The failing cases in this file are all due to typos.
These have been fixed.
Tim
=========================================================================
diff --git a/changelog b/changelog
index 6e74f31..0086e15 100644
--- a/changelog
+++ b/changelog
@@ -1,3 +1,4 @@
+20080429 tpd src/input/schaum12.input post-mortem fixes
20080428 tpd src/input/schaum19.input post-mortem fixes
20080428 tpd src/input/schaum23.input post-mortem fixes
20080428 tpd src/input/schaum24.input post-mortem fixes
diff --git a/src/input/schaum12.input.pamphlet
b/src/input/schaum12.input.pamphlet
index 1c0d0b2..7d5d77a 100644
--- a/src/input/schaum12.input.pamphlet
+++ b/src/input/schaum12.input.pamphlet
@@ -25,36 +25,37 @@ $$
)clear all
--S 1
-aa:=integrate(1/sqrt(a*x^2+b*x+c),x)
---R
+aa:=integrate(1/(a*x^2+b*x+c),x)
--R
--R (1)
--R [
--R log
---R +--------------+
---R +-+ +-+ | 2 +-+
---R (2\|a \|c - 2a x)\|a x + b x + c + 2a x\|c
+--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 2 +-+
---R (- 2a x - b x - 2c)\|a
+--R 3
+--R 4a b c - b
--R /
---R +--------------+
---R +-+ | 2
---R 2\|c \|a x + b x + c - b x - 2c
+--R 2
+--R a x + b x + c
--R /
---R +-+
---R \|a
+--R +-----------+
+--R | 2
+--R \|- 4a c + b
--R ,
---R +--------------+
---R +---+ | 2 +---+ +-+
---R \|- a \|a x + b x + c - \|- a \|c
---R 2atan(------------------------------------)
---R a x
---R -------------------------------------------]
---R +---+
---R \|- a
+--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
+--E
--S 2
bb1:=2/sqrt(4*a*c-b^2)*atan((2*a*x+b)/sqrt(4*a*c-b^2))
@@ -99,26 +100,26 @@ cc1:=aa.1-bb1
--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 2 2 | 2 2 2
+--R (2a x + 2a b x - 2a c + b )\|- 4a c + b + (8a c - 2a b )x
--R +
---R 2 +-+
---R (- 2a x - b x - 2c)\|a
+--R 3
+--R 4a b c - b
--R /
---R +--------------+
---R +-+ | 2
---R 2\|c \|a x + b x + c - b x - 2c
+--R 2
+--R a x + b x + c
--R +
---R +-+ 2a x + b
---R - 2\|a atan(------------)
---R +---------+
---R | 2
---R \|4a c - b
+--R +-----------+
+--R | 2 2a x + b
+--R - 2\|- 4a c + b atan(------------)
+--R +---------+
+--R | 2
+--R \|4a c - b
--R /
---R +---------+
---R +-+ | 2
---R \|a \|4a c - b
+--R +-----------+ +---------+
+--R | 2 | 2
+--R \|- 4a c + b \|4a c - b
--R Type: Expression
Integer
--E
@@ -126,81 +127,142 @@ cc1:=aa.1-bb1
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 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 + 2a x + b
---R - \|a log(---------------------------)
---R +-----------+
---R | 2
---R \|- 4a c + b + 2a x + b
+--R +-----------+
+--R | 2
+--R - \|- 4a c + b + 2a x + b
+--R - log(---------------------------)
+--R +-----------+
+--R | 2
+--R \|- 4a c + b + 2a x + b
--R /
--R +-----------+
---R | 2 +-+
---R \|- 4a c + b \|a
+--R | 2
+--R \|- 4a c + b
--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 +---------+
+--R | 2
+--R (2a x + b)\|4a c - b 2a x + b
+--R 2atan(----------------------) - 2atan(------------)
+--R 2 +---------+
+--R 4a c - b | 2
+--R \|4a c - b
+--R (6) ---------------------------------------------------
+--R +---------+
+--R | 2
+--R \|4a c - b
--R Type: Expression
Integer
--E
---S 7 14:265 Axiom cannot simplify these expressions
+--S 7
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 +---------+ | 2
+--R | 2 - \|- 4a c + b + 2a x + b
+--R - \|4a c - b 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 +-----------+ | 2
+--R | 2 (2a x + b)\|4a c - b
+--R 2\|- 4a c + b atan(----------------------)
+--R 2
+--R 4a c - b
--R /
---R +-----------+
---R | 2 +---+
---R \|- 4a c + b \|- a
+--R +-----------+ +---------+
+--R | 2 | 2
+--R \|- 4a c + b \|4a c - b
--R Type: Expression
Integer
--E
+
+--S 8
+atanrule:=rule(atan(x) == -%i/2*log((1+%i*x)/(1-%i*x)))
+--R
+--R - x + %i
+--R %i log(--------)
+--R x + %i
+--R (8) atan(x) == - ----------------
+--R 2
+--R Type: RewriteRule(Integer,Complex Integer,Expression Complex
Integer)
+--E
+
+--S 9
+dd3:=atanrule cc3
+--R
+--R (9)
+--R +---------+
+--R | 2
+--R \|4a c - b + 2%i a x + %i b
+--R %i log(-----------------------------)
+--R +---------+
+--R | 2
+--R \|4a c - b - 2%i a x - %i b
+--R +
+--R +---------+
+--R | 2 2
+--R (- 2a x - b)\|4a c - b + 4%i a c - %i b
+--R - %i log(------------------------------------------)
+--R +---------+
+--R | 2 2
+--R (2a x + b)\|4a c - b + 4%i a c - %i b
+--R /
+--R +---------+
+--R | 2
+--R \|4a c - b
+--R Type: Expression Complex
Integer
+--E
+
+--S 10
+ee3:=expandLog dd3
+--R
+--R (10)
+--R +---------+
+--R | 2 2
+--R %i log((2a x + b)\|4a c - b + 4%i a c - %i b )
+--R +
+--R +---------+
+--R | 2 2
+--R - %i log((2a x + b)\|4a c - b - 4%i a c + %i b )
+--R +
+--R +---------+
+--R | 2
+--R %i log(\|4a c - b + 2%i a x + %i b)
+--R +
+--R +---------+
+--R | 2
+--R - %i log(\|4a c - b - 2%i a x - %i b) - %i log(- 1)
+--R /
+--R +---------+
+--R | 2
+--R \|4a c - b
+--R Type: Expression Complex
Integer
+--E
+
+--S 11 14:265 Schaums and Axiom agree
+ff3:=complexNormalize ee3
+--R
+--R (11) 0
+--R Type: Expression Complex
Integer
+--E
@
\section{\cite{1}:14.266~~~~~$\displaystyle
@@ -211,59 +273,46 @@ $$
<<*>>=
)clear all
---S 8
-aa:=integrate(x/sqrt(a*x^2+b*x+c),x)
---R
+--S 12
+aa:=integrate(x/(a*x^2+b*x+c),x)
--R
--R (1)
--R [
---R +--------------+
---R +-+ | 2 2
---R (2b\|c \|a x + b x + c - b x - 2b c)
+--R b
--R *
--R log
---R +--------------+
---R +-+ +-+ | 2 +-+
---R (2\|a \|c + 2a x)\|a x + b x + c - 2a x\|c
+--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 2 +-+
---R (- 2a x - b x - 2c)\|a
+--R 3
+--R - 4a b c + b
--R /
---R +--------------+
---R +-+ | 2
---R 2\|c \|a x + b x + c - b x - 2c
+--R 2
+--R a x + b x + c
--R +
---R +--------------+
---R +-+ | 2 2 +-+ +-+
---R - 2b x\|a \|a x + b x + c + (4a x + 2b x)\|a \|c
+--R +-----------+
+--R 2 | 2
+--R log(a x + b x + c)\|- 4a c + b
--R /
---R +--------------+
---R +-+ +-+ | 2 +-+
---R 4a\|a \|c \|a x + b x + c + (- 2a b x - 4a c)\|a
+--R +-----------+
+--R | 2
+--R 2a\|- 4a c + b
--R ,
---R
---R +--------------+
---R +-+ | 2 2
---R (- 2b\|c \|a x + b x + c + b x + 2b c)
---R *
---R +--------------+
---R +---+ | 2 +---+ +-+
---R \|- a \|a x + b x + c - \|- a \|c
---R atan(------------------------------------)
---R a x
---R +
---R +--------------+
---R +---+ | 2 2 +---+ +-+
---R - b x\|- a \|a x + b x + c + (2a x + b x)\|- a \|c
---R /
---R +--------------+
---R +---+ +-+ | 2 +---+
---R 2a\|- a \|c \|a x + b x + c + (- a b x - 2a c)\|- a
---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 -------------------------------------------------------------------]
+--R +---------+
+--R | 2
+--R 2a\|4a c - b
--R Type: Union(List Expression
Integer,...)
--E
---S 9
+--S 13
t1:=integrate(1/(a*x^2+b*x+c),x)
--R
--R
@@ -297,7 +346,7 @@ t1:=integrate(1/(a*x^2+b*x+c),x)
--R Type: Union(List Expression
Integer,...)
--E
---S 10
+--S 14
bb1:=1/(2*a)*log(a*x^2+b*x+c)-b/(2*a)*t1.1
--R
--R
@@ -326,7 +375,7 @@ bb1:=1/(2*a)*log(a*x^2+b*x+c)-b/(2*a)*t1.1
--R Type: Expression
Integer
--E
---S 11
+--S 15
bb2:=1/(2*a)*log(a*x^2+b*x+c)-b/(2*a)*t1.2
--R
--R
@@ -343,29 +392,11 @@ bb2:=1/(2*a)*log(a*x^2+b*x+c)-b/(2*a)*t1.2
--R Type: Expression
Integer
--E
---S 12
+--S 16
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 b
--R *
--R log
--R +-----------+
@@ -378,39 +409,32 @@ cc1:=aa.1-bb1
--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 b
--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 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 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 +-----------+
+--R | 2
+--R 2a\|- 4a c + b
--R Type: Expression
Integer
--E
---S 13
+--S 17
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 | 2
+--R b\|4a c - b
--R *
--R log
--R +-----------+
@@ -423,53 +447,26 @@ cc2:=aa.2-bb1
--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 +-----------+ | 2
+--R | 2 (2a x + b)\|4a c - b
+--R - 2b\|- 4a c + b atan(----------------------)
+--R 2
+--R 4a c - b
--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 +-----------+ +---------+
+--R | 2 | 2
+--R 2a\|- 4a c + b \|4a c - b
--R Type: Expression
Integer
--E
---S 14
+--S 18
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 | 2
+--R b\|4a c - b
--R *
--R log
--R +-----------+
@@ -482,93 +479,23 @@ cc3:=aa.2-bb1
--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 +-----------+ | 2
+--R | 2 (2a x + b)\|4a c - b
+--R - 2b\|- 4a c + b atan(----------------------)
+--R 2
+--R 4a c - b
--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 +-----------+ +---------+
+--R | 2 | 2
+--R 2a\|- 4a c + b \|4a c - b
--R Type: Expression
Integer
--E
---S 15 14:266 Axiom cannot simplify these expressions
+--S 19 14:266 Schaums and Axiom agree
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 (8) 0
--R Type: Expression
Integer
--E
@@ -583,7 +510,7 @@ $$
<<*>>=
)clear all
---S 16
+--S 20
aa:=integrate(x^2/(a*x^2+b*x+c),x)
--R
--R
@@ -630,7 +557,7 @@ aa:=integrate(x^2/(a*x^2+b*x+c),x)
--R Type: Union(List Expression
Integer,...)
--E
---S 17
+--S 21
t1:=integrate(1/(a*x^2+b*x+c),x)
--R
--R
@@ -664,7 +591,7 @@ t1:=integrate(1/(a*x^2+b*x+c),x)
--R Type: Union(List Expression
Integer,...)
--E
---S 18
+--S 22
bb1:=x/a-b/(2*a^2)*log(a*x^2+b*x+c)+(b^2-2*a*c)/(2*a^2)*t1.1
--R
--R
@@ -693,7 +620,7 @@ bb1:=x/a-b/(2*a^2)*log(a*x^2+b*x+c)+(b^2-2*a*c)/(2*a^2)*t1.1
--R Type: Expression
Integer
--E
---S 19
+--S 23
bb2:=x/a-b/(2*a^2)*log(a*x^2+b*x+c)+(b^2-2*a*c)/(2*a^2)*t1.2
--R
--R
@@ -715,7 +642,7 @@ bb2:=x/a-b/(2*a^2)*log(a*x^2+b*x+c)+(b^2-2*a*c)/(2*a^2)*t1.2
--R Type: Expression
Integer
--E
---S 20
+--S 24
cc1:=bb1-aa.1
--R
--R (5)
@@ -753,7 +680,7 @@ cc1:=bb1-aa.1
--R Type: Expression
Integer
--E
---S 21 14:267 Schaums and Axiom differ by a constant
+--S 25 14:267 Schaums and Axiom differ by a constant
dd1:=complexNormalize cc1
--R
--R 2 3 2 2
@@ -775,7 +702,7 @@ $$
<<*>>=
)clear all
---S 22 14:268 Axiom cannot compute this integral
+--S 26 14:268 Axiom cannot compute this integral
aa:=integrate(x^m/(a*x^2+b*x+c),x)
--R
--R
@@ -798,7 +725,7 @@ $$
<<*>>=
)clear all
---S 23
+--S 27
aa:=integrate(1/(x*(a*x^2+b*x+c)),x)
--R
--R
@@ -844,7 +771,7 @@ aa:=integrate(1/(x*(a*x^2+b*x+c)),x)
--R Type: Union(List Expression
Integer,...)
--E
---S 24
+--S 28
t1:=integrate(1/(a*x^2+b*x+c),x)
--R
--R
@@ -878,7 +805,7 @@ t1:=integrate(1/(a*x^2+b*x+c),x)
--R Type: Union(List Expression
Integer,...)
--E
---S 25
+--S 29
bb1:=1/(2*c)*log(x^2/(a*x^2+b*x+c))-b/(2*c)*t1.1
--R
--R
@@ -909,7 +836,7 @@ bb1:=1/(2*c)*log(x^2/(a*x^2+b*x+c))-b/(2*c)*t1.1
--R Type: Expression
Integer
--E
---S 26
+--S 30
bb2:=1/(2*c)*log(x^2/(a*x^2+b*x+c))-b/(2*c)*t1.2
--R
--R
@@ -926,7 +853,7 @@ bb2:=1/(2*c)*log(x^2/(a*x^2+b*x+c))-b/(2*c)*t1.2
--R Type: Expression
Integer
--E
---S 27
+--S 31
cc1:=bb1-aa.1
--R
--R (5)
@@ -970,7 +897,7 @@ cc1:=bb1-aa.1
--R Type: Expression
Integer
--E
---S 28
+--S 32
dd1:=expandLog cc1
--R
--R (6)
@@ -1005,7 +932,7 @@ dd1:=expandLog cc1
--R Type: Expression
Integer
--E
---S 29 14:269 Schaums and Axiom differ by a constant
+--S 33 14:269 Schaums and Axiom differ by a constant
ee1:=complexNormalize dd1
--R
--R 3 2 2
@@ -1027,7 +954,7 @@ $$
<<*>>=
)clear all
---S 30
+--S 34
aa:=integrate(1/(x^2*(a*x^2+b*x+c)),x)
--R
--R
@@ -1074,7 +1001,7 @@ aa:=integrate(1/(x^2*(a*x^2+b*x+c)),x)
--R Type: Union(List Expression
Integer,...)
--E
---S 31
+--S 35
t1:=integrate(1/(a*x^2+b*x+c),x)
--R
--R
@@ -1108,7 +1035,7 @@ t1:=integrate(1/(a*x^2+b*x+c),x)
--R Type: Union(List Expression
Integer,...)
--E
---S 32
+--S 36
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
@@ -1139,7 +1066,7 @@
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 Type: Expression
Integer
--E
---S 33
+--S 37
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
@@ -1163,7 +1090,7 @@
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 Type: Expression
Integer
--E
---S 34
+--S 38
cc1:=bb1-aa.1
--R
--R (5)
@@ -1207,7 +1134,7 @@ cc1:=bb1-aa.1
--R Type: Expression
Integer
--E
---S 35
+--S 39
dd1:=expandLog cc1
--R
--R (6)
@@ -1242,7 +1169,7 @@ dd1:=expandLog cc1
--R Type: Expression
Integer
--E
---S 36 14:270 Schaums and Axiom differ by a constant
+--S 40 14:270 Schaums and Axiom differ by a constant
ee1:=complexNormalize dd1
--R
--R 2 3 2 2
@@ -1265,7 +1192,7 @@ $$
<<*>>=
)clear all
---S 37 14:271 Axiom cannot compute this integral
+--S 41 14:271 Axiom cannot compute this integral
aa:=integrate(1/(x^n*(a*x^2+b*x+c)),x)
--R
--R
@@ -1288,7 +1215,7 @@ $$
<<*>>=
)clear all
---S 38
+--S 42
aa:=integrate(1/(a*x^2+b*x+c)^2,x)
--R
--R
@@ -1330,7 +1257,7 @@ aa:=integrate(1/(a*x^2+b*x+c)^2,x)
--R Type: Union(List Expression
Integer,...)
--E
---S 39
+--S 43
t1:=integrate(1/(a*x^2+b*x+c),x)
--R
--R
@@ -1364,13 +1291,12 @@ t1:=integrate(1/(a*x^2+b*x+c),x)
--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
+--S 44
+bb1:=(2*a*x+b)/((4*a*c-b^2)*(a*x^2+b*x+c))+(2*a)/(4*a*c-b^2)*t1.1
--R
--R (3)
---R 2 2
---R (- a b x - b x - b c)
+--R 2 2
+--R (2a x + 2a b x + 2a c)
--R *
--R log
--R +-----------+
@@ -1385,7 +1311,7 @@
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 | 2
---R (b x + 2c)\|- 4a c + b
+--R (2a x + b)\|- 4a c + b
--R /
--R +-----------+
--R 2 2 2 3 2 2 | 2
@@ -1393,99 +1319,70 @@
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 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
+--S 45
+bb2:=(2*a*x+b)/((4*a*c-b^2)*(a*x^2+b*x+c))+(2*a)/(4*a*c-b^2)*t1.2
--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 +---------+
+--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 Type: Expression
Integer
--E
---S 42
+--S 46
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 (5) 0
--R Type: Expression
Integer
--E
---S 43
+--S 47
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
+--R 2a\|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
---R ((- b + 2a)x - 2c + b)\|- 4a c + b \|4a c - b
+--R +---------+
+--R +-----------+ | 2
+--R | 2 (2a x + b)\|4a c - b
+--R 4a\|- 4a c + b atan(----------------------)
+--R 2
+--R 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 +-----------+ +---------+
+--R 2 | 2 | 2
+--R (4a c - b )\|- 4a c + b \|4a c - b
--R Type: Expression
Integer
--E
---S 44
+--S 48
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 | 2
+--R 2a\|4a c - b
--R *
--R log
--R +-----------+
@@ -1498,41 +1395,23 @@ cc3:=aa.1-bb2
--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 +-----------+ | 2
+--R | 2 (2a x + b)\|4a c - b
+--R - 4a\|- 4a c + b atan(----------------------)
+--R 2
+--R 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 +-----------+ +---------+
+--R 2 | 2 | 2
+--R (4a c - b )\|- 4a c + b \|4a c - b
--R Type: Expression
Integer
--E
---S 45 14:272 Axiom cannot simplify these expressions
+--S 49 14:272 Schaums and Axiom agree
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 (8) 0
--R Type: Expression
Integer
--E
@@ -1547,7 +1426,7 @@ $$
<<*>>=
)clear all
---S 46
+--S 50
aa:=integrate(x/(a*x^2+b*x+c)^2,x)
--R
--R
@@ -1594,7 +1473,7 @@ aa:=integrate(x/(a*x^2+b*x+c)^2,x)
--R Type: Union(List Expression
Integer,...)
--E
---S 47
+--S 51
t1:=integrate(1/(a*x^2+b*x+c),x)
--R
--R
@@ -1628,7 +1507,7 @@ t1:=integrate(1/(a*x^2+b*x+c),x)
--R Type: Union(List Expression
Integer,...)
--E
---S 48
+--S 52
bb1:=-(b*x+2*c)/((4*a*c-b^2)*(a*x^2+b*x+c))-b/(4*a*c-b^2)*t1.1
--R
--R
@@ -1657,7 +1536,7 @@
bb1:=-(b*x+2*c)/((4*a*c-b^2)*(a*x^2+b*x+c))-b/(4*a*c-b^2)*t1.1
--R Type: Expression
Integer
--E
---S 49
+--S 53
bb2:=-(b*x+2*c)/((4*a*c-b^2)*(a*x^2+b*x+c))-b/(4*a*c-b^2)*t1.2
--R
--R
@@ -1679,7 +1558,7 @@
bb2:=-(b*x+2*c)/((4*a*c-b^2)*(a*x^2+b*x+c))-b/(4*a*c-b^2)*t1.2
--R Type: Expression
Integer
--E
---S 50
+--S 54
cc1:=bb1-aa.1
--R
--R (5)
@@ -1717,7 +1596,7 @@ cc1:=bb1-aa.1
--R Type: Expression
Integer
--E
---S 51
+--S 55
dd1:=expandLog cc1
--R
--R (6)
@@ -1752,7 +1631,7 @@ dd1:=expandLog cc1
--R Type: Expression
Integer
--E
---S 52 14:273 Schaums and Axiom differ by a constant
+--S 56 14:273 Schaums and Axiom differ by a constant
ee1:=complexNormalize dd1
--R
--R 3 2 2
@@ -1775,7 +1654,7 @@ $$
<<*>>=
)clear all
---S 53
+--S 57
aa:=integrate(x^2/(a*x^2+b*x+c)^2,x)
--R
--R
@@ -1822,7 +1701,7 @@ aa:=integrate(x^2/(a*x^2+b*x+c)^2,x)
--R Type: Union(List Expression
Integer,...)
--E
---S 54
+--S 58
t1:=integrate(1/(a*x^2+b*x+c),x)
--R
--R
@@ -1856,7 +1735,7 @@ t1:=integrate(1/(a*x^2+b*x+c),x)
--R Type: Union(List Expression
Integer,...)
--E
---S 55
+--S 59
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)
@@ -1884,7 +1763,7 @@
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 Type: Expression
Integer
--E
---S 56
+--S 60
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)
@@ -1905,7 +1784,7 @@
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 Type: Expression
Integer
--E
---S 57 14:274 Schaums and Axiom agree
+--S 61 14:274 Schaums and Axiom agree
cc1:=aa.1-bb1
--R
--R (5) 0
@@ -1928,7 +1807,7 @@ $$
<<*>>=
)clear all
---S 58 14:275 Axiom cannot compute this integral
+--S 62 14:275 Axiom cannot compute this integral
aa:=integrate(x^m/(a*x^2+b*x+c)^n,x)
--R
--R
@@ -1956,7 +1835,7 @@ $$
<<*>>=
)clear all
---S 59 14:276 Axiom cannot compute this integral
+--S 63 14:276 Axiom cannot compute this integral
aa:=integrate(x^(2*n-1)/(a*x^2+b*x+c)^n,x)
--R
--R
@@ -1984,7 +1863,7 @@ $$
<<*>>=
)clear all
---S 60
+--S 64
aa:=integrate(1/(x*(a*x^2+b*x+c)^2),x)
--R
--R
@@ -2058,7 +1937,7 @@ aa:=integrate(1/(x*(a*x^2+b*x+c)^2),x)
--R Type: Union(List Expression
Integer,...)
--E
---S 61
+--S 65
t1:=integrate(1/(a*x^2+b*x+c)^2,x)
--R
--R
@@ -2100,7 +1979,7 @@ t1:=integrate(1/(a*x^2+b*x+c)^2,x)
--R Type: Union(List Expression
Integer,...)
--E
---S 62
+--S 66
t2:=integrate(1/(x*(a*x^2+b*x+c)),x)
--R
--R
@@ -2146,7 +2025,7 @@ t2:=integrate(1/(x*(a*x^2+b*x+c)),x)
--R Type: Union(List Expression
Integer,...)
--E
---S 63
+--S 67
bb1:=1/(2*c*(a*x^2+b*x+c))-b/(2*c)*t1.1+1/c*t2.1
--R
--R
@@ -2201,7 +2080,7 @@ bb1:=1/(2*c*(a*x^2+b*x+c))-b/(2*c)*t1.1+1/c*t2.1
--R Type: Expression
Integer
--E
---S 64
+--S 68
bb2:=1/(2*c*(a*x^2+b*x+c))-b/(2*c)*t1.2+1/c*t2.1
--R
--R
@@ -2258,7 +2137,7 @@ bb2:=1/(2*c*(a*x^2+b*x+c))-b/(2*c)*t1.2+1/c*t2.1
--R Type: Expression
Integer
--E
---S 65
+--S 69
bb3:=1/(2*c*(a*x^2+b*x+c))-b/(2*c)*t1.1+1/c*t2.2
--R
--R
@@ -2314,7 +2193,7 @@ bb3:=1/(2*c*(a*x^2+b*x+c))-b/(2*c)*t1.1+1/c*t2.2
--R Type: Expression
Integer
--E
---S 66
+--S 70
bb4:=1/(2*c*(a*x^2+b*x+c))-b/(2*c)*t1.2+1/c*t2.2
--R
--R
@@ -2351,7 +2230,7 @@ bb4:=1/(2*c*(a*x^2+b*x+c))-b/(2*c)*t1.2+1/c*t2.2
--R Type: Expression
Integer
--E
---S 67
+--S 71
cc1:=aa.1-bb1
--R
--R (8)
@@ -2387,7 +2266,7 @@ cc1:=aa.1-bb1
--R Type: Expression
Integer
--E
---S 68
+--S 72
dd1:=expandLog cc1
--R
--R (9)
@@ -2420,7 +2299,7 @@ dd1:=expandLog cc1
--R Type: Expression
Integer
--E
---S 69 14:277 Schaums and Axiom differ by a constant
+--S 73 14:277 Schaums and Axiom differ by a constant
ee1:=complexNormalize dd1
--R
--R 3 2 2
@@ -2447,7 +2326,7 @@ $$
<<*>>=
)clear all
---S 70
+--S 74
aa:=integrate(1/(x^2*(a*x^2+b*x+c)^2),x)
--R
--R
@@ -2531,7 +2410,7 @@ aa:=integrate(1/(x^2*(a*x^2+b*x+c)^2),x)
--R Type: Union(List Expression
Integer,...)
--E
---S 71
+--S 75
t1:=integrate(1/(a*x^2+b*x+c)^2,x)
--R
--R
@@ -2573,7 +2452,7 @@ t1:=integrate(1/(a*x^2+b*x+c)^2,x)
--R Type: Union(List Expression
Integer,...)
--E
---S 72
+--S 76
t2:=integrate(1/(x*(a*x^2+b*x+c)^2),x)
--R
--R
@@ -2647,7 +2526,7 @@ t2:=integrate(1/(x*(a*x^2+b*x+c)^2),x)
--R Type: Union(List Expression
Integer,...)
--E
---S 73
+--S 77
bb1:=-1/(c*x*(a*x^2+b*x+c))-((3*a)/c)*t1.1-(2*b)/c*t2.1
--R
--R
@@ -2704,7 +2583,7 @@ bb1:=-1/(c*x*(a*x^2+b*x+c))-((3*a)/c)*t1.1-(2*b)/c*t2.1
--R Type: Expression
Integer
--E
---S 74
+--S 78
bb2:=-1/(c*x*(a*x^2+b*x+c))-((3*a)/c)*t1.2-(2*b)/c*t2.1
--R
--R
@@ -2766,7 +2645,7 @@ bb2:=-1/(c*x*(a*x^2+b*x+c))-((3*a)/c)*t1.2-(2*b)/c*t2.1
--R Type: Expression
Integer
--E
---S 75
+--S 79
bb3:=-1/(c*x*(a*x^2+b*x+c))-((3*a)/c)*t1.1-(2*b)/c*t2.2
--R
--R
@@ -2824,7 +2703,7 @@ bb3:=-1/(c*x*(a*x^2+b*x+c))-((3*a)/c)*t1.1-(2*b)/c*t2.2
--R Type: Expression
Integer
--E
---S 76
+--S 80
bb4:=-1/(c*x*(a*x^2+b*x+c))-((3*a)/c)*t1.2-(2*b)/c*t2.2
--R
--R
@@ -2866,7 +2745,7 @@ bb4:=-1/(c*x*(a*x^2+b*x+c))-((3*a)/c)*t1.2-(2*b)/c*t2.2
--R Type: Expression
Integer
--E
---S 77
+--S 81
cc1:=aa.1-bb1
--R
--R (8)
@@ -2904,7 +2783,7 @@ cc1:=aa.1-bb1
--R Type: Expression
Integer
--E
---S 78
+--S 82
dd1:=expandLog cc1
--R
--R (9)
@@ -2939,7 +2818,7 @@ dd1:=expandLog cc1
--R Type: Expression
Integer
--E
---S 79 14:278 Schaums and Axiom differ by a constant
+--S 83 14:278 Schaums and Axiom differ by a constant
ee1:=complexNormalize dd1
--R
--R 2 3 2 2
@@ -2966,7 +2845,7 @@ $$
<<*>>=
)clear all
---S 80 14:279 Axiom cannot compute this integral
+--S 84 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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