--R +---+ | 2
+--R (- 2a b x - 4a c)\|- a \|4a c - b
+--R Type: Expression
Integer
+--E
+
@
\section{\cite{1}:14.267~~~~~$\displaystyle
@@ -127,7 +583,7 @@ $$
<<*>>=
)clear all
---S 3 of 15
+--S 16
aa:=integrate(x^2/(a*x^2+b*x+c),x)
--R
--R
@@ -173,6 +629,141 @@ aa:=integrate(x^2/(a*x^2+b*x+c),x)
--R ]
--R Type: Union(List Expression
Integer,...)
--E
+
+--S 17
+t1:=integrate(1/(a*x^2+b*x+c),x)
+--R
+--R
+--R (2)
+--R [
+--R log
+--R +-----------+
+--R 2 2 2 | 2 2 2
+--R (2a x + 2a b x - 2a c + b )\|- 4a c + b + (8a c - 2a b )x
+--R +
+--R 3
+--R 4a b c - b
+--R /
+--R 2
+--R a x + b x + c
+--R /
+--R +-----------+
+--R | 2
+--R \|- 4a c + b
+--R ,
+--R +---------+
+--R | 2
+--R (2a x + b)\|4a c - b
+--R 2atan(----------------------)
+--R 2
+--R 4a c - b
+--R -----------------------------]
+--R +---------+
+--R | 2
+--R \|4a c - b
+--R Type: Union(List Expression
Integer,...)
+--E
+
+--S 18
+bb1:=x/a-b/(2*a^2)*log(a*x^2+b*x+c)+(b^2-2*a*c)/(2*a^2)*t1.1
+--R
+--R
+--R (3)
+--R 2
+--R (- 2a c + b )
+--R *
+--R log
+--R +-----------+
+--R 2 2 2 | 2 2 2
+--R (2a x + 2a b x - 2a c + b )\|- 4a c + b + (8a c - 2a b )x
+--R +
+--R 3
+--R 4a b c - b
+--R /
+--R 2
+--R a x + b x + c
+--R +
+--R +-----------+
+--R 2 | 2
+--R (- b log(a x + b x + c) + 2a x)\|- 4a c + b
+--R /
+--R +-----------+
+--R 2 | 2
+--R 2a \|- 4a c + b
+--R Type: Expression
Integer
+--E
+
+--S 19
+bb2:=x/a-b/(2*a^2)*log(a*x^2+b*x+c)+(b^2-2*a*c)/(2*a^2)*t1.2
+--R
+--R
+--R (4)
+--R +---------+
+--R | 2
+--R 2 (2a x + b)\|4a c - b
+--R (- 4a c + 2b )atan(----------------------)
+--R 2
+--R 4a c - b
+--R +
+--R +---------+
+--R 2 | 2
+--R (- b log(a x + b x + c) + 2a x)\|4a c - b
+--R /
+--R +---------+
+--R 2 | 2
+--R 2a \|4a c - b
+--R Type: Expression
Integer
+--E
+
+--S 20
+cc1:=bb1-aa.1
+--R
+--R (5)
+--R 2
+--R (- 2a c + b )
+--R *
+--R log
+--R +-----------+
+--R 2 2 2 | 2 2 2
+--R (2a x + 2a b x - 2a c + b )\|- 4a c + b + (8a c - 2a b )x
+--R +
+--R 3
+--R 4a b c - b
+--R /
+--R 2
+--R a x + b x + c
+--R +
+--R 2
+--R (- 2a c + b )
+--R *
+--R log
+--R +-----------+
+--R 2 2 2 | 2 2 2
+--R (2a x + 2a b x - 2a c + b )\|- 4a c + b + (- 8a c + 2a b
)x
+--R +
+--R 3
+--R - 4a b c + b
+--R /
+--R 2
+--R a x + b x + c
+--R /
+--R +-----------+
+--R 2 | 2
+--R 2a \|- 4a c + b
+--R Type: Expression
Integer
+--E
+
+--S 21 14:267 Schaums and Axiom differ by a constant
+dd1:=complexNormalize cc1
+--R
+--R 2 3 2 2
+--R (- 2a c + b )log(- 16a c + 4a b )
+--R (6) ---------------------------------
+--R +-----------+
+--R 2 | 2
+--R 2a \|- 4a c + b
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.268~~~~~$\displaystyle
@@ -184,7 +775,7 @@ $$
<<*>>=
)clear all
---S 4 of 15
+--S 22 14:268 Axiom cannot compute this integral
aa:=integrate(x^m/(a*x^2+b*x+c),x)
--R
--R
@@ -195,6 +786,7 @@ aa:=integrate(x^m/(a*x^2+b*x+c),x)
--I c + %N b + %N a
--R Type: Union(Expression
Integer,...)
--E
+
@
\section{\cite{1}:14.269~~~~~$\displaystyle
@@ -206,7 +798,7 @@ $$
<<*>>=
)clear all
---S 5 of 15
+--S 23
aa:=integrate(1/(x*(a*x^2+b*x+c)),x)
--R
--R
@@ -251,6 +843,179 @@ aa:=integrate(1/(x*(a*x^2+b*x+c)),x)
--R ]
--R Type: Union(List Expression
Integer,...)
--E
+
+--S 24
+t1:=integrate(1/(a*x^2+b*x+c),x)
+--R
+--R
+--R (2)
+--R [
+--R log
+--R +-----------+
+--R 2 2 2 | 2 2 2
+--R (2a x + 2a b x - 2a c + b )\|- 4a c + b + (8a c - 2a b )x
+--R +
+--R 3
+--R 4a b c - b
+--R /
+--R 2
+--R a x + b x + c
+--R /
+--R +-----------+
+--R | 2
+--R \|- 4a c + b
+--R ,
+--R +---------+
+--R | 2
+--R (2a x + b)\|4a c - b
+--R 2atan(----------------------)
+--R 2
+--R 4a c - b
+--R -----------------------------]
+--R +---------+
+--R | 2
+--R \|4a c - b
+--R Type: Union(List Expression
Integer,...)
+--E
+
+--S 25
+bb1:=1/(2*c)*log(x^2/(a*x^2+b*x+c))-b/(2*c)*t1.1
+--R
+--R
+--R (3)
+--R -
+--R b
+--R *
+--R log
+--R +-----------+
+--R 2 2 2 | 2 2
2
+--R (2a x + 2a b x - 2a c + b )\|- 4a c + b + (8a c - 2a
b )x
+--R +
+--R 3
+--R 4a b c - b
+--R /
+--R 2
+--R a x + b x + c
+--R +
+--R 2 +-----------+
+--R x | 2
+--R log(--------------)\|- 4a c + b
+--R 2
+--R a x + b x + c
+--R /
+--R +-----------+
+--R | 2
+--R 2c\|- 4a c + b
+--R Type: Expression
Integer
+--E
+
+--S 26
+bb2:=1/(2*c)*log(x^2/(a*x^2+b*x+c))-b/(2*c)*t1.2
+--R
+--R
+--R +---------+
+--R | 2 2 +---------+
+--R (2a x + b)\|4a c - b x | 2
+--R - 2b atan(----------------------) + log(--------------)\|4a c - b
+--R 2 2
+--R 4a c - b a x + b x + c
+--R (4) -------------------------------------------------------------------
+--R +---------+
+--R | 2
+--R 2c\|4a c - b
+--R Type: Expression
Integer
+--E
+
+--S 27
+cc1:=bb1-aa.1
+--R
+--R (5)
+--R -
+--R b
+--R *
+--R log
+--R +-----------+
+--R 2 2 2 | 2 2
2
+--R (2a x + 2a b x - 2a c + b )\|- 4a c + b + (8a c - 2a
b )x
+--R +
+--R 3
+--R 4a b c - b
+--R /
+--R 2
+--R a x + b x + c
+--R +
+--R -
+--R b
+--R *
+--R log
+--R +-----------+
+--R 2 2 2 | 2
+--R (2a x + 2a b x - 2a c + b )\|- 4a c + b
+--R +
+--R 2 2 3
+--R (- 8a c + 2a b )x - 4a b c + b
+--R /
+--R 2
+--R a x + b x + c
+--R +
+--R 2 +-----------+
+--R 2 x | 2
+--R (log(a x + b x + c) - 2log(x) + log(--------------))\|- 4a c + b
+--R 2
+--R a x + b x + c
+--R /
+--R +-----------+
+--R | 2
+--R 2c\|- 4a c + b
+--R Type: Expression
Integer
+--E
+
+--S 28
+dd1:=expandLog cc1
+--R
+--R (6)
+--R -
+--R b
+--R *
+--R log
+--R +-----------+
+--R 2 2 2 | 2 2 2
+--R (2a x + 2a b x - 2a c + b )\|- 4a c + b + (8a c - 2a b
)x
+--R +
+--R 3
+--R 4a b c - b
+--R +
+--R -
+--R b
+--R *
+--R log
+--R +-----------+
+--R 2 2 2 | 2 2
2
+--R (2a x + 2a b x - 2a c + b )\|- 4a c + b + (- 8a c + 2a
b )x
+--R +
+--R 3
+--R - 4a b c + b
+--R +
+--R 2
+--R 2b log(a x + b x + c)
+--R /
+--R +-----------+
+--R | 2
+--R 2c\|- 4a c + b
+--R Type: Expression
Integer
+--E
+
+--S 29 14:269 Schaums and Axiom differ by a constant
+ee1:=complexNormalize dd1
+--R
+--R 3 2 2
+--R b log(- 16a c + 4a b )
+--R (7) - ----------------------
+--R +-----------+
+--R | 2
+--R 2c\|- 4a c + b
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.270~~~~~$\displaystyle
@@ -262,7 +1027,7 @@ $$
<<*>>=
)clear all
---S 6 of 15
+--S 30
aa:=integrate(1/(x^2*(a*x^2+b*x+c)),x)
--R
--R
@@ -308,6 +1073,186 @@ aa:=integrate(1/(x^2*(a*x^2+b*x+c)),x)
--R ]
--R Type: Union(List Expression
Integer,...)
--E
+
+--S 31
+t1:=integrate(1/(a*x^2+b*x+c),x)
+--R
+--R
+--R (2)
+--R [
+--R log
+--R +-----------+
+--R 2 2 2 | 2 2 2
+--R (2a x + 2a b x - 2a c + b )\|- 4a c + b + (8a c - 2a b )x
+--R +
+--R 3
+--R 4a b c - b
+--R /
+--R 2
+--R a x + b x + c
+--R /
+--R +-----------+
+--R | 2
+--R \|- 4a c + b
+--R ,
+--R +---------+
+--R | 2
+--R (2a x + b)\|4a c - b
+--R 2atan(----------------------)
+--R 2
+--R 4a c - b
+--R -----------------------------]
+--R +---------+
+--R | 2
+--R \|4a c - b
+--R Type: Union(List Expression
Integer,...)
+--E
+
+--S 32
+bb1:=b/(2*c^2)*log((a*x^2+b*x+c)/x^2)-1/(c*x)+(b^2-2*a*c)/(2*c^2)*t1.1
+--R
+--R
+--R (3)
+--R 2
+--R (- 2a c + b )x
+--R *
+--R log
+--R +-----------+
+--R 2 2 2 | 2 2 2
+--R (2a x + 2a b x - 2a c + b )\|- 4a c + b + (8a c - 2a b )x
+--R +
+--R 3
+--R 4a b c - b
+--R /
+--R 2
+--R a x + b x + c
+--R +
+--R 2 +-----------+
+--R a x + b x + c | 2
+--R (b x log(--------------) - 2c)\|- 4a c + b
+--R 2
+--R x
+--R /
+--R +-----------+
+--R 2 | 2
+--R 2c x\|- 4a c + b
+--R Type: Expression
Integer
+--E
+
+--S 33
+bb2:=b/(2*c^2)*log((a*x^2+b*x+c)/x^2)-1/(c*x)+(b^2-2*a*c)/(2*c^2)*t1.2
+--R
+--R
+--R (4)
+--R +---------+
+--R | 2
+--R 2 (2a x + b)\|4a c - b
+--R (- 4a c + 2b )x atan(----------------------)
+--R 2
+--R 4a c - b
+--R +
+--R 2 +---------+
+--R a x + b x + c | 2
+--R (b x log(--------------) - 2c)\|4a c - b
+--R 2
+--R x
+--R /
+--R +---------+
+--R 2 | 2
+--R 2c x\|4a c - b
+--R Type: Expression
Integer
+--E
+
+--S 34
+cc1:=bb1-aa.1
+--R
+--R (5)
+--R 2
+--R (- 2a c + b )
+--R *
+--R log
+--R +-----------+
+--R 2 2 2 | 2 2 2
+--R (2a x + 2a b x - 2a c + b )\|- 4a c + b + (8a c - 2a b )x
+--R +
+--R 3
+--R 4a b c - b
+--R /
+--R 2
+--R a x + b x + c
+--R +
+--R 2
+--R (- 2a c + b )
+--R *
+--R log
+--R +-----------+
+--R 2 2 2 | 2 2 2
+--R (2a x + 2a b x - 2a c + b )\|- 4a c + b + (- 8a c + 2a b
)x
+--R +
+--R 3
+--R - 4a b c + b
+--R /
+--R 2
+--R a x + b x + c
+--R +
+--R 2
+-----------+
+--R 2 a x + b x + c |
2
+--R (- b log(a x + b x + c) + 2b log(x) + b log(--------------))\|- 4a c
+ b
+--R 2
+--R x
+--R /
+--R +-----------+
+--R 2 | 2
+--R 2c \|- 4a c + b
+--R Type: Expression
Integer
+--E
+
+--S 35
+dd1:=expandLog cc1
+--R
+--R (6)
+--R 2
+--R (- 2a c + b )
+--R *
+--R log
+--R +-----------+
+--R 2 2 2 | 2 2 2
+--R (2a x + 2a b x - 2a c + b )\|- 4a c + b + (8a c - 2a b )x
+--R +
+--R 3
+--R 4a b c - b
+--R +
+--R 2
+--R (- 2a c + b )
+--R *
+--R log
+--R +-----------+
+--R 2 2 2 | 2 2 2
+--R (2a x + 2a b x - 2a c + b )\|- 4a c + b + (- 8a c + 2a b )x
+--R +
+--R 3
+--R - 4a b c + b
+--R +
+--R 2 2
+--R (4a c - 2b )log(a x + b x + c)
+--R /
+--R +-----------+
+--R 2 | 2
+--R 2c \|- 4a c + b
+--R Type: Expression
Integer
+--E
+
+--S 36 14:270 Schaums and Axiom differ by a constant
+ee1:=complexNormalize dd1
+--R
+--R 2 3 2 2
+--R (- 2a c + b )log(- 16a c + 4a b )
+--R (7) ---------------------------------
+--R +-----------+
+--R 2 | 2
+--R 2c \|- 4a c + b
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.271~~~~~$\displaystyle
@@ -320,7 +1265,7 @@ $$
<<*>>=
)clear all
---S 7 of 15
+--S 37 14:271 Axiom cannot compute this integral
aa:=integrate(1/(x^n*(a*x^2+b*x+c)),x)
--R
--R
@@ -331,6 +1276,7 @@ aa:=integrate(1/(x^n*(a*x^2+b*x+c)),x)
--I (c + %N b + %N a)%N
--R Type: Union(Expression
Integer,...)
--E
+
@
\section{\cite{1}:14.272~~~~~$\displaystyle
@@ -342,7 +1288,7 @@ $$
<<*>>=
)clear all
---S 8 of 15
+--S 38
aa:=integrate(1/(a*x^2+b*x+c)^2,x)
--R
--R
@@ -383,6 +1329,213 @@ aa:=integrate(1/(a*x^2+b*x+c)^2,x)
--R ]
--R Type: Union(List Expression
Integer,...)
--E
+
+--S 39
+t1:=integrate(1/(a*x^2+b*x+c),x)
+--R
+--R
+--R (2)
+--R [
+--R log
+--R +-----------+
+--R 2 2 2 | 2 2 2
+--R (2a x + 2a b x - 2a c + b )\|- 4a c + b + (8a c - 2a b )x
+--R +
+--R 3
+--R 4a b c - b
+--R /
+--R 2
+--R a x + b x + c
+--R /
+--R +-----------+
+--R | 2
+--R \|- 4a c + b
+--R ,
+--R +---------+
+--R | 2
+--R (2a x + b)\|4a c - b
+--R 2atan(----------------------)
+--R 2
+--R 4a c - b
+--R -----------------------------]
+--R +---------+
+--R | 2
+--R \|4a c - b
+--R Type: Union(List Expression
Integer,...)
+--E
+
+--S 40
+bb1:=(b*x+2*c)/((4*a*c-b^2)*(a*x^2+b*x+c))-b/(4*a*c-b^2)*t1.1
+--R
+--R
+--R (3)
+--R 2 2
+--R (- a b x - b x - b c)
+--R *
+--R log
+--R +-----------+
+--R 2 2 2 | 2 2 2
+--R (2a x + 2a b x - 2a c + b )\|- 4a c + b + (8a c - 2a b )x
+--R +
+--R 3
+--R 4a b c - b
+--R /
+--R 2
+--R a x + b x + c
+--R +
+--R +-----------+
+--R | 2
+--R (b x + 2c)\|- 4a c + b
+--R /
+--R +-----------+
+--R 2 2 2 3 2 2 | 2
+--R ((4a c - a b )x + (4a b c - b )x + 4a c - b c)\|- 4a c + b
+--R Type: Expression
Integer
+--E
+
+--S 41
+bb2:=(b*x+2*c)/((4*a*c-b^2)*(a*x^2+b*x+c))-b/(4*a*c-b^2)*t1.2
+--R
+--R
+--R (4)
+--R +---------+
+--R | 2
+--R 2 2 (2a x + b)\|4a c - b
+--R (- 2a b x - 2b x - 2b c)atan(----------------------)
+--R 2
+--R 4a c - b
+--R +
+--R +---------+
+--R | 2
+--R (b x + 2c)\|4a c - b
+--R /
+--R +---------+
+--R 2 2 2 3 2 2 | 2
+--R ((4a c - a b )x + (4a b c - b )x + 4a c - b c)\|4a c - b
+--R Type: Expression
Integer
+--E
+
+--S 42
+cc1:=aa.1-bb1
+--R
+--R (5)
+--R 2 2 2
+--R ((a b + 2a )x + (b + 2a b)x + (b + 2a)c)
+--R *
+--R log
+--R +-----------+
+--R 2 2 2 | 2 2 2
+--R (2a x + 2a b x - 2a c + b )\|- 4a c + b + (8a c - 2a b )x
+--R +
+--R 3
+--R 4a b c - b
+--R /
+--R 2
+--R a x + b x + c
+--R +
+--R +-----------+
+--R | 2
+--R ((- b + 2a)x - 2c + b)\|- 4a c + b
+--R /
+--R +-----------+
+--R 2 2 2 3 2 2 | 2
+--R ((4a c - a b )x + (4a b c - b )x + 4a c - b c)\|- 4a c + b
+--R Type: Expression
Integer
+--E
+
+--S 43
+cc2:=aa.2-bb1
+--R
+--R (6)
+--R +---------+
+--R 2 2 | 2
+--R (a b x + b x + b c)\|4a c - b
+--R *
+--R log
+--R +-----------+
+--R 2 2 2 | 2 2 2
+--R (2a x + 2a b x - 2a c + b )\|- 4a c + b + (8a c - 2a b )x
+--R +
+--R 3
+--R 4a b c - b
+--R /
+--R 2
+--R a x + b x + c
+--R +
+--R +---------+
+--R +-----------+ | 2
+--R 2 2 | 2 (2a x + b)\|4a c - b
+--R (4a x + 4a b x + 4a c)\|- 4a c + b atan(----------------------)
+--R 2
+--R 4a c - b
+--R +
+--R +-----------+ +---------+
+--R | 2 | 2
+--R ((- b + 2a)x - 2c + b)\|- 4a c + b \|4a c - b
+--R /
+--R +-----------+
+---------+
+--R 2 2 2 3 2 2 | 2 |
2
+--R ((4a c - a b )x + (4a b c - b )x + 4a c - b c)\|- 4a c + b \|4a c -
b
+--R Type: Expression
Integer
+--E
+
+--S 44
+cc3:=aa.1-bb2
+--R
+--R (7)
+--R +---------+
+--R 2 2 | 2
+--R (2a x + 2a b x + 2a c)\|4a c - b
+--R *
+--R log
+--R +-----------+
+--R 2 2 2 | 2 2 2
+--R (2a x + 2a b x - 2a c + b )\|- 4a c + b + (8a c - 2a b )x
+--R +
+--R 3
+--R 4a b c - b
+--R /
+--R 2
+--R a x + b x + c
+--R +
+--R +---------+
+--R +-----------+ | 2
+--R 2 2 | 2 (2a x + b)\|4a c - b
+--R (2a b x + 2b x + 2b c)\|- 4a c + b atan(----------------------)
+--R 2
+--R 4a c - b
+--R +
+--R +-----------+ +---------+
+--R | 2 | 2
+--R ((- b + 2a)x - 2c + b)\|- 4a c + b \|4a c - b
+--R /
+--R +-----------+
+---------+
+--R 2 2 2 3 2 2 | 2 |
2
+--R ((4a c - a b )x + (4a b c - b )x + 4a c - b c)\|- 4a c + b \|4a c -
b
+--R Type: Expression
Integer
+--E
+
+--S 45 14:272 Axiom cannot simplify these expressions
+cc4:=aa.2-bb2
+--R
+--R (8)
+--R
+---------+
+--R |
2
+--R 2 2 2 (2a x + b)\|4a c -
b
+--R ((2a b + 4a )x + (2b + 4a b)x + (2b +
4a)c)atan(----------------------)
+--R 2
+--R 4a c - b
+--R +
+--R +---------+
+--R | 2
+--R ((- b + 2a)x - 2c + b)\|4a c - b
+--R /
+--R +---------+
+--R 2 2 2 3 2 2 | 2
+--R ((4a c - a b )x + (4a b c - b )x + 4a c - b c)\|4a c - b
+--R Type: Expression
Integer
+--E
+
@
\section{\cite{1}:14.273~~~~~$\displaystyle
@@ -394,7 +1547,7 @@ $$
<<*>>=
)clear all
---S 9 of 15
+--S 46
aa:=integrate(x/(a*x^2+b*x+c)^2,x)
--R
--R
@@ -440,6 +1593,177 @@ aa:=integrate(x/(a*x^2+b*x+c)^2,x)
--R ]
--R Type: Union(List Expression
Integer,...)
--E
+
+--S 47
+t1:=integrate(1/(a*x^2+b*x+c),x)
+--R
+--R
+--R (2)
+--R [
+--R log
+--R +-----------+
+--R 2 2 2 | 2 2 2
+--R (2a x + 2a b x - 2a c + b )\|- 4a c + b + (8a c - 2a b )x
+--R +
+--R 3
+--R 4a b c - b
+--R /
+--R 2
+--R a x + b x + c
+--R /
+--R +-----------+
+--R | 2
+--R \|- 4a c + b
+--R ,
+--R +---------+
+--R | 2
+--R (2a x + b)\|4a c - b
+--R 2atan(----------------------)
+--R 2
+--R 4a c - b
+--R -----------------------------]
+--R +---------+
+--R | 2
+--R \|4a c - b
+--R Type: Union(List Expression
Integer,...)
+--E
+
+--S 48
+bb1:=-(b*x+2*c)/((4*a*c-b^2)*(a*x^2+b*x+c))-b/(4*a*c-b^2)*t1.1
+--R
+--R
+--R (3)
+--R 2 2
+--R (- a b x - b x - b c)
+--R *
+--R log
+--R +-----------+
+--R 2 2 2 | 2 2 2
+--R (2a x + 2a b x - 2a c + b )\|- 4a c + b + (8a c - 2a b )x
+--R +
+--R 3
+--R 4a b c - b
+--R /
+--R 2
+--R a x + b x + c
+--R +
+--R +-----------+
+--R | 2
+--R (- b x - 2c)\|- 4a c + b
+--R /
+--R +-----------+
+--R 2 2 2 3 2 2 | 2
+--R ((4a c - a b )x + (4a b c - b )x + 4a c - b c)\|- 4a c + b
+--R Type: Expression
Integer
+--E
+
+--S 49
+bb2:=-(b*x+2*c)/((4*a*c-b^2)*(a*x^2+b*x+c))-b/(4*a*c-b^2)*t1.2
+--R
+--R
+--R (4)
+--R +---------+
+--R | 2
+--R 2 2 (2a x + b)\|4a c - b
+--R (- 2a b x - 2b x - 2b c)atan(----------------------)
+--R 2
+--R 4a c - b
+--R +
+--R +---------+
+--R | 2
+--R (- b x - 2c)\|4a c - b
+--R /
+--R +---------+
+--R 2 2 2 3 2 2 | 2
+--R ((4a c - a b )x + (4a b c - b )x + 4a c - b c)\|4a c - b
+--R Type: Expression
Integer
+--E
+
+--S 50
+cc1:=bb1-aa.1
+--R
+--R (5)
+--R -
+--R b
+--R *
+--R log
+--R +-----------+
+--R 2 2 2 | 2 2
2
+--R (2a x + 2a b x - 2a c + b )\|- 4a c + b + (8a c - 2a
b )x
+--R +
+--R 3
+--R 4a b c - b
+--R /
+--R 2
+--R a x + b x + c
+--R +
+--R -
+--R b
+--R *
+--R log
+--R +-----------+
+--R 2 2 2 | 2
+--R (2a x + 2a b x - 2a c + b )\|- 4a c + b
+--R +
+--R 2 2 3
+--R (- 8a c + 2a b )x - 4a b c + b
+--R /
+--R 2
+--R a x + b x + c
+--R /
+--R +-----------+
+--R 2 | 2
+--R (4a c - b )\|- 4a c + b
+--R Type: Expression
Integer
+--E
+
+--S 51
+dd1:=expandLog cc1
+--R
+--R (6)
+--R -
+--R b
+--R *
+--R log
+--R +-----------+
+--R 2 2 2 | 2 2 2
+--R (2a x + 2a b x - 2a c + b )\|- 4a c + b + (8a c - 2a b
)x
+--R +
+--R 3
+--R 4a b c - b
+--R +
+--R -
+--R b
+--R *
+--R log
+--R +-----------+
+--R 2 2 2 | 2 2
2
+--R (2a x + 2a b x - 2a c + b )\|- 4a c + b + (- 8a c + 2a
b )x
+--R +
+--R 3
+--R - 4a b c + b
+--R +
+--R 2
+--R 2b log(a x + b x + c)
+--R /
+--R +-----------+
+--R 2 | 2
+--R (4a c - b )\|- 4a c + b
+--R Type: Expression
Integer
+--E
+
+--S 52 14:273 Schaums and Axiom differ by a constant
+ee1:=complexNormalize dd1
+--R
+--R 3 2 2
+--R b log(- 16a c + 4a b )
+--R (7) - -------------------------
+--R +-----------+
+--R 2 | 2
+--R (4a c - b )\|- 4a c + b
+--R Type: Expression
Integer
+--E
+
@
\section{\cite{1}:14.274~~~~~$\displaystyle
@@ -451,7 +1775,7 @@ $$
<<*>>=
)clear all
---S 10 of 15
+--S 53
aa:=integrate(x^2/(a*x^2+b*x+c)^2,x)
--R
--R
@@ -497,6 +1821,96 @@ aa:=integrate(x^2/(a*x^2+b*x+c)^2,x)
--R ]
--R Type: Union(List Expression
Integer,...)
--E
+
+--S 54
+t1:=integrate(1/(a*x^2+b*x+c),x)
+--R
+--R
+--R (2)
+--R [
+--R log
+--R +-----------+
+--R 2 2 2 | 2 2 2
+--R (2a x + 2a b x - 2a c + b )\|- 4a c + b + (8a c - 2a b )x
+--R +
+--R 3
+--R 4a b c - b
+--R /
+--R 2
+--R a x + b x + c
+--R /
+--R +-----------+
+--R | 2
+--R \|- 4a c + b
+--R ,
+--R +---------+
+--R | 2
+--R (2a x + b)\|4a c - b
+--R 2atan(----------------------)
+--R 2
+--R 4a c - b
+--R -----------------------------]
+--R +---------+
+--R | 2
+--R \|4a c - b
+--R Type: Union(List Expression
Integer,...)
+--E
+
+--S 55
+bb1:=((b^2-2*a*c)*x+b*c)/(a*(4*a*c-b^2)*(a*x^2+b*x+c))+(2*c)/(4*a*c-b^2)*t1.1
+--R
+--R (3)
+--R 2 2 2
+--R (2a c x + 2a b c x + 2a c )
+--R *
+--R log
+--R +-----------+
+--R 2 2 2 | 2 2 2
+--R (2a x + 2a b x - 2a c + b )\|- 4a c + b + (8a c - 2a b )x
+--R +
+--R 3
+--R 4a b c - b
+--R /
+--R 2
+--R a x + b x + c
+--R +
+--R +-----------+
+--R 2 | 2
+--R ((- 2a c + b )x + b c)\|- 4a c + b
+--R /
+--R +-----------+
+--R 3 2 2 2 2 3 2 2 2 | 2
+--R ((4a c - a b )x + (4a b c - a b )x + 4a c - a b c)\|- 4a c + b
+--R Type: Expression
Integer
+--E
+
+--S 56
+bb2:=((b^2-2*a*c)*x+b*c)/(a*(4*a*c-b^2)*(a*x^2+b*x+c))+(2*c)/(4*a*c-b^2)*t1.2
+--R
+--R (4)
+--R +---------+
+--R | 2
+--R 2 2 2 (2a x + b)\|4a c - b
+--R (4a c x + 4a b c x + 4a c )atan(----------------------)
+--R 2
+--R 4a c - b
+--R +
+--R +---------+
+--R 2 | 2
+--R ((- 2a c + b )x + b c)\|4a c - b
+--R /
+--R +---------+
+--R 3 2 2 2 2 3 2 2 2 | 2
+--R ((4a c - a b )x + (4a b c - a b )x + 4a c - a b c)\|4a c - b
+--R Type: Expression
Integer
+--E
+
+--S 57 14:274 Schaums and Axiom agree
+cc1:=aa.1-bb1
+--R
+--R (5) 0
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.275~~~~~$\displaystyle
@@ -514,7 +1928,7 @@ $$
<<*>>=
)clear all
---S 11 of 15
+--S 58 14:275 Axiom cannot compute this integral
aa:=integrate(x^m/(a*x^2+b*x+c)^n,x)
--R
--R
@@ -525,6 +1939,7 @@ aa:=integrate(x^m/(a*x^2+b*x+c)^n,x)
--I (c + %N b + %N a)
--R Type: Union(Expression
Integer,...)
--E
+
@
\section{\cite{1}:14.276~~~~~$\displaystyle
@@ -541,7 +1956,7 @@ $$
<<*>>=
)clear all
---S 12 of 15
+--S 59 14:276 Axiom cannot compute this integral
aa:=integrate(x^(2*n-1)/(a*x^2+b*x+c)^n,x)
--R
--R
@@ -552,6 +1967,7 @@ aa:=integrate(x^(2*n-1)/(a*x^2+b*x+c)^n,x)
--I (c + %N b + %N a)
--R Type: Union(Expression
Integer,...)
--E
+
@
\section{\cite{1}:14.277~~~~~$\displaystyle
@@ -568,7 +1984,7 @@ $$
<<*>>=
)clear all
---S 13 of 15
+--S 60
aa:=integrate(1/(x*(a*x^2+b*x+c)^2),x)
--R
--R
@@ -641,6 +2057,380 @@ aa:=integrate(1/(x*(a*x^2+b*x+c)^2),x)
--R ]
--R Type: Union(List Expression
Integer,...)
--E
+
+--S 61
+t1:=integrate(1/(a*x^2+b*x+c)^2,x)
+--R
+--R
+--R (2)
+--R [
+--R 2 2
+--R (2a x + 2a b x + 2a c)
+--R *
+--R log
+--R +-----------+
+--R 2 2 2 | 2 2 2
+--R (2a x + 2a b x - 2a c + b )\|- 4a c + b + (8a c - 2a b
)x
+--R +
+--R 3
+--R 4a b c - b
+--R /
+--R 2
+--R a x + b x + c
+--R +
+--R +-----------+
+--R | 2
+--R (2a x + b)\|- 4a c + b
+--R /
+--R +-----------+
+--R 2 2 2 3 2 2 | 2
+--R ((4a c - a b )x + (4a b c - b )x + 4a c - b c)\|- 4a c + b
+--R ,
+--R +---------+
+--R | 2
+---------+
+--R 2 2 (2a x + b)\|4a c - b |
2
+--R (4a x + 4a b x + 4a c)atan(----------------------) + (2a x + b)\|4a c
- b
+--R 2
+--R 4a c - b
+--R
----------------------------------------------------------------------------
+--R +---------+
+--R 2 2 2 3 2 2 | 2
+--R ((4a c - a b )x + (4a b c - b )x + 4a c - b c)\|4a c - b
+--R ]
+--R Type: Union(List Expression
Integer,...)
+--E
+
+--S 62
+t2:=integrate(1/(x*(a*x^2+b*x+c)),x)
+--R
+--R
+--R (3)
+--R [
+--R b
+--R *
+--R log
+--R +-----------+
+--R 2 2 2 | 2 2
2
+--R (2a x + 2a b x - 2a c + b )\|- 4a c + b + (- 8a c + 2a
b )x
+--R +
+--R 3
+--R - 4a b c + b
+--R /
+--R 2
+--R a x + b x + c
+--R +
+--R +-----------+
+--R 2 | 2
+--R (- log(a x + b x + c) + 2log(x))\|- 4a c + b
+--R /
+--R +-----------+
+--R | 2
+--R 2c\|- 4a c + b
+--R ,
+--R
+--R +---------+
+--R | 2
+--R (2a x + b)\|4a c - b
+--R - 2b atan(----------------------)
+--R 2
+--R 4a c - b
+--R +
+--R +---------+
+--R 2 | 2
+--R (- log(a x + b x + c) + 2log(x))\|4a c - b
+--R /
+--R +---------+
+--R | 2
+--R 2c\|4a c - b
+--R ]
+--R Type: Union(List Expression
Integer,...)
+--E
+
+--S 63
+bb1:=1/(2*c*(a*x^2+b*x+c))-b/(2*c)*t1.1+1/c*t2.1
+--R
+--R
+--R (4)
+--R 2 2 2 2
+--R (- 2a b c x - 2a b c x - 2a b c )
+--R *
+--R log
+--R +-----------+
+--R 2 2 2 | 2 2 2
+--R (2a x + 2a b x - 2a c + b )\|- 4a c + b + (8a c - 2a b )x
+--R +
+--R 3
+--R 4a b c - b
+--R /
+--R 2
+--R a x + b x + c
+--R +
+--R 2 3 2 2 4 2 3
+--R ((4a b c - a b )x + (4a b c - b )x + 4a b c - b c)
+--R *
+--R log
+--R +-----------+
+--R 2 2 2 | 2 2 2
+--R (2a x + 2a b x - 2a c + b )\|- 4a c + b + (- 8a c + 2a b
)x
+--R +
+--R 3
+--R - 4a b c + b
+--R /
+--R 2
+--R a x + b x + c
+--R +
+--R 2 2 2 3 2 2
+--R ((- 4a c + a b )x + (- 4a b c + b )x - 4a c + b c)
+--R *
+--R 2
+--R log(a x + b x + c)
+--R +
+--R 2 2 2 3 2 2
+--R ((8a c - 2a b )x + (8a b c - 2b )x + 8a c - 2b c)log(x) - 2a b
c x
+--R +
+--R 2 2
+--R 4a c - 2b c
+--R *
+--R +-----------+
+--R | 2
+--R \|- 4a c + b
+--R /
+--R
+-----------+
+--R 2 3 2 2 2 3 3 2 4 2 3 | 2
+--R ((8a c - 2a b c )x + (8a b c - 2b c )x + 8a c - 2b c )\|- 4a c + b
+--R Type: Expression
Integer
+--E
+
+--S 64
+bb2:=1/(2*c*(a*x^2+b*x+c))-b/(2*c)*t1.2+1/c*t2.1
+--R
+--R
+--R (5)
+--R +---------+
+--R 2 3 2 2 4 2 3 | 2
+--R ((4a b c - a b )x + (4a b c - b )x + 4a b c - b c)\|4a c - b
+--R *
+--R log
+--R +-----------+
+--R 2 2 2 | 2 2 2
+--R (2a x + 2a b x - 2a c + b )\|- 4a c + b + (- 8a c + 2a b
)x
+--R +
+--R 3
+--R - 4a b c + b
+--R /
+--R 2
+--R a x + b x + c
+--R +
+--R +-----------+
+--R 2 2 2 2 | 2
+--R (- 4a b c x - 4a b c x - 4a b c )\|- 4a c + b
+--R *
+--R +---------+
+--R | 2
+--R (2a x + b)\|4a c - b
+--R atan(----------------------)
+--R 2
+--R 4a c - b
+--R +
+--R 2 2 2 3 2 2
+--R ((- 4a c + a b )x + (- 4a b c + b )x - 4a c + b c)
+--R *
+--R 2
+--R log(a x + b x + c)
+--R +
+--R 2 2 2 3 2 2
+--R ((8a c - 2a b )x + (8a b c - 2b )x + 8a c - 2b c)log(x) - 2a b
c x
+--R +
+--R 2 2
+--R 4a c - 2b c
+--R *
+--R +-----------+ +---------+
+--R | 2 | 2
+--R \|- 4a c + b \|4a c - b
+--R /
+--R
+-----------+
+--R 2 3 2 2 2 3 3 2 4 2 3 |
2
+--R ((8a c - 2a b c )x + (8a b c - 2b c )x + 8a c - 2b c )\|- 4a c +
b
+--R *
+--R +---------+
+--R | 2
+--R \|4a c - b
+--R Type: Expression
Integer
+--E
+
+--S 65
+bb3:=1/(2*c*(a*x^2+b*x+c))-b/(2*c)*t1.1+1/c*t2.2
+--R
+--R
+--R (6)
+--R +---------+
+--R 2 2 2 2 | 2
+--R (- 2a b c x - 2a b c x - 2a b c )\|4a c - b
+--R *
+--R log
+--R +-----------+
+--R 2 2 2 | 2 2 2
+--R (2a x + 2a b x - 2a c + b )\|- 4a c + b + (8a c - 2a b )x
+--R +
+--R 3
+--R 4a b c - b
+--R /
+--R 2
+--R a x + b x + c
+--R +
+--R 2 3 2 2 4 2 3
+--R ((- 8a b c + 2a b )x + (- 8a b c + 2b )x - 8a b c + 2b c)
+--R *
+--R +---------+
+--R +-----------+ | 2
+--R | 2 (2a x + b)\|4a c - b
+--R \|- 4a c + b atan(----------------------)
+--R 2
+--R 4a c - b
+--R +
+--R 2 2 2 3 2 2
+--R ((- 4a c + a b )x + (- 4a b c + b )x - 4a c + b c)
+--R *
+--R 2
+--R log(a x + b x + c)
+--R +
+--R 2 2 2 3 2 2
+--R ((8a c - 2a b )x + (8a b c - 2b )x + 8a c - 2b c)log(x) - 2a b
c x
+--R +
+--R 2 2
+--R 4a c - 2b c
+--R *
+--R +-----------+ +---------+
+--R | 2 | 2
+--R \|- 4a c + b \|4a c - b
+--R /
+--R
+-----------+
+--R 2 3 2 2 2 3 3 2 4 2 3 |
2
+--R ((8a c - 2a b c )x + (8a b c - 2b c )x + 8a c - 2b c )\|- 4a c +
b
+--R *
+--R +---------+
+--R | 2
+--R \|4a c - b
+--R Type: Expression
Integer
+--E
+
+--S 66
+bb4:=1/(2*c*(a*x^2+b*x+c))-b/(2*c)*t1.2+1/c*t2.2
+--R
+--R
+--R (7)
+--R 2 3 2 2 4 2 3
+--R ((- 12a b c + 2a b )x + (- 12a b c + 2b )x - 12a b c + 2b c)
+--R *
+--R +---------+
+--R | 2
+--R (2a x + b)\|4a c - b
+--R atan(----------------------)
+--R 2
+--R 4a c - b
+--R +
+--R 2 2 2 3 2 2
+--R ((- 4a c + a b )x + (- 4a b c + b )x - 4a c + b c)
+--R *
+--R 2
+--R log(a x + b x + c)
+--R +
+--R 2 2 2 3 2 2
+--R ((8a c - 2a b )x + (8a b c - 2b )x + 8a c - 2b c)log(x) - 2a b
c x
+--R +
+--R 2 2
+--R 4a c - 2b c
+--R *
+--R +---------+
+--R | 2
+--R \|4a c - b
+--R /
+--R +---------+
+--R 2 3 2 2 2 3 3 2 4 2 3 | 2
+--R ((8a c - 2a b c )x + (8a b c - 2b c )x + 8a c - 2b c )\|4a c - b
+--R Type: Expression
Integer
+--E
+
+--S 67
+cc1:=aa.1-bb1
+--R
+--R (8)
+--R a b
+--R *
+--R log
+--R +-----------+
+--R 2 2 2 | 2 2 2
+--R (2a x + 2a b x - 2a c + b )\|- 4a c + b + (8a c - 2a b )x
+--R +
+--R 3
+--R 4a b c - b
+--R /
+--R 2
+--R a x + b x + c
+--R +
+--R a b
+--R *
+--R log
+--R +-----------+
+--R 2 2 2 | 2 2 2
+--R (2a x + 2a b x - 2a c + b )\|- 4a c + b + (- 8a c + 2a b
)x
+--R +
+--R 3
+--R - 4a b c + b
+--R /
+--R 2
+--R a x + b x + c
+--R /
+--R +-----------+
+--R 2 2 | 2
+--R (4a c - b c)\|- 4a c + b
+--R Type: Expression
Integer
+--E
+
+--S 68
+dd1:=expandLog cc1
+--R
+--R (9)
+--R a b
+--R *
+--R log
+--R +-----------+
+--R 2 2 2 | 2 2 2
+--R (2a x + 2a b x - 2a c + b )\|- 4a c + b + (8a c - 2a b )x
+--R +
+--R 3
+--R 4a b c - b
+--R +
+--R a b
+--R *
+--R log
+--R +-----------+
+--R 2 2 2 | 2 2 2
+--R (2a x + 2a b x - 2a c + b )\|- 4a c + b + (- 8a c + 2a b )x
+--R +
+--R 3
+--R - 4a b c + b
+--R +
+--R 2
+--R - 2a b log(a x + b x + c)
+--R /
+--R +-----------+
+--R 2 2 | 2
+--R (4a c - b c)\|- 4a c + b
+--R Type: Expression
Integer
+--E
+
+--S 69 14:277 Schaums and Axiom differ by a constant
+ee1:=complexNormalize dd1
+--R
+--R 3 2 2
+--R a b log(- 16a c + 4a b )
+--R (10) ---------------------------
+--R +-----------+
+--R 2 2 | 2
+--R (4a c - b c)\|- 4a c + b
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.278~~~~~$\displaystyle
@@ -657,7 +2447,7 @@ $$
<<*>>=
)clear all
---S 14 of 15
+--S 70
aa:=integrate(1/(x^2*(a*x^2+b*x+c)^2),x)
--R
--R
@@ -740,6 +2530,426 @@ aa:=integrate(1/(x^2*(a*x^2+b*x+c)^2),x)
--R ]
--R Type: Union(List Expression
Integer,...)
--E
+
+--S 71
+t1:=integrate(1/(a*x^2+b*x+c)^2,x)
+--R
+--R
+--R (2)
+--R [
+--R 2 2
+--R (2a x + 2a b x + 2a c)
+--R *
+--R log
+--R +-----------+
+--R 2 2 2 | 2 2 2
+--R (2a x + 2a b x - 2a c + b )\|- 4a c + b + (8a c - 2a b
)x
+--R +
+--R 3
+--R 4a b c - b
+--R /
+--R 2
+--R a x + b x + c
+--R +
+--R +-----------+
+--R | 2
+--R (2a x + b)\|- 4a c + b
+--R /
+--R +-----------+
+--R 2 2 2 3 2 2 | 2
+--R ((4a c - a b )x + (4a b c - b )x + 4a c - b c)\|- 4a c + b
+--R ,
+--R +---------+
+--R | 2
+---------+
+--R 2 2 (2a x + b)\|4a c - b |
2
+--R (4a x + 4a b x + 4a c)atan(----------------------) + (2a x + b)\|4a c
- b
+--R 2
+--R 4a c - b
+--R
----------------------------------------------------------------------------
+--R +---------+
+--R 2 2 2 3 2 2 | 2
+--R ((4a c - a b )x + (4a b c - b )x + 4a c - b c)\|4a c - b
+--R ]
+--R Type: Union(List Expression
Integer,...)
+--E
+
+--S 72
+t2:=integrate(1/(x*(a*x^2+b*x+c)^2),x)
+--R
+--R
+--R (3)
+--R [
+--R 2 3 2 2 4 2 3
+--R ((6a b c - a b )x + (6a b c - b )x + 6a b c - b c)
+--R *
+--R log
+--R +-----------+
+--R 2 2 2 | 2 2
2
+--R (2a x + 2a b x - 2a c + b )\|- 4a c + b + (- 8a c + 2a
b )x
+--R +
+--R 3
+--R - 4a b c + b
+--R /
+--R 2
+--R a x + b x + c
+--R +
+--R 2 2 2 3 2 2
+--R ((- 4a c + a b )x + (- 4a b c + b )x - 4a c + b c)
+--R *
+--R 2
+--R log(a x + b x + c)
+--R +
+--R 2 2 2 3 2 2
+--R ((8a c - 2a b )x + (8a b c - 2b )x + 8a c - 2b c)log(x)
+--R +
+--R 2 2
+--R - 2a b c x + 4a c - 2b c
+--R *
+--R +-----------+
+--R | 2
+--R \|- 4a c + b
+--R /
+--R
+-----------+
+--R 2 3 2 2 2 3 3 2 4 2 3 |
2
+--R ((8a c - 2a b c )x + (8a b c - 2b c )x + 8a c - 2b c )\|- 4a c +
b
+--R ,
+--R
+--R 2 3 2 2 4 2 3
+--R ((- 12a b c + 2a b )x + (- 12a b c + 2b )x - 12a b c + 2b c)
+--R *
+--R +---------+
+--R | 2
+--R (2a x + b)\|4a c - b
+--R atan(----------------------)
+--R 2
+--R 4a c - b
+--R +
+--R 2 2 2 3 2 2
+--R ((- 4a c + a b )x + (- 4a b c + b )x - 4a c + b c)
+--R *
+--R 2
+--R log(a x + b x + c)
+--R +
+--R 2 2 2 3 2 2
+--R ((8a c - 2a b )x + (8a b c - 2b )x + 8a c - 2b c)log(x)
+--R +
+--R 2 2
+--R - 2a b c x + 4a c - 2b c
+--R *
+--R +---------+
+--R | 2
+--R \|4a c - b
+--R /
+--R
+---------+
+--R 2 3 2 2 2 3 3 2 4 2 3 | 2
+--R ((8a c - 2a b c )x + (8a b c - 2b c )x + 8a c - 2b c )\|4a c - b
+--R ]
+--R Type: Union(List Expression
Integer,...)
+--E
+
+--S 73
+bb1:=-1/(c*x*(a*x^2+b*x+c))-((3*a)/c)*t1.1-(2*b)/c*t2.1
+--R
+--R
+--R (4)
+--R 3 2 3 2 2 2 2 3
+--R (- 6a c x - 6a b c x - 6a c x)
+--R *
+--R log
+--R +-----------+
+--R 2 2 2 | 2 2 2
+--R (2a x + 2a b x - 2a c + b )\|- 4a c + b + (8a c - 2a b )x
+--R +
+--R 3
+--R 4a b c - b
+--R /
+--R 2
+--R a x + b x + c
+--R +
+--R 2 2 4 3 3 5 2 2 2 4
+--R ((- 6a b c + a b )x + (- 6a b c + b )x + (- 6a b c + b c)x)
+--R *
+--R log
+--R +-----------+
+--R 2 2 2 | 2 2 2
+--R (2a x + 2a b x - 2a c + b )\|- 4a c + b + (- 8a c + 2a b
)x
+--R +
+--R 3
+--R - 4a b c + b
+--R /
+--R 2
+--R a x + b x + c
+--R +
+--R 2 3 3 2 4 2 2 3
+--R ((4a b c - a b )x + (4a b c - b )x + (4a b c - b c)x)
+--R *
+--R 2
+--R log(a x + b x + c)
+--R +
+--R 2 3 3 2 4 2 2 3
+--R ((- 8a b c + 2a b )x + (- 8a b c + 2b )x + (- 8a b c + 2b
c)x)
+--R *
+--R log(x)
+--R +
+--R 2 2 2 2 2 3 3 2 2
+--R (- 6a c + 2a b c)x + (- 7a b c + 2b c)x - 4a c + b c
+--R *
+--R +-----------+
+--R | 2
+--R \|- 4a c + b
+--R /
+--R
+-----------+
+--R 2 4 2 3 3 4 3 3 2 5 2 4 |
2
+--R ((4a c - a b c )x + (4a b c - b c )x + (4a c - b c )x)\|- 4a c + b
+--R Type: Expression
Integer
+--E
+
+--S 74
+bb2:=-1/(c*x*(a*x^2+b*x+c))-((3*a)/c)*t1.2-(2*b)/c*t2.1
+--R
+--R
+--R (5)
+--R 2 2 4 3 3 5 2 2 2 4
+--R ((- 6a b c + a b )x + (- 6a b c + b )x + (- 6a b c + b c)x)
+--R *
+--R +---------+
+--R | 2
+--R \|4a c - b
+--R *
+--R log
+--R +-----------+
+--R 2 2 2 | 2 2 2
+--R (2a x + 2a b x - 2a c + b )\|- 4a c + b + (- 8a c + 2a b
)x
+--R +
+--R 3
+--R - 4a b c + b
+--R /
+--R 2
+--R a x + b x + c
+--R +
+--R +-----------+
+--R 3 2 3 2 2 2 2 3 | 2
+--R (- 12a c x - 12a b c x - 12a c x)\|- 4a c + b
+--R *
+--R +---------+
+--R | 2
+--R (2a x + b)\|4a c - b
+--R atan(----------------------)
+--R 2
+--R 4a c - b
+--R +
+--R 2 3 3 2 4 2 2 3
+--R ((4a b c - a b )x + (4a b c - b )x + (4a b c - b c)x)
+--R *
+--R 2
+--R log(a x + b x + c)
+--R +
+--R 2 3 3 2 4 2 2 3
+--R ((- 8a b c + 2a b )x + (- 8a b c + 2b )x + (- 8a b c + 2b
c)x)
+--R *
+--R log(x)
+--R +
+--R 2 2 2 2 2 3 3 2 2
+--R (- 6a c + 2a b c)x + (- 7a b c + 2b c)x - 4a c + b c
+--R *
+--R +-----------+ +---------+
+--R | 2 | 2
+--R \|- 4a c + b \|4a c - b
+--R /
+--R
+-----------+
+--R 2 4 2 3 3 4 3 3 2 5 2 4 |
2
+--R ((4a c - a b c )x + (4a b c - b c )x + (4a c - b c )x)\|- 4a c
+ b
+--R *
+--R +---------+
+--R | 2
+--R \|4a c - b
+--R Type: Expression
Integer
+--E
+
+--S 75
+bb3:=-1/(c*x*(a*x^2+b*x+c))-((3*a)/c)*t1.1-(2*b)/c*t2.2
+--R
+--R
+--R (6)
+--R +---------+
+--R 3 2 3 2 2 2 2 3 | 2
+--R (- 6a c x - 6a b c x - 6a c x)\|4a c - b
+--R *
+--R log
+--R +-----------+
+--R 2 2 2 | 2 2 2
+--R (2a x + 2a b x - 2a c + b )\|- 4a c + b + (8a c - 2a b )x
+--R +
+--R 3
+--R 4a b c - b
+--R /
+--R 2
+--R a x + b x + c
+--R +
+--R 2 2 4 3 3 5 2 2 2 4
+--R ((12a b c - 2a b )x + (12a b c - 2b )x + (12a b c - 2b c)x)
+--R *
+--R +---------+
+--R +-----------+ | 2
+--R | 2 (2a x + b)\|4a c - b
+--R \|- 4a c + b atan(----------------------)
+--R 2
+--R 4a c - b
+--R +
+--R 2 3 3 2 4 2 2 3
+--R ((4a b c - a b )x + (4a b c - b )x + (4a b c - b c)x)
+--R *
+--R 2
+--R log(a x + b x + c)
+--R +
+--R 2 3 3 2 4 2 2 3
+--R ((- 8a b c + 2a b )x + (- 8a b c + 2b )x + (- 8a b c + 2b
c)x)
+--R *
+--R log(x)
+--R +
+--R 2 2 2 2 2 3 3 2 2
+--R (- 6a c + 2a b c)x + (- 7a b c + 2b c)x - 4a c + b c
+--R *
+--R +-----------+ +---------+
+--R | 2 | 2
+--R \|- 4a c + b \|4a c - b
+--R /
+--R
+-----------+
+--R 2 4 2 3 3 4 3 3 2 5 2 4 |
2
+--R ((4a c - a b c )x + (4a b c - b c )x + (4a c - b c )x)\|- 4a c
+ b
+--R *
+--R +---------+
+--R | 2
+--R \|4a c - b
+--R Type: Expression
Integer
+--E
+
+--S 76
+bb4:=-1/(c*x*(a*x^2+b*x+c))-((3*a)/c)*t1.2-(2*b)/c*t2.2
+--R
+--R
+--R (7)
+--R 3 2 2 2 4 3 2 2 3 5 2
+--R (- 12a c + 12a b c - 2a b )x + (- 12a b c + 12a b c - 2b )x
+--R +
+--R 2 3 2 2 4
+--R (- 12a c + 12a b c - 2b c)x
+--R *
+--R +---------+
+--R | 2
+--R (2a x + b)\|4a c - b
+--R atan(----------------------)
+--R 2
+--R 4a c - b
+--R +
+--R 2 3 3 2 4 2 2 3
+--R ((4a b c - a b )x + (4a b c - b )x + (4a b c - b c)x)
+--R *
+--R 2
+--R log(a x + b x + c)
+--R +
+--R 2 3 3 2 4 2 2 3
+--R ((- 8a b c + 2a b )x + (- 8a b c + 2b )x + (- 8a b c + 2b
c)x)
+--R *
+--R log(x)
+--R +
+--R 2 2 2 2 2 3 3 2 2
+--R (- 6a c + 2a b c)x + (- 7a b c + 2b c)x - 4a c + b c
+--R *
+--R +---------+
+--R | 2
+--R \|4a c - b
+--R /
+--R +---------+
+--R 2 4 2 3 3 4 3 3 2 5 2 4 | 2
+--R ((4a c - a b c )x + (4a b c - b c )x + (4a c - b c )x)\|4a c - b
+--R Type: Expression
Integer
+--E
+
+--S 77
+cc1:=aa.1-bb1
+--R
+--R (8)
+--R 2
+--R 6a
+--R *
+--R log
+--R +-----------+
+--R 2 2 2 | 2 2 2
+--R (2a x + 2a b x - 2a c + b )\|- 4a c + b + (8a c - 2a b )x
+--R +
+--R 3
+--R 4a b c - b
+--R /
+--R 2
+--R a x + b x + c
+--R +
+--R 2
+--R 6a
+--R *
+--R log
+--R +-----------+
+--R 2 2 2 | 2 2 2
+--R (2a x + 2a b x - 2a c + b )\|- 4a c + b + (- 8a c + 2a b
)x
+--R +
+--R 3
+--R - 4a b c + b
+--R /
+--R 2
+--R a x + b x + c
+--R /
+--R +-----------+
+--R 2 2 | 2
+--R (4a c - b c)\|- 4a c + b
+--R Type: Expression
Integer
+--E
+
+--S 78
+dd1:=expandLog cc1
+--R
+--R (9)
+--R 2
+--R 6a
+--R *
+--R log
+--R +-----------+
+--R 2 2 2 | 2 2 2
+--R (2a x + 2a b x - 2a c + b )\|- 4a c + b + (8a c - 2a b )x
+--R +
+--R 3
+--R 4a b c - b
+--R +
+--R 2
+--R 6a
+--R *
+--R log
+--R +-----------+
+--R 2 2 2 | 2 2 2
+--R (2a x + 2a b x - 2a c + b )\|- 4a c + b + (- 8a c + 2a b )x
+--R +
+--R 3
+--R - 4a b c + b
+--R +
+--R 2 2
+--R - 12a log(a x + b x + c)
+--R /
+--R +-----------+
+--R 2 2 | 2
+--R (4a c - b c)\|- 4a c + b
+--R Type: Expression
Integer
+--E
+
+--S 79 14:278 Schaums and Axiom differ by a constant
+ee1:=complexNormalize dd1
+--R
+--R 2 3 2 2
+--R 6a log(- 16a c + 4a b )
+--R (10) ---------------------------
+--R +-----------+
+--R 2 2 | 2
+--R (4a c - b c)\|- 4a c + b
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.279~~~~~$\displaystyle
@@ -756,7 +2966,7 @@ $$
<<*>>=
)clear all
---S 15 of 15
+--S 80 14:279 Axiom cannot compute this integral
aa:=integrate(1/(x^m*(a*x^2+b*x+c)^n),x)
--R
--R
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