Sun Jan 3 23:45:56 MET 1999
REDUCE 3.7, 15-Jan-99 ...
1: 1:
2: 2: 2: 2: 2: 2: 2: 2: 2:
3: 3: COMMENT
THE REDUCE INTEGRATION TEST PACKAGE
Edited By
Anthony C. Hearn
The RAND Corporation
This file is designed to provide a set of representative tests of the
Reduce integration package. Not all examples go through, even when an
integral exists, since some of the arguments are outside the domain of
applicability of the current package. However, future improvements to
the package will result in more closed-form evaluations in later
releases. We would appreciate any additional contributions to this test
file either because they illustrate some feature (good or bad) of the
current package, or suggest domains which future versions should handle.
Any suggestions for improved organization of this test file (e.g., in a
way which corresponds more directly to the organization of a standard
integration table book such as Gradshteyn and Ryznik) are welcome.
Acknowledgments:
The examples in this file have been contributed by the following.
Any omissions to this list should be reported to the Editor.
David M. Dahm
James H. Davenport
John P. Fitch
Steven Harrington
Anthony C. Hearn
K. Siegfried Koelbig
Ernst Krupnikov
Arthur C. Norman
Herbert Stoyan
;
Comment we first set up a suitable testing functions;
fluid '(gcknt!*);
global '(faillist!* gcnumber!* inittime number!-of!-integrals
unintlist!*);
symbolic operator time;
symbolic procedure initialize!-integral!-test;
begin
faillist!* := unintlist!* := nil;
number!-of!-integrals := 0;
gcnumber!* := gcknt!*;
inittime := time()
end;
initialize!-integral!-test
symbolic procedure summarize!-integral!-test;
begin scalar totaltime;
totaltime := time()-inittime;
prin2t
" ***** SUMMARY OF INTEGRAL TESTS *****";
terpri();
prin2 "Number of integrals tested: ";
prin2t number!-of!-integrals;
terpri();
prin2 "Total time taken: ";
prin2 totaltime;
prin2t " ms";
terpri();
if gcnumber!*
then <<prin2 "Number of garbage collections: ";
prin2t (gcknt!* - gcnumber!*);
terpri()>>;
prin2 "Number of incorrect integrals: ";
prin2t length faillist!*;
terpri();
prin2 "Number of unevaluated integrals: ";
prin2t length unintlist!*;
terpri();
if faillist!*
then <<prin2t "Integrands of incorrect integrals are:";
for each x in reverse faillist!* do mathprint car x>>;
if unintlist!*
then <<prin2t "Integrands of unevaluated integrals are:";
terpri();
for each x in reverse unintlist!* do mathprint car x>>
end;
summarize!-integral!-test
procedure testint(a,b);
begin scalar der,diffce,res,tt;
tt:=time();
symbolic (number!-of!-integrals := number!-of!-integrals + 1);
res:=int(a,b);
% write "time for integral: ",time()-tt," ms";
off precise;
der := df(res,b);
diffce := der-a;
if diffce neq 0
then begin for all x let cot x=cos x/sin x,
sec x=1/cos x,
sin x**2=1-cos x**2,
tan(x/2)=sin x/(1+cos x),
tan x=sin x/cos x,
tanh x=
(e**(x)-e**(-x))/(e**x+e**(-x)),
coth x= 1/tanh x;
diffce := diffce;
for all x clear cot x,sec x,sin x**2,tan x,tan(x/2),
tanh x,coth x
end;
%hopefully, difference appeared non-zero due to absence of
%above transformations;
if diffce neq 0
then <<on combineexpt; diffce := diffce; off combineexpt>>;
if diffce neq 0
then begin scalar !*reduced;
symbolic(!*reduced := t);
for all x let cos(2x)= 1-2sin x**2, sin x**2=1-cos x**2;
diffce := diffce;
for all x clear cos(2x),sin x**2
end;
if diffce neq 0
then <<write
" ***** DERIVATIVE OF INTEGRAL NOT EQUAL TO INTEGRAND *****";
symbolic(faillist!* := list(a,b,res,der) . faillist!*)>>;
symbolic if smemq('int,res)
then unintlist!* := list(a,b,res) . unintlist!*;
on precise;
return res
end;
testint
symbolic initialize!-integral!-test();
% References are to Gradshteyn and Ryznik.
testint(1+x+x**2,x);
2
x*(2*x + 3*x + 6)
--------------------
6
testint(x**2*(2*x**2+x)**2,x);
5 2
x *(60*x + 70*x + 21)
------------------------
105
testint(x*(x**2+2*x+1),x);
2 2
x *(3*x + 8*x + 6)
---------------------
12
testint(1/x,x);
log(x)
% 2.01 #2;
testint((x+1)**3/(x-1)**4,x);
3 2 3
3*log(x - 1)*x - 9*log(x - 1)*x + 9*log(x - 1)*x - 3*log(x - 1) - 6*x - 2
------------------------------------------------------------------------------
3 2
3*(x - 3*x + 3*x - 1)
testint(1/(x*(x-1)*(x+1)**2),x);
(log(x - 1)*x + log(x - 1) + 3*log(x + 1)*x + 3*log(x + 1) - 4*log(x)*x
- 4*log(x) + 2*x)/(4*(x + 1))
testint((a*x+b)/((x-p)*(x-q)),x);
log(p - x)*a*p + log(p - x)*b - log(q - x)*a*q - log(q - x)*b
---------------------------------------------------------------
p - q
testint(1/(a*x**2+b*x+c),x);
2 2*a*x + b
2*sqrt(4*a*c - b )*atan(------------------)
2
sqrt(4*a*c - b )
---------------------------------------------
2
4*a*c - b
testint((a*x+b)/(1+x**2),x);
2
2*atan(x)*b + log(x + 1)*a
-----------------------------
2
testint(1/(x**2-2*x+3),x);
x - 1
sqrt(2)*atan(---------)
sqrt(2)
-------------------------
2
% Rational function examples from Hardy, Pure Mathematics, p 253 et seq.
testint(1/((x-1)*(x**2+1))**2,x);
3 2 2 3 2 2
(atan(x)*x - atan(x)*x + atan(x)*x - atan(x) + log(x + 1)*x - log(x + 1)*x
2 2 3 2
+ log(x + 1)*x - log(x + 1) - 2*log(x - 1)*x + 2*log(x - 1)*x
3 3 2
- 2*log(x - 1)*x + 2*log(x - 1) - x - 2*x + 1)/(4*(x - x + x - 1))
testint(x/((x-a)*(x-b)*(x-c)),x);
(log(a - x)*a*b - log(a - x)*a*c - log(b - x)*a*b + log(b - x)*b*c
2 2 2 2 2 2
+ log(c - x)*a*c - log(c - x)*b*c)/(a *b - a *c - a*b + a*c + b *c - b*c )
testint(x/((x**2+a**2)*(x**2+b**2)),x);
2 2 2 2
- log(a + x ) + log(b + x )
--------------------------------
2 2
2*(a - b )
testint(x**2/((x**2+a**2)*(x**2+b**2)),x);
x x
atan(---)*a - atan(---)*b
a b
---------------------------
2 2
a - b
testint(x/((x-1)*(x**2+1)),x);
2
2*atan(x) - log(x + 1) + 2*log(x - 1)
----------------------------------------
4
testint(x/(1+x**3),x);
2*x - 1 2
2*sqrt(3)*atan(---------) + log(x - x + 1) - 2*log(x + 1)
sqrt(3)
------------------------------------------------------------
6
testint(x**3/((x-1)**2*(x**3+1)),x);
2 2
( - 4*log(x - x + 1)*x + 4*log(x - x + 1) + 9*log(x - 1)*x - 9*log(x - 1)
- log(x + 1)*x + log(x + 1) - 6*x)/(12*(x - 1))
testint(1/(1+x**4),x);
sqrt(2) - 2*x sqrt(2) + 2*x
(sqrt(2)*( - 2*atan(---------------) + 2*atan(---------------)
sqrt(2) sqrt(2)
2 2
- log( - sqrt(2)*x + x + 1) + log(sqrt(2)*x + x + 1)))/8
testint(x**2/(1+x**4),x);
sqrt(2) - 2*x sqrt(2) + 2*x
(sqrt(2)*( - 2*atan(---------------) + 2*atan(---------------)
sqrt(2) sqrt(2)
2 2
+ log( - sqrt(2)*x + x + 1) - log(sqrt(2)*x + x + 1)))/8
testint(1/(1+x**2+x**4),x);
2*x - 1 2*x + 1 2
(2*sqrt(3)*atan(---------) + 2*sqrt(3)*atan(---------) - 3*log(x - x + 1)
sqrt(3) sqrt(3)
2
+ 3*log(x + x + 1))/12
% Examples involving a+b*x.
z := a+b*x;
z := a + b*x
testint(z**p,x);
p
(a + b*x) *(a + b*x)
----------------------
b*(p + 1)
testint(x*z**p,x);
p 2 2 2 2 2
(a + b*x) *( - a + a*b*p*x + b *p*x + b *x )
------------------------------------------------
2 2
b *(p + 3*p + 2)
testint(x**2*z**p,x);
p
((a + b*x)
3 2 2 2 2 2 2 3 2 3 3 3 3 3
*(2*a - 2*a *b*p*x + a*b *p *x + a*b *p*x + b *p *x + 3*b *p*x + 2*b *x ))
3 3 2
/(b *(p + 6*p + 11*p + 6))
testint(1/z,x);
log(a + b*x)
--------------
b
testint(1/z**2,x);
x
-------------
a*(a + b*x)
testint(x/z,x);
- log(a + b*x)*a + b*x
-------------------------
2
b
testint(x**2/z,x);
2 2 2
2*log(a + b*x)*a - 2*a*b*x + b *x
-------------------------------------
3
2*b
testint(1/(x*z),x);
- log(a + b*x) + log(x)
--------------------------
a
testint(1/(x**2*z),x);
log(a + b*x)*b*x - log(x)*b*x - a
-----------------------------------
2
a *x
testint(1/(x*z)**2,x);
2 2 2 2
(2*log(a + b*x)*a*b*x + 2*log(a + b*x)*b *x - 2*log(x)*a*b*x - 2*log(x)*b *x
2 2 2 3
- a + 2*b *x )/(a *x*(a + b*x))
testint(1/(c**2+x**2),x);
x
atan(---)
c
-----------
c
testint(1/(c**2-x**2),x);
log( - c - x) - log(c - x)
----------------------------
2*c
% More complicated rational function examples, mostly contributed
% by David M. Dahm, who also developed the code to integrate them.
testint(1/(2*x**3-1),x);
1/3
2/3 2*2 *x + 1 2/3 2 1/3
(2 *( - 2*sqrt(3)*atan(--------------) - log(2 *x + 2 *x + 1)
sqrt(3)
1/3
+ 2*log(2 *x - 1)))/12
testint(1/(x**3-2),x);
1/3
1/3 2 + 2*x 2/3 1/3 2
(2 *( - 2*sqrt(3)*atan(--------------) - log(2 + 2 *x + x )
1/3
2 *sqrt(3)
1/3
+ 2*log( - 2 + x)))/12
testint(1/(a*x**3-b),x);
1/3 1/3
1/3 2/3 2*a *x + b
(b *a *( - 2*sqrt(3)*atan(-----------------)
1/3
b *sqrt(3)
2/3 2 1/3 1/3 2/3 1/3 1/3
- log(a *x + b *a *x + b ) + 2*log(a *x - b )))/(6*a*b
)
testint(1/(x**4-2),x);
1/4 x 1/4 1/4
2 *( - 2*atan(------) - log(2 + x) + log( - 2 + x))
1/4
2
-------------------------------------------------------------
8
testint(1/(5*x**4-1),x);
3/4 sqrt(5)*x 1/4 1/4
5 *( - 2*atan(-----------) + log(5 *x - 1) - log(5 *x + 1))
1/4
5
-------------------------------------------------------------------
20
testint(1/(3*x**4+7),x);
1/4
3/4 1/4 sqrt(2)*21 - 2*sqrt(3)*x
(sqrt(2)*3 *7 *( - 2*atan(-----------------------------)
1/4
sqrt(2)*21
1/4
sqrt(2)*21 + 2*sqrt(3)*x
+ 2*atan(-----------------------------)
1/4
sqrt(2)*21
1/4 2
- log( - sqrt(2)*21 *x + sqrt(7) + sqrt(3)*x )
1/4 2
+ log(sqrt(2)*21 *x + sqrt(7) + sqrt(3)*x )))/168
testint(1/(x**4+3*x**2-1),x);
2*x
(sqrt(2)*(6*sqrt(sqrt(13) + 3)*sqrt(13)*atan(----------------------------)
sqrt(sqrt(13) + 3)*sqrt(2)
2*x
- 26*sqrt(sqrt(13) + 3)*atan(----------------------------) + 3
sqrt(sqrt(13) + 3)*sqrt(2)
*sqrt(sqrt(13) - 3)*sqrt(13)*log( - sqrt(sqrt(13) - 3) + sqrt(2)*x)
- 3*sqrt(sqrt(13) - 3)*sqrt(13)*log(sqrt(sqrt(13) - 3) + sqrt(2)*x)
+ 13*sqrt(sqrt(13) - 3)*log( - sqrt(sqrt(13) - 3) + sqrt(2)*x)
- 13*sqrt(sqrt(13) - 3)*log(sqrt(sqrt(13) - 3) + sqrt(2)*x)))/104
testint(1/(x**4-3*x**2-1),x);
2*x
(sqrt(2)*( - 6*sqrt(sqrt(13) - 3)*sqrt(13)*atan(----------------------------)
sqrt(sqrt(13) - 3)*sqrt(2)
2*x
- 26*sqrt(sqrt(13) - 3)*atan(----------------------------) - 3
sqrt(sqrt(13) - 3)*sqrt(2)
*sqrt(sqrt(13) + 3)*sqrt(13)*log( - sqrt(sqrt(13) + 3) + sqrt(2)*x)
+ 3*sqrt(sqrt(13) + 3)*sqrt(13)*log(sqrt(sqrt(13) + 3) + sqrt(2)*x)
+ 13*sqrt(sqrt(13) + 3)*log( - sqrt(sqrt(13) + 3) + sqrt(2)*x)
- 13*sqrt(sqrt(13) + 3)*log(sqrt(sqrt(13) + 3) + sqrt(2)*x)))/104
testint(1/(x**4-3*x**2+1),x);
( - sqrt(5)*log( - sqrt(5) + 2*x - 1) - sqrt(5)*log( - sqrt(5) + 2*x + 1)
+ sqrt(5)*log(sqrt(5) + 2*x - 1) + sqrt(5)*log(sqrt(5) + 2*x + 1)
+ 5*log( - sqrt(5) + 2*x - 1) - 5*log( - sqrt(5) + 2*x + 1)
+ 5*log(sqrt(5) + 2*x - 1) - 5*log(sqrt(5) + 2*x + 1))/20
testint(1/(x**4-4*x**2+1),x);
2*x 2*x
(sqrt(2)*(2*sqrt(3)*atanh(-------------------) + 6*atanh(-------------------)
sqrt(6) - sqrt(2) sqrt(6) - sqrt(2)
- sqrt(6) - sqrt(2) + 2*x
- sqrt(3)*log(----------------------------)
2
sqrt(6) + sqrt(2) + 2*x
+ sqrt(3)*log(-------------------------)
2
- sqrt(6) - sqrt(2) + 2*x
+ 3*log(----------------------------)
2
sqrt(6) + sqrt(2) + 2*x
- 3*log(-------------------------)))/24
2
testint(1/(x**4+4*x**2+1),x);
2*x 2*x
(sqrt(2)*(2*sqrt(3)*atan(-------------------) - 6*atan(-------------------)
sqrt(6) + sqrt(2) sqrt(6) + sqrt(2)
- sqrt(6)*i + sqrt(2)*i + 2*x
- sqrt(3)*log(--------------------------------)*i
2
sqrt(6)*i - sqrt(2)*i + 2*x
+ sqrt(3)*log(-----------------------------)*i
2
- sqrt(6)*i + sqrt(2)*i + 2*x
- 3*log(--------------------------------)*i
2
sqrt(6)*i - sqrt(2)*i + 2*x
+ 3*log(-----------------------------)*i))/24
2
testint(1/(x**4+x**2+2),x);
sqrt(2*sqrt(2) - 1) - 2*x
(2*sqrt(2*sqrt(2) + 1)*sqrt(2)*atan(---------------------------)
sqrt(2*sqrt(2) + 1)
sqrt(2*sqrt(2) - 1) - 2*x
- 8*sqrt(2*sqrt(2) + 1)*atan(---------------------------)
sqrt(2*sqrt(2) + 1)
sqrt(2*sqrt(2) - 1) + 2*x
- 2*sqrt(2*sqrt(2) + 1)*sqrt(2)*atan(---------------------------)
sqrt(2*sqrt(2) + 1)
sqrt(2*sqrt(2) - 1) + 2*x
+ 8*sqrt(2*sqrt(2) + 1)*atan(---------------------------)
sqrt(2*sqrt(2) + 1)
2
- sqrt(2*sqrt(2) - 1)*sqrt(2)*log( - sqrt(2*sqrt(2) - 1)*x + sqrt(2) + x )
2
+ sqrt(2*sqrt(2) - 1)*sqrt(2)*log(sqrt(2*sqrt(2) - 1)*x + sqrt(2) + x )
2
- 4*sqrt(2*sqrt(2) - 1)*log( - sqrt(2*sqrt(2) - 1)*x + sqrt(2) + x )
2
+ 4*sqrt(2*sqrt(2) - 1)*log(sqrt(2*sqrt(2) - 1)*x + sqrt(2) + x ))/56
testint(1/(x**4-x**2+2),x);
sqrt(2*sqrt(2) + 1) - 2*x
( - 2*sqrt(2*sqrt(2) - 1)*sqrt(2)*atan(---------------------------)
sqrt(2*sqrt(2) - 1)
sqrt(2*sqrt(2) + 1) - 2*x
- 8*sqrt(2*sqrt(2) - 1)*atan(---------------------------)
sqrt(2*sqrt(2) - 1)
sqrt(2*sqrt(2) + 1) + 2*x
+ 2*sqrt(2*sqrt(2) - 1)*sqrt(2)*atan(---------------------------)
sqrt(2*sqrt(2) - 1)
sqrt(2*sqrt(2) + 1) + 2*x
+ 8*sqrt(2*sqrt(2) - 1)*atan(---------------------------)
sqrt(2*sqrt(2) - 1)
2
+ sqrt(2*sqrt(2) + 1)*sqrt(2)*log( - sqrt(2*sqrt(2) + 1)*x + sqrt(2) + x )
2
- sqrt(2*sqrt(2) + 1)*sqrt(2)*log(sqrt(2*sqrt(2) + 1)*x + sqrt(2) + x )
2
- 4*sqrt(2*sqrt(2) + 1)*log( - sqrt(2*sqrt(2) + 1)*x + sqrt(2) + x )
2
+ 4*sqrt(2*sqrt(2) + 1)*log(sqrt(2*sqrt(2) + 1)*x + sqrt(2) + x ))/56
testint(1/(x**6-1),x);
2*x - 1 2*x + 1 2
( - 2*sqrt(3)*atan(---------) - 2*sqrt(3)*atan(---------) + log(x - x + 1)
sqrt(3) sqrt(3)
2
- log(x + x + 1) + 2*log(x - 1) - 2*log(x + 1))/12
testint(1/(x**6-2),x);
1/6 1/6
1/6 2 - 2*x 2 + 2*x
(2 *(2*sqrt(3)*atan(--------------) - 2*sqrt(3)*atan(--------------)
1/6 1/6
2 *sqrt(3) 2 *sqrt(3)
1/6 1/6 1/6 1/3 2
- 2*log(2 + x) + 2*log( - 2 + x) + log( - 2 *x + 2 + x )
1/6 1/3 2
- log(2 *x + 2 + x )))/24
testint(1/(x**6+2),x);
1/6 1/6
1/6 2 *sqrt(3) - 2*x 2 *sqrt(3) + 2*x
(2 *( - 2*atan(--------------------) + 2*atan(--------------------)
1/6 1/6
2 2
x 1/6 1/3 2
+ 4*atan(------) - sqrt(3)*log( - 2 *sqrt(3)*x + 2 + x )
1/6
2
1/6 1/3 2
+ sqrt(3)*log(2 *sqrt(3)*x + 2 + x )))/24
testint(1/(x**8+1),x);
sqrt( - sqrt(2) + 2) - 2*x
( - 2*sqrt(sqrt(2) + 2)*atan(----------------------------)
sqrt(sqrt(2) + 2)
sqrt( - sqrt(2) + 2) + 2*x
+ 2*sqrt(sqrt(2) + 2)*atan(----------------------------)
sqrt(sqrt(2) + 2)
sqrt(sqrt(2) + 2) - 2*x
- 2*sqrt( - sqrt(2) + 2)*atan(-------------------------)
sqrt( - sqrt(2) + 2)
sqrt(sqrt(2) + 2) + 2*x
+ 2*sqrt( - sqrt(2) + 2)*atan(-------------------------)
sqrt( - sqrt(2) + 2)
2
- sqrt( - sqrt(2) + 2)*log( - sqrt( - sqrt(2) + 2)*x + x + 1)
2
+ sqrt( - sqrt(2) + 2)*log(sqrt( - sqrt(2) + 2)*x + x + 1)
2
- sqrt(sqrt(2) + 2)*log( - sqrt(sqrt(2) + 2)*x + x + 1)
2
+ sqrt(sqrt(2) + 2)*log(sqrt(sqrt(2) + 2)*x + x + 1))/16
testint(1/(x**8-1),x);
sqrt(2) - 2*x sqrt(2) + 2*x
(2*sqrt(2)*atan(---------------) - 2*sqrt(2)*atan(---------------) - 4*atan(x)
sqrt(2) sqrt(2)
2 2
+ sqrt(2)*log( - sqrt(2)*x + x + 1) - sqrt(2)*log(sqrt(2)*x + x + 1)
+ 2*log(x - 1) - 2*log(x + 1))/16
testint(1/(x**8-x**4+1),x);
sqrt(6) + sqrt(2) - 4*x
( - 2*sqrt( - sqrt(3) + 2)*sqrt(3)*atan(-------------------------)
2*sqrt( - sqrt(3) + 2)
sqrt(6) + sqrt(2) - 4*x
- 6*sqrt( - sqrt(3) + 2)*atan(-------------------------)
2*sqrt( - sqrt(3) + 2)
sqrt(6) + sqrt(2) + 4*x
+ 2*sqrt( - sqrt(3) + 2)*sqrt(3)*atan(-------------------------)
2*sqrt( - sqrt(3) + 2)
sqrt(6) + sqrt(2) + 4*x
+ 6*sqrt( - sqrt(3) + 2)*atan(-------------------------)
2*sqrt( - sqrt(3) + 2)
2*sqrt( - sqrt(3) + 2) - 4*x
- 2*sqrt(6)*atan(------------------------------)
sqrt(6) + sqrt(2)
2*sqrt( - sqrt(3) + 2) + 4*x
+ 2*sqrt(6)*atan(------------------------------)
sqrt(6) + sqrt(2)
2
- sqrt( - sqrt(3) + 2)*sqrt(3)*log( - sqrt( - sqrt(3) + 2)*x + x + 1)
2
+ sqrt( - sqrt(3) + 2)*sqrt(3)*log(sqrt( - sqrt(3) + 2)*x + x + 1)
2
- 3*sqrt( - sqrt(3) + 2)*log( - sqrt( - sqrt(3) + 2)*x + x + 1)
2
+ 3*sqrt( - sqrt(3) + 2)*log(sqrt( - sqrt(3) + 2)*x + x + 1)
2
- sqrt(6)*x - sqrt(2)*x + 2*x + 2
- sqrt(6)*log(-------------------------------------)
2
2
sqrt(6)*x + sqrt(2)*x + 2*x + 2
+ sqrt(6)*log(----------------------------------))/24
2
testint(x**7/(x**12+1),x);
sqrt(6) + sqrt(2) - 4*x
( - sqrt( - sqrt(3) + 2)*sqrt(6)*atan(-------------------------)
2*sqrt( - sqrt(3) + 2)
sqrt(6) + sqrt(2) - 4*x
- 3*sqrt( - sqrt(3) + 2)*sqrt(2)*atan(-------------------------)
2*sqrt( - sqrt(3) + 2)
sqrt(6) + sqrt(2) + 4*x
- sqrt( - sqrt(3) + 2)*sqrt(6)*atan(-------------------------)
2*sqrt( - sqrt(3) + 2)
sqrt(6) + sqrt(2) + 4*x
- 3*sqrt( - sqrt(3) + 2)*sqrt(2)*atan(-------------------------)
2*sqrt( - sqrt(3) + 2)
2*sqrt( - sqrt(3) + 2) - 4*x
+ sqrt( - sqrt(3) + 2)*sqrt(6)*atan(------------------------------)
sqrt(6) + sqrt(2)
2*sqrt( - sqrt(3) + 2) - 4*x
+ 3*sqrt( - sqrt(3) + 2)*sqrt(2)*atan(------------------------------)
sqrt(6) + sqrt(2)
2*sqrt( - sqrt(3) + 2) + 4*x
+ sqrt( - sqrt(3) + 2)*sqrt(6)*atan(------------------------------)
sqrt(6) + sqrt(2)
2*sqrt( - sqrt(3) + 2) + 4*x
+ 3*sqrt( - sqrt(3) + 2)*sqrt(2)*atan(------------------------------)
sqrt(6) + sqrt(2)
2 2
+ log( - sqrt( - sqrt(3) + 2)*x + x + 1) - 2*log( - sqrt(2)*x + x + 1)
2 2
+ log(sqrt( - sqrt(3) + 2)*x + x + 1) - 2*log(sqrt(2)*x + x + 1)
2
- sqrt(6)*x - sqrt(2)*x + 2*x + 2
+ log(-------------------------------------)
2
2
sqrt(6)*x + sqrt(2)*x + 2*x + 2
+ log(----------------------------------))/24
2
% Examples involving logarithms.
testint(log x,x);
x*(log(x) - 1)
testint(x*log x,x);
2
x *(2*log(x) - 1)
-------------------
4
testint(x**2*log x,x);
3
x *(3*log(x) - 1)
-------------------
9
testint(x**p*log x,x);
p
x *x*(log(x)*p + log(x) - 1)
------------------------------
2
p + 2*p + 1
testint((log x)**2,x);
2
x*(log(x) - 2*log(x) + 2)
testint(x**9*log x**11,x);
10 11 10 9
(x *(15625000*log(x) - 17187500*log(x) + 17187500*log(x)
8 7 6 5
- 15468750*log(x) + 12375000*log(x) - 8662500*log(x) + 5197500*log(x)
4 3 2
- 2598750*log(x) + 1039500*log(x) - 311850*log(x) + 62370*log(x)
- 6237))/156250000
testint(log x**2/x,x);
3
log(x)
---------
3
testint(1/log x,x);
ei(log(x))
testint(1/log(x+1),x);
ei(log(x + 1))
testint(1/(x*log x),x);
log(log(x))
testint(1/(x*log x)**2,x);
- (ei( - log(x))*log(x)*x + 1)
---------------------------------
log(x)*x
testint((log x)**p/x,x);
p
log(x) *log(x)
----------------
p + 1
testint(log x *(a*x+b),x);
x*(2*log(x)*a*x + 4*log(x)*b - a*x - 4*b)
-------------------------------------------
4
testint((a*x+b)**2*log x,x);
2 2 2 2 2 2
(x*(6*log(x)*a *x + 18*log(x)*a*b*x + 18*log(x)*b - 2*a *x - 9*a*b*x - 18*b )
)/18
testint(log x/(a*x+b)**2,x);
- log(a*x + b)*a*x - log(a*x + b)*b + log(x)*a*x
---------------------------------------------------
a*b*(a*x + b)
testint(x*log (a*x+b),x);
2 2 2 2 2
2*log(a*x + b)*a *x - 2*log(a*x + b)*b - a *x + 2*a*b*x
------------------------------------------------------------
2
4*a
testint(x**2*log(a*x+b),x);
3 3 3 3 3 2 2 2
6*log(a*x + b)*a *x + 6*log(a*x + b)*b - 2*a *x + 3*a *b*x - 6*a*b *x
---------------------------------------------------------------------------
3
18*a
testint(log(x**2+a**2),x);
x 2 2
2*atan(---)*a + log(a + x )*x - 2*x
a
testint(x*log(x**2+a**2),x);
2 2 2 2 2 2 2
log(a + x )*a + log(a + x )*x - x
----------------------------------------
2
testint(x**2*log(x**2+a**2),x);
x 3 2 2 3 2 3
- 6*atan(---)*a + 3*log(a + x )*x + 6*a *x - 2*x
a
-------------------------------------------------------
9
testint(x**4*log(x**2+a**2),x);
x 5 2 2 5 4 2 3 5
30*atan(---)*a + 15*log(a + x )*x - 30*a *x + 10*a *x - 6*x
a
------------------------------------------------------------------
75
testint(log(x**2-a**2),x);
2 2 2 2
- log( - a + x )*a + log( - a + x )*x + 2*log( - a - x)*a - 2*x
testint(log(log(log(log(x)))),x);
1
- int(-------------------------------------,x) + log(log(log(log(x))))*x
log(log(log(x)))*log(log(x))*log(x)
% Examples involving circular functions.
testint(sin x,x);
- cos(x)
% 2.01 #5;
testint(cos x,x);
sin(x)
% #6;
testint(tan x,x);
2
log(tan(x) + 1)
------------------
2
% #11;
testint(1/tan(x),x);
2
- log(tan(x) + 1) + 2*log(tan(x))
-------------------------------------
2
% 2.01 #12;
testint(1/(1+tan(x))**2,x);
2 2
( - log(tan(x) + 1)*tan(x) - log(tan(x) + 1) + 2*log(tan(x) + 1)*tan(x)
+ 2*log(tan(x) + 1) + 2*tan(x))/(4*(tan(x) + 1))
testint(1/cos x,x);
x x
- log(tan(---) - 1) + log(tan(---) + 1)
2 2
testint(1/sin x,x);
x
log(tan(---))
2
testint(sin x**2,x);
- cos(x)*sin(x) + x
----------------------
2
testint(x**3*sin(x**2),x);
2 2 2
- cos(x )*x + sin(x )
-------------------------
2
testint(sin x**3,x);
2
- cos(x)*sin(x) - 2*cos(x) + 2
----------------------------------
3
testint(sin x**p,x);
p
int(sin(x) ,x)
testint((sin x**2+1)**2*cos x,x);
4 2
sin(x)*(3*sin(x) + 10*sin(x) + 15)
--------------------------------------
15
testint(cos x**2,x);
cos(x)*sin(x) + x
-------------------
2
testint(cos x**3,x);
2
sin(x)*( - sin(x) + 3)
-------------------------
3
testint(sin(a*x+b),x);
- cos(a*x + b)
-----------------
a
testint(1/cos x**2,x);
sin(x)
--------
cos(x)
testint(sin x*sin(2*x),x);
- 2*cos(2*x)*sin(x) + cos(x)*sin(2*x)
----------------------------------------
3
testint(x*sin x,x);
- cos(x)*x + sin(x)
testint(x**2*sin x,x);
2
- cos(x)*x + 2*cos(x) + 2*sin(x)*x
testint(x*sin x**2,x);
2 2
- 2*cos(x)*sin(x)*x + sin(x) + x - 2
-----------------------------------------
4
testint(x**2*sin x**2,x);
2 2 3
- 6*cos(x)*sin(x)*x + 3*cos(x)*sin(x) + 6*sin(x) *x + 2*x - 3*x
--------------------------------------------------------------------
12
testint(x*sin x**3,x);
2 3
- 3*cos(x)*sin(x) *x - 6*cos(x)*x + sin(x) + 6*sin(x)
---------------------------------------------------------
9
testint(x*cos x,x);
cos(x) + sin(x)*x
testint(x**2*cos x,x);
2
2*cos(x)*x + sin(x)*x - 2*sin(x)
testint(x*cos x**2,x);
2 2
2*cos(x)*sin(x)*x - sin(x) + x + 2
--------------------------------------
4
testint(x**2*cos x**2,x);
2 2 3
6*cos(x)*sin(x)*x - 3*cos(x)*sin(x) - 6*sin(x) *x + 2*x + 3*x
-----------------------------------------------------------------
12
testint(x*cos x**3,x);
2 3
- cos(x)*sin(x) + 7*cos(x) - 3*sin(x) *x + 9*sin(x)*x + 1
-------------------------------------------------------------
9
testint(sin x/x,x);
si(x)
testint(cos x/x,x);
ci(x)
testint(sin x/x**2,x);
ci(x)*x - sin(x)
------------------
x
testint(sin x**2/x,x);
- ci(2*x) + log(x)
---------------------
2
testint(tan x**3,x);
2 2
- log(tan(x) + 1) + tan(x)
-------------------------------
2
% z := a+b*x;
testint(sin z,x);
- cos(a + b*x)
-----------------
b
testint(cos z,x);
sin(a + b*x)
--------------
b
testint(tan z,x);
2
log(tan(a + b*x) + 1)
------------------------
2*b
testint(1/tan z,x);
2
- log(tan(a + b*x) + 1) + 2*log(tan(a + b*x))
-------------------------------------------------
2*b
testint(1/sin z,x);
a + b*x
log(tan(---------))
2
---------------------
b
testint(1/cos z,x);
a + b*x a + b*x
- log(tan(---------) - 1) + log(tan(---------) + 1)
2 2
------------------------------------------------------
b
testint(sin z**2,x);
- cos(a + b*x)*sin(a + b*x) + b*x
------------------------------------
2*b
testint(sin z**3,x);
2
- cos(a + b*x)*sin(a + b*x) - 2*cos(a + b*x) + 2
----------------------------------------------------
3*b
testint(cos z**2,x);
cos(a + b*x)*sin(a + b*x) + b*x
---------------------------------
2*b
testint(cos z**3,x);
2
sin(a + b*x)*( - sin(a + b*x) + 3)
-------------------------------------
3*b
testint(1/cos z**2,x);
sin(a + b*x)
----------------
cos(a + b*x)*b
testint(1/(1+cos x),x);
x
tan(---)
2
testint(1/(1-cos x),x);
- 1
----------
x
tan(---)
2
testint(1/(1+sin x),x);
x
2*tan(---)
2
--------------
x
tan(---) + 1
2
testint(1/(1-sin x),x);
x
- 2*tan(---)
2
---------------
x
tan(---) - 1
2
testint(1/(a+b*sin x),x);
x
tan(---)*a + b
2 2 2
2*sqrt(a - b )*atan(----------------)
2 2
sqrt(a - b )
----------------------------------------
2 2
a - b
testint(1/(a+b*sin x+cos x),x);
x x
tan(---)*a - tan(---) + b
2 2 2 2
2*sqrt(a - b - 1)*atan(---------------------------)
2 2
sqrt(a - b - 1)
-------------------------------------------------------
2 2
a - b - 1
testint(x**2*sin z**2,x);
2 2
( - 6*cos(a + b*x)*sin(a + b*x)*b *x + 3*cos(a + b*x)*sin(a + b*x)
2 3 3 3
+ 6*sin(a + b*x) *b*x + 9*a + 2*b *x - 3*b*x)/(12*b )
testint(cos x*cos(2*x),x);
- cos(2*x)*sin(x) + 2*cos(x)*sin(2*x)
----------------------------------------
3
testint(x**2*cos z**2,x);
2 2
(6*cos(a + b*x)*sin(a + b*x)*b *x - 3*cos(a + b*x)*sin(a + b*x)
2 3 3 3
- 6*sin(a + b*x) *b*x + 2*b *x + 3*b*x)/(12*b )
testint(1/tan x**3,x);
2 2 2
log(tan(x) + 1)*tan(x) - 2*log(tan(x))*tan(x) - 1
------------------------------------------------------
2
2*tan(x)
testint(x**3*tan(x)**4,x);
2 2 3 3 2 2
(48*int(tan(x)*x ,x) - 6*log(tan(x) + 1) + 4*tan(x) *x - 6*tan(x) *x
3 4 2
- 12*tan(x)*x + 12*tan(x)*x + 3*x - 6*x )/12
testint(x**3*tan(x)**6,x);
2 2 5 3 4 2
( - 276*int(tan(x)*x ,x) + 60*log(tan(x) + 1) + 12*tan(x) *x - 9*tan(x) *x
3 3 3 2 2 2 3
- 20*tan(x) *x + 6*tan(x) *x + 48*tan(x) *x - 3*tan(x) + 60*tan(x)*x
4 2
- 114*tan(x)*x - 15*x + 57*x )/60
testint(x*tan(x)**2,x);
2 2
- log(tan(x) + 1) + 2*tan(x)*x - x
---------------------------------------
2
testint(sin(2*x)*cos(3*x),x);
2*cos(3*x)*cos(2*x) + 3*sin(3*x)*sin(2*x)
-------------------------------------------
5
testint(sin x**2*cos x**2,x);
3
2*cos(x)*sin(x) - cos(x)*sin(x) + x
--------------------------------------
8
testint(1/(sin x**2*cos x**2),x);
2
2*sin(x) - 1
---------------
cos(x)*sin(x)
testint(d**x*sin x,x);
x
d *( - cos(x) + log(d)*sin(x))
--------------------------------
2
log(d) + 1
testint(d**x*cos x,x);
x
d *(cos(x)*log(d) + sin(x))
-----------------------------
2
log(d) + 1
testint(x*d**x*sin x,x);
x 2 3
(d *( - cos(x)*log(d) *x + 2*cos(x)*log(d) - cos(x)*x + log(d) *sin(x)*x
2 4 2
- log(d) *sin(x) + log(d)*sin(x)*x + sin(x)))/(log(d) + 2*log(d) + 1)
testint(x*d**x*cos x,x);
x 3 2
(d *(cos(x)*log(d) *x - cos(x)*log(d) + cos(x)*log(d)*x + cos(x)
2 4 2
+ log(d) *sin(x)*x - 2*log(d)*sin(x) + sin(x)*x))/(log(d) + 2*log(d) + 1
)
testint(x**2*d**x*sin x,x);
x 4 2 3 2 2
(d *( - cos(x)*log(d) *x + 4*cos(x)*log(d) *x - 2*cos(x)*log(d) *x
2 2
- 6*cos(x)*log(d) + 4*cos(x)*log(d)*x - cos(x)*x + 2*cos(x)
5 2 4 3 2
+ log(d) *sin(x)*x - 2*log(d) *sin(x)*x + 2*log(d) *sin(x)*x
3 2
+ 2*log(d) *sin(x) + log(d)*sin(x)*x - 6*log(d)*sin(x) + 2*sin(x)*x))/(
6 4 2
log(d) + 3*log(d) + 3*log(d) + 1)
testint(x**2*d**x*cos x,x);
x 5 2 4 3 2
(d *(cos(x)*log(d) *x - 2*cos(x)*log(d) *x + 2*cos(x)*log(d) *x
3 2
+ 2*cos(x)*log(d) + cos(x)*log(d)*x - 6*cos(x)*log(d) + 2*cos(x)*x
4 2 3 2 2
+ log(d) *sin(x)*x - 4*log(d) *sin(x)*x + 2*log(d) *sin(x)*x
2 2 6
+ 6*log(d) *sin(x) - 4*log(d)*sin(x)*x + sin(x)*x - 2*sin(x)))/(log(d)
4 2
+ 3*log(d) + 3*log(d) + 1)
testint(x**3*d**x*sin x,x);
x 6 3 5 2 4 3
(d *( - cos(x)*log(d) *x + 6*cos(x)*log(d) *x - 3*cos(x)*log(d) *x
4 3 2 3
- 18*cos(x)*log(d) *x + 12*cos(x)*log(d) *x + 24*cos(x)*log(d)
2 3 2 2
- 3*cos(x)*log(d) *x - 12*cos(x)*log(d) *x + 6*cos(x)*log(d)*x
3 7 3
- 24*cos(x)*log(d) - cos(x)*x + 6*cos(x)*x + log(d) *sin(x)*x
6 2 5 3 5
- 3*log(d) *sin(x)*x + 3*log(d) *sin(x)*x + 6*log(d) *sin(x)*x
4 2 4 3 3
- 3*log(d) *sin(x)*x - 6*log(d) *sin(x) + 3*log(d) *sin(x)*x
3 2 2 2
- 12*log(d) *sin(x)*x + 3*log(d) *sin(x)*x + 36*log(d) *sin(x)
3 2
+ log(d)*sin(x)*x - 18*log(d)*sin(x)*x + 3*sin(x)*x - 6*sin(x)))/(
8 6 4 2
log(d) + 4*log(d) + 6*log(d) + 4*log(d) + 1)
testint(x**3*d**x*cos x,x);
x 7 3 6 2 5 3
(d *(cos(x)*log(d) *x - 3*cos(x)*log(d) *x + 3*cos(x)*log(d) *x
5 4 2 4
+ 6*cos(x)*log(d) *x - 3*cos(x)*log(d) *x - 6*cos(x)*log(d)
3 3 3 2 2
+ 3*cos(x)*log(d) *x - 12*cos(x)*log(d) *x + 3*cos(x)*log(d) *x
2 3 2
+ 36*cos(x)*log(d) + cos(x)*log(d)*x - 18*cos(x)*log(d)*x + 3*cos(x)*x
6 3 5 2 4 3
- 6*cos(x) + log(d) *sin(x)*x - 6*log(d) *sin(x)*x + 3*log(d) *sin(x)*x
4 3 2 3
+ 18*log(d) *sin(x)*x - 12*log(d) *sin(x)*x - 24*log(d) *sin(x)
2 3 2 2
+ 3*log(d) *sin(x)*x + 12*log(d) *sin(x)*x - 6*log(d)*sin(x)*x
3 8 6
+ 24*log(d)*sin(x) + sin(x)*x - 6*sin(x)*x))/(log(d) + 4*log(d)
4 2
+ 6*log(d) + 4*log(d) + 1)
testint(sin x*sin(2*x)*sin(3*x),x);
( - cos(3*x)*cos(2*x)*cos(x) + 6*cos(3*x)*cos(2*x)*sin(x)*x
+ 6*cos(3*x)*cos(x)*sin(2*x)*x - 8*cos(3*x)*sin(2*x)*sin(x)
- 6*cos(2*x)*cos(x)*sin(3*x)*x + 3*cos(2*x)*sin(3*x)*sin(x)
+ 6*sin(3*x)*sin(2*x)*sin(x)*x)/24
testint(cos x*cos(2*x)*cos(3*x),x);
(6*cos(3*x)*cos(2*x)*cos(x)*x + 8*cos(3*x)*cos(2*x)*sin(x)
+ 5*cos(3*x)*cos(x)*sin(2*x) - 6*cos(3*x)*sin(2*x)*sin(x)*x
+ 6*cos(2*x)*sin(3*x)*sin(x)*x + 6*cos(x)*sin(3*x)*sin(2*x)*x
+ 9*sin(3*x)*sin(2*x)*sin(x))/24
testint(sin(x*kx)**3*x**2,x);
2 2 2 2 2 2
( - 9*cos(kx*x)*sin(kx*x) *kx *x + 2*cos(kx*x)*sin(kx*x) - 18*cos(kx*x)*kx *x
3 3
+ 40*cos(kx*x) + 6*sin(kx*x) *kx*x + 36*sin(kx*x)*kx*x + 16)/(27*kx )
testint(x*cos(xi/sin(x))*cos(x)/sin(x)**2,x);
xi
cos(--------)*cos(x)*x
sin(x)
int(------------------------,x)
2
sin(x)
% Mixed angles and half angles.
int(cos(x)/(sin(x)*tan(x/2)),x);
x
- (tan(---)*x + 1)
2
---------------------
x
tan(---)
2
% This integral produces a messy result because the code for
% converting half angle tans to sin and cos is not effective enough.
testint(sin(a*x)/(b+c*sin(a*x))**2,x);
a*x
tan(-----)*b + c
2 2 2 2
( - 2*sqrt(b - c )*atan(------------------)*sin(a*x)*c
2 2
sqrt(b - c )
a*x
tan(-----)*b + c
2 2 2 3 2
- 2*sqrt(b - c )*atan(------------------)*b*c - cos(a*x)*b + cos(a*x)*b*c )/
2 2
sqrt(b - c )
4 2 3 5 5 3 2 4
(a*(sin(a*x)*b *c - 2*sin(a*x)*b *c + sin(a*x)*c + b - 2*b *c + b*c ))
% Examples involving logarithms and circular functions.
testint(sin log x,x);
x*( - cos(log(x)) + sin(log(x)))
----------------------------------
2
testint(cos log x,x);
x*(cos(log(x)) + sin(log(x)))
-------------------------------
2
% Examples involving exponentials.
testint(e**x,x);
x
e
% 2.01 #3;
testint(a**x,x);
x
a
--------
log(a)
% 2.01 #4;
testint(e**(a*x),x);
a*x
e
------
a
testint(e**(a*x)/x,x);
ei(a*x)
testint(1/(a+b*e**(m*x)),x);
m*x
- log(e *b + a) + m*x
--------------------------
a*m
testint(e**(2*x)/(1+e**x),x);
x x
e - log(e + 1)
testint(e**(2*x)*e**(a*x),x);
a*x + 2*x
e
------------
a + 2
testint(1/(a*e**(m*x)+b*e**(-m*x)),x);
m*x
e *a
sqrt(b)*sqrt(a)*atan(-----------------)
sqrt(b)*sqrt(a)
-----------------------------------------
a*b*m
testint(x*e**(a*x),x);
a*x
e *(a*x - 1)
----------------
2
a
testint(x**20*e**x,x);
x 20 19 18 17 16 15 14
e *(x - 20*x + 380*x - 6840*x + 116280*x - 1860480*x + 27907200*x
13 12 11 10
- 390700800*x + 5079110400*x - 60949324800*x + 670442572800*x
9 8 7
- 6704425728000*x + 60339831552000*x - 482718652416000*x
6 5 4
+ 3379030566912000*x - 20274183401472000*x + 101370917007360000*x
3 2
- 405483668029440000*x + 1216451004088320000*x - 2432902008176640000*x
+ 2432902008176640000)
testint(a**x/b**x,x);
x
a
----------------------
x
b *(log(a) - log(b))
testint(a**x*b**x,x);
x x
b *a
-----------------
log(a) + log(b)
testint(a**x/x**2,x);
x
ei(log(a)*x)*log(a)*x - a
----------------------------
x
testint(x*a**x/(1+b*x)**2,x);
x
a *x
int(-----------------------------------------------------------,x)*(log(a) - b)
2 2 3 2 2
log(a)*b *x + 2*log(a)*b*x + log(a) - b *x - 2*b *x - b
testint(x*e**(a*x)/(1+a*x)**2,x);
a*x
e
--------------
2
a *(a*x + 1)
testint(x*k**(x**2),x);
2
x
k
----------
2*log(k)
testint(e**(x**2),x);
- sqrt(pi)*erf(i*x)*i
------------------------
2
testint(x*e**(x**2),x);
2
x
e
-----
2
testint((x+1)*e**(1/x)/x**4,x);
1/x 2
e *( - x + x - 1)
----------------------
2
x
testint((2*x**3+x)*(e**(x**2))**2*e**(1-x*e**(x**2))/(1-x*e**(x**2))**2,
x);
- e
--------------------
2
x 2
e *x x
e *(e *x - 1)
testint(e**(e**(e**(e**x))),x);
x
e
e
e
int(e ,x)
% Examples involving exponentials and logarithms.
testint(e**x*log x,x);
x
- ei(x) + e *log(x)
testint(x*e**x*log x,x);
x x x
ei(x) + e *log(x)*x - e *log(x) - e
testint(e**(2*x)*log(e**x),x);
2*x
e *(2*x - 1)
----------------
4
% Examples involving square roots.
testint(sqrt(2)*x**2 + 2*x,x);
2
x *(sqrt(2)*x + 3)
--------------------
3
testint(log x/sqrt(a*x+b),x);
(2*(sqrt(a*x + b)*log(x) - 2*sqrt(a*x + b)
+ 2*sqrt(b)*log( - sqrt(a*x + b) - sqrt(b)) - sqrt(b)*log(x)))/a
u:=sqrt(a+b*x);
u := sqrt(a + b*x)
v:=sqrt(c+d*x);
v := sqrt(c + d*x)
testint(u*v,x);
2 2
(sqrt(c + d*x)*sqrt(a + b*x)*a*b*d + sqrt(c + d*x)*sqrt(a + b*x)*b *c*d
2 2
+ 2*sqrt(c + d*x)*sqrt(a + b*x)*b *d *x
sqrt(d)*sqrt(a + b*x) + sqrt(b)*sqrt(c + d*x) 2 2
- sqrt(d)*sqrt(b)*log(-----------------------------------------------)*a *d +
sqrt(a*d - b*c)
sqrt(d)*sqrt(a + b*x) + sqrt(b)*sqrt(c + d*x)
2*sqrt(d)*sqrt(b)*log(-----------------------------------------------)*a*b*c*d
sqrt(a*d - b*c)
sqrt(d)*sqrt(a + b*x) + sqrt(b)*sqrt(c + d*x) 2 2
- sqrt(d)*sqrt(b)*log(-----------------------------------------------)*b *c )/
sqrt(a*d - b*c)
2 2
(4*b *d )
testint(u,x);
2*sqrt(a + b*x)*(a + b*x)
---------------------------
3*b
testint(x*u,x);
2 2 2
2*sqrt(a + b*x)*( - 2*a + a*b*x + 3*b *x )
---------------------------------------------
2
15*b
testint(x**2*u,x);
3 2 2 2 3 3
2*sqrt(a + b*x)*(8*a - 4*a *b*x + 3*a*b *x + 15*b *x )
----------------------------------------------------------
3
105*b
testint(u/x,x);
2*sqrt(a + b*x) - sqrt(a)*log( - sqrt(a + b*x) - sqrt(a))
+ sqrt(a)*log( - sqrt(a + b*x) + sqrt(a))
testint(u/x**2,x);
( - 2*sqrt(a + b*x)*a - sqrt(a)*log( - sqrt(a + b*x) - sqrt(a))*b*x
+ sqrt(a)*log( - sqrt(a + b*x) + sqrt(a))*b*x)/(2*a*x)
testint(1/u,x);
2*sqrt(a + b*x)
-----------------
b
testint(x/u,x);
2*sqrt(a + b*x)*( - 2*a + b*x)
--------------------------------
2
3*b
testint(x**2/u,x);
2 2 2
2*sqrt(a + b*x)*(8*a - 4*a*b*x + 3*b *x )
--------------------------------------------
3
15*b
testint(1/(x*u),x);
sqrt(a)*( - log( - sqrt(a + b*x) - sqrt(a)) + log( - sqrt(a + b*x) + sqrt(a)))
--------------------------------------------------------------------------------
a
testint(1/(x**2*u),x);
( - 2*sqrt(a + b*x)*a + sqrt(a)*log( - sqrt(a + b*x) - sqrt(a))*b*x
2
- sqrt(a)*log( - sqrt(a + b*x) + sqrt(a))*b*x)/(2*a *x)
testint(u**p,x);
p/2
2*(a + b*x) *(a + b*x)
--------------------------
b*(p + 2)
testint(x*u**p,x);
p/2 2 2 2 2 2
2*(a + b*x) *( - 2*a + a*b*p*x + b *p*x + 2*b *x )
--------------------------------------------------------
2 2
b *(p + 6*p + 8)
testint(atan((-sqrt(2)+2*x)/sqrt(2)),x);
sqrt(2) - 2*x sqrt(2) - 2*x
(2*sqrt(2)*atan(---------------) - 4*atan(---------------)*x
sqrt(2) sqrt(2)
2
- sqrt(2)*log(sqrt(2)*x - x - 1))/4
testint(1/sqrt(x**2-1),x);
2
log(sqrt(x - 1) + x)
testint(sqrt(x+1)*sqrt x,x);
2*sqrt(x)*sqrt(x + 1)*x + sqrt(x)*sqrt(x + 1) - log(sqrt(x + 1) + sqrt(x))
----------------------------------------------------------------------------
4
testint(sin(sqrt x),x);
2*( - sqrt(x)*cos(sqrt(x)) + sin(sqrt(x)))
testint(x*(1-x^2)^(-9/4),x);
2 1/4
- 2*( - x + 1)
----------------------------
2 2
5*sqrt( - x + 1)*(x - 1)
testint(x/sqrt(1-x^4),x);
2
asin(x )
----------
2
testint(1/(x*sqrt(1+x^4)),x);
4 2 4 2
log(sqrt(x + 1) + x - 1) - log(sqrt(x + 1) + x + 1)
---------------------------------------------------------
2
testint(x/sqrt(1+x^2+x^4),x);
4 2 2
2*sqrt(x + x + 1) + 2*x + 1
log(--------------------------------)
sqrt(3)
---------------------------------------
2
testint(1/(x*sqrt(x^2-1-x^4)),x);
4 2
sqrt( - x + x - 1)
- int(----------------------,x)
5 3
x - x + x
% Examples from James Davenport's thesis:
testint(1/sqrt(x**2-1)+10/sqrt(x**2-4),x);
2
2 sqrt(x - 4) + x
log(sqrt(x - 1) + x) + 10*log(------------------)
2
% p. 173
testint(sqrt(x+sqrt(x**2+a**2))/x,x);
2 2
sqrt(sqrt(a + x ) + x)
int(-------------------------,x)
x
% Examples generated by differentiating various functions.
testint(df(sqrt(1+x**2)/(1-x),x),x);
2
- sqrt(x + 1)
-----------------
x - 1
testint(df(log(x+sqrt(1+x**2)),x),x);
2
log(sqrt(x + 1) + x)
testint(df(sqrt(x)+sqrt(x+1)+sqrt(x+2),x),x);
sqrt(x + 2) + sqrt(x + 1) + sqrt(x)
testint(df(sqrt(x**5-2*x+1)-sqrt(x**3+1),x),x);
5 3
sqrt(x - 2*x + 1) - sqrt(x + 1)
% Another such example from James Davenport's thesis (p. 146).
% It contains a point of order 3, which is found by use of Mazur's
% bound on the torsion of elliptic curves over the rationals;
testint(df(log(1+sqrt(x**3+1)),x),x);
3
sqrt(x + 1)
3*( - int(--------------,x) + log(x))
4
x + x
---------------------------------------
2
% Examples quoted by Joel Moses:
testint(1/sqrt(2*h*r**2-alpha**2),r);
2 2
sqrt( - alpha + 2*h*r ) + sqrt(h)*sqrt(2)*r
sqrt(h)*sqrt(2)*log(----------------------------------------------)
alpha
---------------------------------------------------------------------
2*h
testint(1/(r*sqrt(2*h*r**2-alpha**2-epsilon**2)),r);
2 2
(2*sqrt(alpha + epsilon )
2 2 2
sqrt( - alpha - epsilon + 2*h*r ) + sqrt(h)*sqrt(2)*r 2
*atan(---------------------------------------------------------))/(alpha
2 2
sqrt(alpha + epsilon )
2
+ epsilon )
testint(1/(r*sqrt(2*h*r**2-alpha**2-2*k*r)),r);
2 2
sqrt(h)*sqrt( - alpha + 2*h*r - 2*k*r)*sqrt(2) + 2*h*r
2*atan(----------------------------------------------------------)
sqrt(h)*sqrt(2)*alpha
--------------------------------------------------------------------
alpha
testint(1/(r*sqrt(2*h*r**2-alpha**2-epsilon**2-2*k*r)),r);
2 2
(2*sqrt(alpha + epsilon )
2 2 2
sqrt(h)*sqrt( - alpha - epsilon + 2*h*r - 2*k*r)*sqrt(2) + 2*h*r
*atan(---------------------------------------------------------------------))/(
2 2
sqrt(h)*sqrt(alpha + epsilon )*sqrt(2)
2 2
alpha + epsilon )
testint(r/sqrt(2*e*r**2-alpha**2),r);
2 2
sqrt( - alpha + 2*e*r )
--------------------------
2*e
testint(r/sqrt(2*e*r**2-alpha**2-epsilon**2),r);
2 2 2
sqrt( - alpha + 2*e*r - epsilon )
-------------------------------------
2*e
testint(r/sqrt(2*e*r**2-alpha**2-2*k*r**4),r);
2
e*i - 2*i*k*r
sqrt(k)*sqrt(2)*asinh(--------------------------)*i
2 2
sqrt( - 2*alpha *k + e )
-----------------------------------------------------
4*k
testint(r/sqrt(2*e*r**2-alpha**2-2*k*r),r);
2 2
(2*sqrt( - alpha + 2*e*r - 2*k*r)*e + sqrt(e)*sqrt(2)
2 2
sqrt(e)*sqrt( - alpha + 2*e*r - 2*k*r)*sqrt(2) + 2*e*r - k 2
*log(--------------------------------------------------------------)*k)/(4*e )
2 2
sqrt(2*alpha *e + k )
testint(1/(r*sqrt(2*h*r**2-alpha**2-2*k*r**4)),r);
2 2 4
sqrt( - alpha + 2*h*r - 2*k*r )
- int(-----------------------------------,r)
2 3 5
alpha *r - 2*h*r + 2*k*r
testint(1/(r*sqrt(2*h*r**2-alpha**2-epsilon**2-2*k*r**4)),r);
2 2 2 4
sqrt( - alpha - epsilon + 2*h*r - 2*k*r )
- int(----------------------------------------------,r)
2 2 3 5
alpha *r + epsilon *r - 2*h*r + 2*k*r
Comment many of these integrals used to require Steve Harrington's
code to evaluate. They originated in Novosibirsk as examples
of using Analytik. There are still a few examples that could
be evaluated using better heuristics;
testint(a*sin(3*x+5)**2*cos(3*x+5),x);
3
sin(3*x + 5) *a
-----------------
9
testint(log(x**2)/x**3,x);
2
- (log(x ) + 1)
------------------
2
2*x
testint(x*sin(x+a),x);
- cos(a + x)*x + sin(a + x)
testint((log(x)*(1-x)-1)/(e**x*log(x)**2),x);
x
-----------
x
e *log(x)
testint(x**3*(a*x**2+b)**(-1),x);
2 2
- log(a*x + b)*b + a*x
---------------------------
2
2*a
testint(x**(1/2)*(x+1)**(-7/2),x);
2 2
(2*( - 2*sqrt(x + 1)*x - 4*sqrt(x + 1)*x - 2*sqrt(x + 1) + 2*sqrt(x)*x
2
+ 5*sqrt(x)*x))/(15*sqrt(x + 1)*(x + 2*x + 1))
testint(x**(-1)*(x+1)**(-1),x);
- log(x + 1) + log(x)
testint(x**(-1/2)*(2*x-1)**(-1),x);
sqrt(2)*(log(2*sqrt(x) - sqrt(2)) - log(2*sqrt(x) + sqrt(2)))
---------------------------------------------------------------
2
testint((x**2+1)*x**(1/2),x);
2
2*sqrt(x)*x*(3*x + 7)
------------------------
21
testint(x**(-1)*(x-a)**(1/3),x);
1/6 1/6
2*( - a + x) - a *sqrt(3)
( - 2*sqrt(3)*atan(--------------------------------)*a
1/6
a
1/6 1/6
2*( - a + x) + a *sqrt(3) 2/3 1/3
+ 2*sqrt(3)*atan(--------------------------------)*a + 6*a *( - a + x)
1/6
a
1/3 1/3
- 2*log(( - a + x) + a )*a
1/6 1/6 1/3 1/3
+ log( - a *( - a + x) *sqrt(3) + ( - a + x) + a )*a
1/6 1/6 1/3 1/3 2/3
+ log(a *( - a + x) *sqrt(3) + ( - a + x) + a )*a)/(2*a )
testint(x*sinh(x),x);
cosh(x)*x - sinh(x)
testint(x*cosh(x),x);
- cosh(x) + sinh(x)*x
testint(sinh(2*x)/cosh(2*x),x);
log(cosh(2*x))
----------------
2
testint((i*eps*sinh x-1)/(eps*i*cosh x+i*a-x),x);
log(cosh(x)*eps*i + a*i - x)
testint(sin(2*x+3)*cos(x)**2,x);
2
( - 4*cos(2*x + 3)*cos(x)*sin(x)*x + 2*cos(2*x + 3)*sin(x) - 3*cos(2*x + 3)
2
- 4*sin(2*x + 3)*sin(x) *x + 2*sin(2*x + 3)*x + 3)/8
testint(x*atan(x),x);
2
atan(x)*x + atan(x) - x
--------------------------
2
testint(x*acot(x),x);
2
acot(x)*x + acot(x) + x
--------------------------
2
testint(x*log(x**2+a),x);
2 2 2 2
log(a + x )*a + log(a + x )*x - x
-------------------------------------
2
testint(sin(x+a)*cos(x),x);
- cos(a + x)*cos(x) - cos(a + x)*sin(x)*x + cos(x)*sin(a + x)*x
------------------------------------------------------------------
2
testint(cos(x+a)*sin(x),x);
- cos(a + x)*cos(x) + cos(a + x)*sin(x)*x - cos(x)*sin(a + x)*x
------------------------------------------------------------------
2
testint((1+sin(x))**(1/2),x);
int(sqrt(sin(x) + 1),x)
testint((1-sin(x))**(1/2),x);
int(sqrt( - sin(x) + 1),x)
testint((1+cos(x))**(1/2),x);
int(sqrt(cos(x) + 1),x)
testint((1-cos(x))**(1/2),x);
int(sqrt( - cos(x) + 1),x)
testint(1/(x**(1/2)-(x-1)**(1/2)),x);
2*(sqrt(x - 1)*x - sqrt(x - 1) + sqrt(x)*x)
---------------------------------------------
3
testint(1/(1-(x+1)**(1/2)),x);
- 2*(sqrt(x + 1) + log(sqrt(x + 1) - 1))
testint(x/(x**4+36)**(1/2),x);
4 2
sqrt(x + 36) + x
log(--------------------)
6
---------------------------
2
testint(1/(x**(1/3)+x**(1/2)),x);
1/6 1/3 1/6
6*x - 3*x + 2*sqrt(x) - 6*log(x + 1)
testint(log(2+3*x**2),x);
3*x 2
2*sqrt(6)*atan(---------) + 3*log(3*x + 2)*x - 6*x
sqrt(6)
-----------------------------------------------------
3
testint(cot(x),x);
x 2 x
- log(tan(---) + 1) + log(tan(---))
2 2
testint(cot x**4,x);
3
- cot(x) + 3*cot(x) + 3*x
-----------------------------
3
testint(tanh(x),x);
2*x
log(e + 1) - x
testint(coth(x),x);
x x
log(e - 1) + log(e + 1) - x
testint(b**x,x);
x
b
--------
log(b)
testint((x**4+x**(-4)+2)**(1/2),x);
4
x - 3
--------
3*x
testint((2*x+1)/(3*x+2),x);
- log(3*x + 2) + 6*x
-----------------------
9
testint(x*log(x+(x**2+1)**(1/2)),x);
2 2 2 2
- sqrt(x + 1)*x + 2*log(sqrt(x + 1) + x)*x + log(sqrt(x + 1) + x)
------------------------------------------------------------------------
4
testint(x*(e**x*sin(x)+1)**2,x);
2*x 2*x x x
( - 2*e *cos(x)*sin(x)*x + e *cos(x)*sin(x) - 8*e *cos(x)*x + 8*e *cos(x)
2*x 2 2*x 2*x x 2
+ 2*e *sin(x) *x + e *x - e + 8*e *sin(x)*x + 4*x )/8
testint(x*e**x*cos(x),x);
x
e *(cos(x)*x + sin(x)*x - sin(x))
-----------------------------------
2
Comment the following set came from Herbert Stoyan;
testint(1/(x-3)**4,x);
- 1
---------------------------
3 2
3*(x - 9*x + 27*x - 27)
testint(x/(x**3-1),x);
2*x + 1 2
2*sqrt(3)*atan(---------) - log(x + x + 1) + 2*log(x - 1)
sqrt(3)
------------------------------------------------------------
6
testint(x/(x**4-1),x);
2
- log(x + 1) + log(x - 1) + log(x + 1)
------------------------------------------
4
testint(log(x)*(x**3+1)/(x**4+2),x);
log(x) log(x) 2
- 4*int(----------,x) + 2*int(--------,x) + log(x)
5 4
x + 2*x x + 2
------------------------------------------------------
2
testint(log(x)+log(x+1)+log(x+2),x);
log(x + 2)*x + 2*log(x + 2) + log(x + 1)*x + log(x + 1) + log(x)*x - 3*x
testint(1/(x**3+5),x);
1/3
1/3 5 - 2*x 2/3 1/3 2
(5 *( - 2*sqrt(3)*atan(--------------) - log(5 - 5 *x + x )
1/3
sqrt(3)*5
1/3
+ 2*log(5 + x)))/30
testint(1/sqrt(1+x**2),x);
2
log(sqrt(x + 1) + x)
testint(sqrt(x**2+3),x);
2
2 sqrt(x + 3) + x
sqrt(x + 3)*x + 3*log(------------------)
sqrt(3)
--------------------------------------------
2
testint(x/(x+1)**2,x);
log(x + 1)*x + log(x + 1) - x
-------------------------------
x + 1
COMMENT The following integrals were used among others as a test of
Moses' SIN program;
testint(asin x,x);
2
asin(x)*x + sqrt( - x + 1)
testint(x**2*asin x,x);
2
int(asin(x)*x ,x)
testint(sec x**2/(1+sec x**2-3*tan x),x);
x x
log( - sqrt(5) + 2*tan(---) + 1) - log( - sqrt(2) + tan(---) + 1)
2 2
x x
+ log(sqrt(5) + 2*tan(---) + 1) - log(sqrt(2) + tan(---) + 1)
2 2
testint(1/sec x**2,x);
cos(x)*sin(x) + x
-------------------
2
testint((5*x**2-3*x-2)/(x**2*(x-2)),x);
3*log(x - 2)*x + 2*log(x)*x - 1
---------------------------------
x
testint(1/(4*x**2+9)**(1/2),x);
2
sqrt(4*x + 9) + 2*x
log(----------------------)
3
-----------------------------
2
testint((x**2+4)**(-1/2),x);
2
sqrt(x + 4) + x
log(------------------)
2
testint(1/(9*x**2-12*x+10),x);
3*x - 2
sqrt(6)*atan(---------)
sqrt(6)
-------------------------
18
testint(1/(x**8-2*x**7+2*x**6-2*x**5+x**4),x);
2 4 2 3 4 3
(3*log(x + 1)*x - 3*log(x + 1)*x - 30*log(x - 1)*x + 30*log(x - 1)*x
4 3 4 2 3
+ 24*log(x)*x - 24*log(x)*x - 30*x + 12*x + 8*x + 4)/(12*x *(x - 1))
testint((a*x**3+b*x**2+c*x+d)/((x+1)*x*(x-3)),x);
(27*log(x - 3)*a + 9*log(x - 3)*b + 3*log(x - 3)*c + log(x - 3)*d
- 3*log(x + 1)*a + 3*log(x + 1)*b - 3*log(x + 1)*c + 3*log(x + 1)*d
- 4*log(x)*d + 12*a*x)/12
testint(1/(2-log(x**2+1))**5,x);
2 5 2 4 2 3 2 2
- int(1/(log(x + 1) - 10*log(x + 1) + 40*log(x + 1) - 80*log(x + 1)
2
+ 80*log(x + 1) - 32),x)
% The next integral appeared in Risch's 1968 paper.
testint(2*x*e**(x**2)*log(x)+e**(x**2)/x+(log(x)-2)/(log(x)**2+x)**2+
((2/x)*log(x)+(1/x)+1)/(log(x)**2+x),x);
2 2
x 3 x 2 2 2
(e *log(x) + e *log(x)*x + log(log(x) + x)*log(x) + log(log(x) + x)*x
2
- log(x))/(log(x) + x)
% The following integral would not evaluate in REDUCE 3.3.
testint(exp(x*ze+x/2)*sin(pi*ze)**4*x**4,ze);
(2*x*ze + x)/2 3 3 3
(e *x *( - 16*cos(pi*ze)*sin(pi*ze) *pi *x
3 3 3
- 4*cos(pi*ze)*sin(pi*ze) *pi*x - 24*cos(pi*ze)*sin(pi*ze)*pi *x
4 2 2 4 4 2 2 2 4
+ 4*sin(pi*ze) *pi *x + sin(pi*ze) *x + 12*sin(pi*ze) *pi *x + 24*pi ))/
4 2 2 4
(64*pi + 20*pi *x + x )
% This one evaluates:
testint(erf(x),x);
2
x
e *erf(x)*pi*x + sqrt(pi)
----------------------------
2
x
e *pi
% So why not this one?
testint(erf(x+a),x);
int(erf(a + x),x)
Comment here is an example of using the integrator with pattern
matching;
for all m,n let int(k1**m*log(k1)**n/(p**2-k1**2),k1)=foo(m,n),
int(k1*log(k1)**n/(p**2-k1**2),k1)=foo(1,n),
int(k1**m*log(k1)/(p**2-k1**2),k1)=foo(m,1),
int(k1*log(k1)/(p**2-k1**2),k1)=foo(1,1),
int(log(k1)**n/(k1*(p**2-k1**2)),k1)=foo(-1,n);
int(k1**2*log(k1)/(p**2-k1**2),k1);
*** foo declared operator
foo(2,1)
COMMENT It is interesting to see how much of this one can be done;
let f1s= (12*log(s/mc**2)*s**2*pi**2*mc**3*(-8*s-12*mc**2+3*mc)
+ pi**2*(12*s**4*mc+3*s**4+176*s**3*mc**3-24*s**3*mc**2
-144*s**2*mc**5-48*s*mc**7+24*s*mc**6+4*mc**9-3*mc**8))
/(384*e**(s/y)*s**2);
int(f1s,s);
2 s/y - s 9 s/y - s 8
(pi *( - 4*e *ei(------)*mc *s + 3*e *ei(------)*mc *s
y y
s/y - s 7 s/y - s 6
- 48*e *ei(------)*mc *s*y + 24*e *ei(------)*mc *s*y
y y
s/y - s 5 2 s/y - s 4 2
- 144*e *ei(------)*mc *s*y + 36*e *ei(------)*mc *s*y
y y
s/y - s 3 3 s 5 2
- 96*e *ei(------)*mc *s*y + 144*log(-----)*mc *s*y
y 2
mc
s 4 2 s 3 2 2
- 36*log(-----)*mc *s*y + 96*log(-----)*mc *s *y
2 2
mc mc
s 3 3 9 8 5 2
+ 96*log(-----)*mc *s*y - 4*mc *y + 3*mc *y + 144*mc *s*y
2
mc
3 2 2 3 3 2 2 2 2 3 3 2
- 176*mc *s *y - 80*mc *s*y + 24*mc *s *y + 24*mc *s*y - 12*mc*s *y
2 3 4 3 2 2 3 4 s/y
- 24*mc*s *y - 24*mc*s*y - 3*s *y - 6*s *y - 6*s*y ))/(384*e *s*y)
factor ei,log;
ws;
s/y - s 3 2
(e *ei(------)*mc *pi *s
y
6 5 4 3 2 2 2 3
*( - 4*mc + 3*mc - 48*mc *y + 24*mc *y - 144*mc *y + 36*mc*y - 96*y )
s 3 2 2 2 2 9
+ 12*log(-----)*mc *pi *s*y *(12*mc - 3*mc + 8*s + 8*y) + pi *y*( - 4*mc
2
mc
8 5 3 2 3 2 2 2
+ 3*mc + 144*mc *s*y - 176*mc *s *y - 80*mc *s*y + 24*mc *s *y
2 2 3 2 2 3 3 2 2
+ 24*mc *s*y - 12*mc*s *y - 24*mc*s *y - 24*mc*s*y - 3*s *y - 6*s *y
3 s/y
- 6*s*y ))/(384*e *s*y)
Comment the following integrals reveal deficiencies in the current
integrator;
%high degree denominator;
%testint(1/(2-log(x**2+1))**5,x);
%this example should evaluate;
testint(sin(2*x)/cos(x),x);
sin(2*x)
int(----------,x)
cos(x)
%this example, which appeared in Tobey's thesis, needs factorization
%over algebraic fields. It currently gives an ugly answer and so has
%been suppressed;
% testint((7*x**13+10*x**8+4*x**7-7*x**6-4*x**3-4*x**2+3*x+3)/
% (x**14-2*x**8-2*x**7-2*x**4-4*x**3-x**2+2*x+1),x);
symbolic summarize!-integral!-test();
***** SUMMARY OF INTEGRAL TESTS *****
Number of integrals tested: 278
Total time taken: 19320 ms
Number of garbage collections: 38
Number of incorrect integrals: 0
Number of unevaluated integrals: 21
Integrands of unevaluated integrals are:
log(log(log(log(x))))
p
sin(x)
4 3
tan(x) *x
6 3
tan(x) *x
xi
cos(--------)*cos(x)*x
sin(x)
------------------------
2
sin(x)
x
a *x
-------------------
2 2
b *x + 2*b*x + 1
x
e
e
e
e
1
------------------------
4 2
sqrt( - x + x - 1)*x
2 2
sqrt(sqrt(a + x ) + x)
-------------------------
x
2
3*x
---------------------------
3 3
2*sqrt(x + 1) + 2*x + 2
1
-------------------------------------
2 2 4
sqrt( - alpha + 2*h*r - 2*k*r )*r
1
------------------------------------------------
2 2 2 4
sqrt( - alpha - epsilon + 2*h*r - 2*k*r )*r
sqrt(sin(x) + 1)
sqrt( - sin(x) + 1)
sqrt(cos(x) + 1)
sqrt( - cos(x) + 1)
3
log(x)*x + log(x)
--------------------
4
x + 2
2
asin(x)*x
2 5 2 4 2 3 2 2
( - 1)/(log(x + 1) - 10*log(x + 1) + 40*log(x + 1) - 80*log(x + 1)
2
+ 80*log(x + 1) - 32)
erf(a + x)
sin(2*x)
----------
cos(x)
end;
4: 4: 4: 4: 4: 4: 4: 4: 4:
Time for test: 19340 ms, plus GC time: 1060 ms
5: 5:
Quitting
Sun Jan 3 23:46:20 MET 1999