Sat May 30 16:21:53 PDT 1992
REDUCE 3.4.1, 15-Jul-92 ...
1: 1:
2: 2:
(ALGINT)
3: 3:
Time: 425 ms
4: 4: 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";
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=sin x/cos x,
tan(x/2)=sin x/(1+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 <<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!*;
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*LOG(X - 1)*X - 9*LOG(X - 1)*X + 9*LOG(X - 1)*X - 3*LOG(X - 1)
3 3 2
- 6*X - 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
LOG(X + 1)*A + 2*ATAN(X)*B
-----------------------------
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);
2 3 2 2 2 2
(LOG(X + 1)*X - LOG(X + 1)*X + LOG(X + 1)*X - LOG(X + 1)
3 2
- 2*LOG(X - 1)*X + 2*LOG(X - 1)*X - 2*LOG(X - 1)*X + 2*LOG(X - 1)
3 2 3
+ ATAN(X)*X - ATAN(X)*X + ATAN(X)*X - ATAN(X) - X - 2*X + 1)/(4
3 2
*(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
+ LOG(C - X)*A*C - LOG(C - X)*B*C)/(A *B - A *C - A*B + A*C
2 2
+ 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
- LOG(X + 1) + 2*LOG(X - 1) + 2*ATAN(X)
-------------------------------------------
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);
2 2
(SQRT(2)*( - LOG( - SQRT(2)*X + X + 1) + LOG(SQRT(2)*X + X + 1)
- SQRT(2) + 2*X SQRT(2) + 2*X
+ 2*ATAN(------------------) + 2*ATAN(---------------)))/8
SQRT(2) SQRT(2)
testint(x**2/(1+x**4),x);
2 2
(SQRT(2)*(LOG( - SQRT(2)*X + X + 1) - LOG(SQRT(2)*X + X + 1)
- SQRT(2) + 2*X SQRT(2) + 2*X
+ 2*ATAN(------------------) + 2*ATAN(---------------)))/8
SQRT(2) SQRT(2)
testint(1/(1+x**2+x**4),x);
2*X - 1 2*X + 1
(2*SQRT(3)*ATAN(---------) + 2*SQRT(3)*ATAN(---------)
SQRT(3) SQRT(3)
2 2
- 3*LOG(X - X + 1) + 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 3 2 2 2 2 2 2 3 2 3
((A + B*X) *(2*A - 2*A *B*P*X + A*B *P *X + A*B *P*X + B *P *X
3 3 3 3 3 3 2
+ 3*B *P*X + 2*B *X ))/(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*LOG(A + B*X)*A*B*X + 2*LOG(A + B*X)*B *X - 2*LOG(X)*A*B*X
2 2 2 2 2 3
- 2*LOG(X)*B *X - 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
- LOG(A *X + B *A *X + B )
1/3 1/3
+ 2*LOG(A *X - B )))/(6*A*B)
testint(1/(x**4-2),x);
1/4 1/4 1/4 X
2 *( - LOG(2 + X) + LOG( - 2 + X) - 2*ATAN(------))
1/4
2
-------------------------------------------------------------
8
testint(1/(5*x**4-1),x);
1/4
(SQRT(5)*5
1/4 1/4 SQRT(5)*X
*(LOG(5 *X - 1) - LOG(5 *X + 1) - 2*ATAN(-----------)))/20
1/4
5
testint(1/(3*x**4+7),x);
1/4 1/4 2
(SQRT(6)*21 *( - LOG( - SQRT(2)*21 *X + SQRT(7) + SQRT(3)*X )
1/4 2
+ LOG(SQRT(2)*21 *X + SQRT(7) + SQRT(3)*X )
1/4
- SQRT(2)*21 + 2*SQRT(3)*X
+ 2*ATAN(--------------------------------)
1/4
SQRT(2)*21
1/4
SQRT(2)*21 + 2*SQRT(3)*X
+ 2*ATAN(-----------------------------)))/168
1/4
SQRT(2)*21
testint(1/(x**4+3*x**2-1),x);
(SQRT(2)*(3*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)
+ 6*SQRT(SQRT(13) + 3)*SQRT(13)
2*X
*ATAN(----------------------------)
SQRT(SQRT(13) + 3)*SQRT(2)
2*X
- 26*SQRT(SQRT(13) + 3)*ATAN(----------------------------)
SQRT(SQRT(13) + 3)*SQRT(2)
))/104
testint(1/(x**4-3*x**2-1),x);
(SQRT(2)*( - 6*SQRT(SQRT(13) - 3)*SQRT(13)
2*X
*ATAN(----------------------------)
SQRT(SQRT(13) - 3)*SQRT(2)
2*X
- 26*SQRT(SQRT(13) - 3)*ATAN(----------------------------)
SQRT(SQRT(13) - 3)*SQRT(2)
- 3*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);
X
(4*SQRT(SQRT(3) - 2)*SQRT(3)*ATAN(-------------------)
SQRT(SQRT(3) - 2)
X
+ 6*SQRT(SQRT(3) - 2)*ATAN(-------------------)
SQRT(SQRT(3) - 2)
+ 2*SQRT(SQRT(3) + 2)*SQRT(3)*LOG( - SQRT(SQRT(3) + 2) + X)
- 2*SQRT(SQRT(3) + 2)*SQRT(3)*LOG(SQRT(SQRT(3) + 2) + X)
- 3*SQRT(SQRT(3) + 2)*LOG( - SQRT(SQRT(3) + 2) + X)
+ 3*SQRT(SQRT(3) + 2)*LOG(SQRT(SQRT(3) + 2) + X))/12
testint(1/(x**4+4*x**2+1),x);
( - 2*SQRT(SQRT(3) - 2)*SQRT(3)*LOG( - SQRT(SQRT(3) - 2) + X)
+ 2*SQRT(SQRT(3) - 2)*SQRT(3)*LOG(SQRT(SQRT(3) - 2) + X)
- 3*SQRT(SQRT(3) - 2)*LOG( - SQRT(SQRT(3) - 2) + X)
+ 3*SQRT(SQRT(3) - 2)*LOG(SQRT(SQRT(3) - 2) + X)
X
- 4*SQRT(SQRT(3) + 2)*SQRT(3)*ATAN(-------------------)
SQRT(SQRT(3) + 2)
X
+ 6*SQRT(SQRT(3) + 2)*ATAN(-------------------))/12
SQRT(SQRT(3) + 2)
testint(1/(x**4+x**2+2),x);
( - SQRT(2*SQRT(2) - 1)*SQRT(2)
2
*LOG( - SQRT(2*SQRT(2) - 1)*X + SQRT(2) + X ) + SQRT(2*SQRT(2) - 1)
2
*SQRT(2)*LOG(SQRT(2*SQRT(2) - 1)*X + SQRT(2) + X ) - 4
2
*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 ) -
- 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
- 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
+ 8*SQRT(2*SQRT(2) + 1)*ATAN(---------------------------))/56
SQRT(2*SQRT(2) + 1)
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
+ 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
+ 8*SQRT(2*SQRT(2) - 1)*ATAN(---------------------------) +
SQRT(2*SQRT(2) - 1)
SQRT(2*SQRT(2) + 1)*SQRT(2)
2
*LOG( - SQRT(2*SQRT(2) + 1)*X + SQRT(2) + X ) - SQRT(2*SQRT(2) + 1)
2
*SQRT(2)*LOG(SQRT(2*SQRT(2) + 1)*X + SQRT(2) + X ) - 4
2
*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*SQRT(3)*ATAN(---------) - 2*SQRT(3)*ATAN(---------)
SQRT(3) SQRT(3)
2 2
+ LOG(X - X + 1) - LOG(X + X + 1) + 2*LOG(X - 1) - 2*LOG(X + 1))/
12
testint(1/(x**6-2),x);
1/6
1/6 2 + 2*X
(2 *( - 2*SQRT(3)*ATAN(--------------)
1/6
2 *SQRT(3)
1/6
- 2 + 2*X 1/6
- 2*SQRT(3)*ATAN(---------------) - 2*LOG(2 + X)
1/6
2 *SQRT(3)
1/6 1/6 1/3 2
+ 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 1/3 2
(2 *( - SQRT(3)*LOG( - 2 *SQRT(3)*X + 2 + X )
1/6 1/3 2
+ SQRT(3)*LOG(2 *SQRT(3)*X + 2 + X )
1/6
- 2 *SQRT(3) + 2*X
+ 2*ATAN(-----------------------)
1/6
2
1/6
2 *SQRT(3) + 2*X X
+ 2*ATAN(--------------------) + 4*ATAN(------)))/24
1/6 1/6
2 2
testint(1/(x**8+1),x);
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)
- 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)
- 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(----------------------------))/16
SQRT(SQRT(2) + 2)
testint(1/(x**8-1),x);
2 2
(SQRT(2)*LOG( - SQRT(2)*X + X + 1) - SQRT(2)*LOG(SQRT(2)*X + X + 1)
- SQRT(2) + 2*X
- 2*SQRT(2)*ATAN(------------------)
SQRT(2)
SQRT(2) + 2*X
- 2*SQRT(2)*ATAN(---------------) + 2*LOG(X - 1) - 2*LOG(X + 1)
SQRT(2)
- 4*ATAN(X))/16
testint(1/(x**8-x**4+1),x);
( - SQRT( - SQRT(3) + 2)*SQRT(3)
2
*LOG( - SQRT( - SQRT(3) + 2)*X + X + 1)
2
+ SQRT( - SQRT(3) + 2)*SQRT(3)*LOG(SQRT( - SQRT(3) + 2)*X + X + 1)
- SQRT(SQRT(3) + 2) + 2*X
+ 2*SQRT( - SQRT(3) + 2)*SQRT(3)*ATAN(----------------------------)
SQRT( - SQRT(3) + 2)
SQRT(SQRT(3) + 2) + 2*X
+ 2*SQRT( - SQRT(3) + 2)*SQRT(3)*ATAN(-------------------------)
SQRT( - SQRT(3) + 2)
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)
- SQRT(SQRT(3) + 2) + 2*X
+ 6*SQRT( - SQRT(3) + 2)*ATAN(----------------------------)
SQRT( - SQRT(3) + 2)
SQRT(SQRT(3) + 2) + 2*X
+ 6*SQRT( - SQRT(3) + 2)*ATAN(-------------------------)
SQRT( - SQRT(3) + 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)
- SQRT( - SQRT(3) + 2) + 2*X
- 2*SQRT(SQRT(3) + 2)*SQRT(3)*ATAN(-------------------------------)
SQRT(SQRT(3) + 2)
SQRT( - SQRT(3) + 2) + 2*X
- 2*SQRT(SQRT(3) + 2)*SQRT(3)*ATAN(----------------------------)
SQRT(SQRT(3) + 2)
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)
- SQRT( - SQRT(3) + 2) + 2*X
+ 6*SQRT(SQRT(3) + 2)*ATAN(-------------------------------)
SQRT(SQRT(3) + 2)
SQRT( - SQRT(3) + 2) + 2*X
+ 6*SQRT(SQRT(3) + 2)*ATAN(----------------------------))/24
SQRT(SQRT(3) + 2)
testint(x**7/(x**12+1),x);
( - 2*SQRT(SQRT(3) + 2)*SQRT( - SQRT(3) + 2)*SQRT(3)
- SQRT( - SQRT(3) + 2) + 2*X
*ATAN(-------------------------------) + 2*SQRT(SQRT(3) + 2)
SQRT(SQRT(3) + 2)
- SQRT(SQRT(3) + 2) + 2*X
*SQRT( - SQRT(3) + 2)*SQRT(3)*ATAN(----------------------------) + 2
SQRT( - SQRT(3) + 2)
*SQRT(SQRT(3) + 2)*SQRT( - SQRT(3) + 2)*SQRT(3)
SQRT( - SQRT(3) + 2) + 2*X
*ATAN(----------------------------) - 2*SQRT(SQRT(3) + 2)
SQRT(SQRT(3) + 2)
SQRT(SQRT(3) + 2) + 2*X
*SQRT( - SQRT(3) + 2)*SQRT(3)*ATAN(-------------------------)
SQRT( - SQRT(3) + 2)
2
+ LOG( - SQRT( - SQRT(3) + 2)*X + X + 1)
2
+ LOG( - SQRT(SQRT(3) + 2)*X + X + 1)
2
- 2*LOG( - SQRT(2)*X + X + 1)
2
+ LOG(SQRT( - SQRT(3) + 2)*X + X + 1)
2 2
+ LOG(SQRT(SQRT(3) + 2)*X + X + 1) - 2*LOG(SQRT(2)*X + X + 1))/24
% 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
- 15468750*LOG(X) + 12375000*LOG(X) - 8662500*LOG(X)
5 4 3
+ 5197500*LOG(X) - 2598750*LOG(X) + 1039500*LOG(X)
2
- 311850*LOG(X) + 62370*LOG(X) - 6237))/156250000
testint(log x**2/x,x);
3
LOG(X)
---------
3
testint(1/log x,x);
1
INT(--------,X)
LOG(X)
testint(1/log(x+1),x);
X
INT(---------------------------,X) + LOG(LOG(X + 1))
LOG(X + 1)*X + LOG(X + 1)
testint(1/(x*log x),x);
LOG(LOG(X))
testint(1/(x*log x)**2,x);
1
- (INT(-----------,X)*LOG(X)*X + 1)
2
LOG(X)*X
--------------------------------------
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
(X*(6*LOG(X)*A *X + 18*LOG(X)*A*B*X + 18*LOG(X)*B - 2*A *X
2
- 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
(6*LOG(A*X + B)*A *X + 6*LOG(A*X + B)*B - 2*A *X + 3*A *B*X
2 3
- 6*A*B *X)/(18*A )
testint(log(x**2+a**2),x);
2 2 X
LOG(A + X )*X + 2*ATAN(---)*A - 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);
2 2 3 X 3 2 3
3*LOG(A + X )*X - 6*ATAN(---)*A + 6*A *X - 2*X
A
----------------------------------------------------
9
testint(x**4*log(x**2+a**2),x);
2 2 5 X 5 4 2 3 5
15*LOG(A + X )*X + 30*ATAN(---)*A - 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(X)))*LOG(LOG(X))*LOG(X)
+ LOG(LOG(LOG(LOG(X))))*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);
- SIN(X)*COS(X) + X
----------------------
2
testint(x**3*sin(x**2),x);
2 2 2
SIN(X ) - COS(X )*X
----------------------
2
testint(sin x**3,x);
2
- SIN(X) *COS(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);
SIN(X)*COS(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);
SIN(2*X)*COS(X) - 2*SIN(X)*COS(2*X)
-------------------------------------
3
testint(x*sin x,x);
SIN(X) - COS(X)*X
testint(x**2*sin x,x);
2
2*SIN(X)*X - COS(X)*X + 2*COS(X)
testint(x*sin x**2,x);
2 2
SIN(X) - 2*SIN(X)*COS(X)*X + X - 2
--------------------------------------
4
testint(x**2*sin x**2,x);
2 2 3
6*SIN(X) *X - 6*SIN(X)*COS(X)*X + 3*SIN(X)*COS(X) + 2*X - 3*X
-----------------------------------------------------------------
12
testint(x*sin x**3,x);
3 2
SIN(X) - 3*SIN(X) *COS(X)*X + 6*SIN(X) - 6*COS(X)*X
------------------------------------------------------
9
testint(x*cos x,x);
SIN(X)*X + COS(X)
testint(x**2*cos x,x);
2
SIN(X)*X - 2*SIN(X) + 2*COS(X)*X
testint(x*cos x**2,x);
2 2
- SIN(X) + 2*SIN(X)*COS(X)*X + X + 2
-----------------------------------------
4
testint(x**2*cos x**2,x);
2 2 3
- 6*SIN(X) *X + 6*SIN(X)*COS(X)*X - 3*SIN(X)*COS(X) + 2*X + 3*X
--------------------------------------------------------------------
12
testint(x*cos x**3,x);
3 2
- 3*SIN(X) *X - SIN(X) *COS(X) + 9*SIN(X)*X + 7*COS(X) + 1
-------------------------------------------------------------
9
testint(sin x/x,x);
SIN(X)
INT(--------,X)
X
testint(cos x/x,x);
COS(X)
INT(--------,X)
X
testint(sin x/x**2,x);
SIN(X)
INT(--------,X)
2
X
testint(sin x**2/x,x);
2
SIN(X)
INT(---------,X)
X
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);
- SIN(A + B*X)*COS(A + B*X) + B*X
------------------------------------
2*B
testint(sin z**3,x);
2
- SIN(A + B*X) *COS(A + B*X) - 2*COS(A + B*X) + 2
----------------------------------------------------
3*B
testint(cos z**2,x);
SIN(A + B*X)*COS(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 2
(6*SIN(A + B*X) *B*X - 6*SIN(A + B*X)*COS(A + B*X)*B *X
3 3 3
+ 3*SIN(A + B*X)*COS(A + B*X) + 9*A + 2*B *X - 3*B*X)/(12*B )
testint(cos x*cos(2*x),x);
2*SIN(2*X)*COS(X) - SIN(X)*COS(2*X)
-------------------------------------
3
testint(x**2*cos z**2,x);
2 2 2
( - 6*SIN(A + B*X) *B*X + 6*SIN(A + B*X)*COS(A + B*X)*B *X
3 3 3
- 3*SIN(A + B*X)*COS(A + 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
(48*INT(TAN(X)*X ,X) - 6*LOG(TAN(X) + 1) + 4*TAN(X) *X
2 2 3 4 2
- 6*TAN(X) *X - 12*TAN(X)*X + 12*TAN(X)*X + 3*X - 6*X )/12
testint(x**3*tan(x)**6,x);
2 2 5 3
( - 276*INT(TAN(X)*X ,X) + 60*LOG(TAN(X) + 1) + 12*TAN(X) *X
4 2 3 3 3 2 2
- 9*TAN(X) *X - 20*TAN(X) *X + 6*TAN(X) *X + 48*TAN(X) *X
2 3 4 2
- 3*TAN(X) + 60*TAN(X)*X - 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);
3*SIN(3*X)*SIN(2*X) + 2*COS(3*X)*COS(2*X)
-------------------------------------------
5
testint(sin x**2*cos x**2,x);
3
2*SIN(X) *COS(X) - SIN(X)*COS(X) + X
--------------------------------------
8
testint(1/(sin x**2*cos x**2),x);
2
2*SIN(X) - 1
---------------
SIN(X)*COS(X)
testint(d**x*sin x,x);
X
D *(SIN(X)*LOG(D) - COS(X))
-----------------------------
2
LOG(D) + 1
testint(d**x*cos x,x);
X
D *(SIN(X) + COS(X)*LOG(D))
-----------------------------
2
LOG(D) + 1
testint(x*d**x*sin x,x);
X 3 2
(D *(SIN(X)*LOG(D) *X - SIN(X)*LOG(D) + SIN(X)*LOG(D)*X + SIN(X)
2 4
- COS(X)*LOG(D) *X + 2*COS(X)*LOG(D) - COS(X)*X))/(LOG(D)
2
+ 2*LOG(D) + 1)
testint(x*d**x*cos x,x);
X 2 3
(D *(SIN(X)*LOG(D) *X - 2*SIN(X)*LOG(D) + SIN(X)*X + COS(X)*LOG(D) *X
2 4
- COS(X)*LOG(D) + COS(X)*LOG(D)*X + COS(X)))/(LOG(D)
2
+ 2*LOG(D) + 1)
testint(x**2*d**x*sin x,x);
X 5 2 4 3 2
(D *(SIN(X)*LOG(D) *X - 2*SIN(X)*LOG(D) *X + 2*SIN(X)*LOG(D) *X
3 2
+ 2*SIN(X)*LOG(D) + SIN(X)*LOG(D)*X - 6*SIN(X)*LOG(D)
4 2 3
+ 2*SIN(X)*X - COS(X)*LOG(D) *X + 4*COS(X)*LOG(D) *X
2 2 2
- 2*COS(X)*LOG(D) *X - 6*COS(X)*LOG(D) + 4*COS(X)*LOG(D)*X
2 6 4 2
- COS(X)*X + 2*COS(X)))/(LOG(D) + 3*LOG(D) + 3*LOG(D) + 1)
testint(x**2*d**x*cos x,x);
X 4 2 3 2 2
(D *(SIN(X)*LOG(D) *X - 4*SIN(X)*LOG(D) *X + 2*SIN(X)*LOG(D) *X
2 2
+ 6*SIN(X)*LOG(D) - 4*SIN(X)*LOG(D)*X + SIN(X)*X - 2*SIN(X)
5 2 4 3 2
+ 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)
6 4 2
+ 2*COS(X)*X))/(LOG(D) + 3*LOG(D) + 3*LOG(D) + 1)
testint(x**3*d**x*sin x,x);
X 7 3 6 2 5 3
(D *(SIN(X)*LOG(D) *X - 3*SIN(X)*LOG(D) *X + 3*SIN(X)*LOG(D) *X
5 4 2 4
+ 6*SIN(X)*LOG(D) *X - 3*SIN(X)*LOG(D) *X - 6*SIN(X)*LOG(D)
3 3 3
+ 3*SIN(X)*LOG(D) *X - 12*SIN(X)*LOG(D) *X
2 2 2 3
+ 3*SIN(X)*LOG(D) *X + 36*SIN(X)*LOG(D) + SIN(X)*LOG(D)*X
2
- 18*SIN(X)*LOG(D)*X + 3*SIN(X)*X - 6*SIN(X)
6 3 5 2 4 3
- COS(X)*LOG(D) *X + 6*COS(X)*LOG(D) *X - 3*COS(X)*LOG(D) *X
4 3 2
- 18*COS(X)*LOG(D) *X + 12*COS(X)*LOG(D) *X
3 2 3 2
+ 24*COS(X)*LOG(D) - 3*COS(X)*LOG(D) *X - 12*COS(X)*LOG(D) *X
2 3
+ 6*COS(X)*LOG(D)*X - 24*COS(X)*LOG(D) - COS(X)*X
8 6 4 2
+ 6*COS(X)*X))/(LOG(D) + 4*LOG(D) + 6*LOG(D) + 4*LOG(D) + 1
)
testint(x**3*d**x*cos x,x);
X 6 3 5 2 4 3
(D *(SIN(X)*LOG(D) *X - 6*SIN(X)*LOG(D) *X + 3*SIN(X)*LOG(D) *X
4 3 2
+ 18*SIN(X)*LOG(D) *X - 12*SIN(X)*LOG(D) *X
3 2 3 2
- 24*SIN(X)*LOG(D) + 3*SIN(X)*LOG(D) *X + 12*SIN(X)*LOG(D) *X
2 3
- 6*SIN(X)*LOG(D)*X + 24*SIN(X)*LOG(D) + SIN(X)*X
7 3 6 2
- 6*SIN(X)*X + COS(X)*LOG(D) *X - 3*COS(X)*LOG(D) *X
5 3 5
+ 3*COS(X)*LOG(D) *X + 6*COS(X)*LOG(D) *X
4 2 4 3 3
- 3*COS(X)*LOG(D) *X - 6*COS(X)*LOG(D) + 3*COS(X)*LOG(D) *X
3 2 2 2
- 12*COS(X)*LOG(D) *X + 3*COS(X)*LOG(D) *X + 36*COS(X)*LOG(D)
3 2
+ COS(X)*LOG(D)*X - 18*COS(X)*LOG(D)*X + 3*COS(X)*X
8 6 4 2
- 6*COS(X)))/(LOG(D) + 4*LOG(D) + 6*LOG(D) + 4*LOG(D) + 1)
testint(sin x*sin(2*x)*sin(3*x),x);
(6*SIN(3*X)*SIN(2*X)*SIN(X)*X - 8*SIN(3*X)*SIN(2*X)*COS(X)
- 5*SIN(3*X)*SIN(X)*COS(2*X) - 6*SIN(3*X)*COS(2*X)*COS(X)*X
+ 6*SIN(2*X)*COS(3*X)*COS(X)*X + 6*SIN(X)*COS(3*X)*COS(2*X)*X
- 9*COS(3*X)*COS(2*X)*COS(X))/24
testint(cos x*cos(2*x)*cos(3*x),x);
(SIN(3*X)*SIN(2*X)*SIN(X) + 6*SIN(3*X)*SIN(2*X)*COS(X)*X
+ 6*SIN(3*X)*SIN(X)*COS(2*X)*X + 8*SIN(3*X)*COS(2*X)*COS(X)
- 6*SIN(2*X)*SIN(X)*COS(3*X)*X - 3*SIN(2*X)*COS(3*X)*COS(X)
+ 6*COS(3*X)*COS(2*X)*COS(X)*X)/24
testint(sin(x*kx)**3*x**2,x);
3 2 2 2
(6*SIN(X*KX) *X*KX - 9*SIN(X*KX) *COS(X*KX)*X *KX
2 2 2
+ 2*SIN(X*KX) *COS(X*KX) + 36*SIN(X*KX)*X*KX - 18*COS(X*KX)*X *KX
3
+ 40*COS(X*KX) + 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 )*SIN(A*X)*ATAN(------------------)*C
2 2
SQRT(B - C )
A*X
TAN(-----)*B + C
2 2 2 3
- 2*SQRT(B - C )*ATAN(------------------)*B*C - COS(A*X)*B
2 2
SQRT(B - C )
2 4 2 3 5
+ COS(A*X)*B*C )/(A*(SIN(A*X)*B *C - 2*SIN(A*X)*B *C + SIN(A*X)*C
5 3 2 4
+ B - 2*B *C + B*C ))
% Examples involving logarithms and circular functions.
testint(sin log x,x);
X*(SIN(LOG(X)) - COS(LOG(X)))
-------------------------------
2
testint(cos log x,x);
X*(SIN(LOG(X)) + COS(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);
A*X
E
INT(------,X)
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
E *(X - 20*X + 380*X - 6840*X + 116280*X - 1860480*X
14 13 12
+ 27907200*X - 390700800*X + 5079110400*X
11 10 9
- 60949324800*X + 670442572800*X - 6704425728000*X
8 7 6
+ 60339831552000*X - 482718652416000*X + 3379030566912000*X
5 4
- 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
A
INT(----,X)
2
X
testint(x*a**x/(1+b*x)**2,x);
X
X A
(A - INT(-------------------,X)*LOG(A)*B*X
2 2
B *X + 2*B*X + 1
X
A
- INT(-------------------,X)*LOG(A)
2 2
B *X + 2*B*X + 1
X X
A 2 A
+ INT(-------------------,X)*B *X + INT(-------------------,X)*B)/(
2 2 2 2
B *X + 2*B*X + 1 B *X + 2*B*X + 1
LOG(A)*B*(B*X + 1))
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);
2
X
INT(E ,X)
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
X E
E *LOG(X) - INT(----,X)
X
testint(x*e**x*log x,x);
X
X X X E
E *LOG(X)*X - E *LOG(X) - E + INT(----,X)
X
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)
- SQRT(B)*LOG(SQRT(A*X + B) - SQRT(B))
+ SQRT(B)*LOG(SQRT(A*X + B) + SQRT(B))))/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
2
+ 2*SQRT(C + D*X)*SQRT(A + B*X)*B *C*D
2 2
+ 4*SQRT(C + D*X)*SQRT(A + B*X)*B *D *X + SQRT(D)*SQRT(B)*LOG(
2*SQRT(D)*SQRT(B)*SQRT(C + D*X)*SQRT(A + B*X) - A*D - B*C
2 2
- 2*B*D*X)*A *D - 2*SQRT(D)*SQRT(B)*LOG(
2*SQRT(D)*SQRT(B)*SQRT(C + D*X)*SQRT(A + B*X) - A*D - B*C
- 2*B*D*X)*A*B*C*D + SQRT(D)*SQRT(B)*LOG(
2*SQRT(D)*SQRT(B)*SQRT(C + D*X)*SQRT(A + B*X) - A*D - B*C
2 2 2 2
- 2*B*D*X)*B *C + SQRT(D)*SQRT(B)*LOG(-1)*A *D
2 2
- 2*SQRT(D)*SQRT(B)*LOG(-1)*A*B*C*D + SQRT(D)*SQRT(B)*LOG(-1)*B *C
2 2
)/(8*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);
2
( - SQRT(2)*LOG( - SQRT(2)*X + X + 1)
- SQRT(2) + 2*X - SQRT(2) + 2*X
- 2*SQRT(2)*ATAN(------------------) + 4*ATAN(------------------)*X
SQRT(2) SQRT(2)
)/4
testint(1/sqrt(x**2-1),x);
2 2
- LOG(SQRT(X - 1) - X) + LOG(SQRT(X - 1) + X)
--------------------------------------------------
2
testint(sqrt(x+1)*sqrt x,x);
(4*SQRT(X)*SQRT(X + 1)*X + 2*SQRT(X)*SQRT(X + 1)
+ LOG(SQRT(X + 1) - SQRT(X)) - LOG(SQRT(X + 1) + SQRT(X)))/8
% Examples from James Davenport's thesis:
testint(1/sqrt(x**2-1)+10/sqrt(x**2-4),x);
2 2
( - 10*LOG(SQRT(X - 4) - X) + 10*LOG(SQRT(X - 4) + X)
2 2
- LOG(SQRT(X - 1) - X) + LOG(SQRT(X - 1) + X))/2
% p. 173
testint(sqrt(x+sqrt(x**2+a**2))/x,x);
2 2
2*SQRT(SQRT(A + X ) + X)
2 2
+ SQRT(A)*LOG(SQRT(SQRT(A + X ) + X) - SQRT(A))
2 2
- SQRT(A)*LOG(SQRT(SQRT(A + X ) + X) + SQRT(A)) + SQRT(A)*ATAN((
2 2 2 2
SQRT(A)*SQRT(SQRT(A + X ) + X)*SQRT(A + X )
2 2
- SQRT(A)*SQRT(SQRT(A + X ) + X)*A
2 2 2
- SQRT(A)*SQRT(SQRT(A + X ) + X)*X)/(2*A ))
% 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 2
- LOG(SQRT(X + 1) - X) + LOG(SQRT(X + 1) + X)
--------------------------------------------------
2
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 3
- LOG(SQRT(X + 1) - 1) + LOG(SQRT(X + 1) + 1) + 3*LOG(X)
-------------------------------------------------------------
2
% Examples quoted by Joel Moses:
testint(1/sqrt(2*h*r**2-alpha**2),r);
2 2
(SQRT(H)*SQRT(2)*( - LOG(SQRT(2*H*R - ALPHA ) - SQRT(H)*SQRT(2)*R)
2 2
+ LOG(SQRT(2*H*R - ALPHA ) + SQRT(H)*SQRT(2)*R)))/(4*H)
testint(1/(r*sqrt(2*h*r**2-alpha**2-epsilon**2)),r);
2 2 2 2 2
(SQRT( - ALPHA - EPSILON )*ATAN((SQRT(2*H*R - ALPHA - EPSILON )
2 2 2
*SQRT( - ALPHA - EPSILON )*H*I*R -
2 2 2 2 2
SQRT(2*H*R - ALPHA - EPSILON )*SQRT( - ALPHA - EPSILON )*I
2 2 2 2
*ALPHA - SQRT(2*H*R - ALPHA - EPSILON )
2 2 2 2 2
*SQRT( - ALPHA - EPSILON )*I*EPSILON )/(2*H*R *ALPHA
2 2 4 2 2 4
+ 2*H*R *EPSILON - ALPHA - 2*ALPHA *EPSILON - EPSILON ))*I
2 2
)/(2*(ALPHA + EPSILON ))
testint(1/(r*sqrt(2*h*r**2-alpha**2-2*k*r)),r);
2 2
ATAN(( - SQRT(2*H*R - 2*K*R - ALPHA )*K*R
2 2 2 2
- SQRT(2*H*R - 2*K*R - ALPHA )*ALPHA )/(2*H*R *ALPHA
3
- 2*K*R*ALPHA - ALPHA ))/ALPHA
testint(1/(r*sqrt(2*h*r**2-alpha**2-epsilon**2-2*k*r)),r);
2 2
(SQRT( - ALPHA - EPSILON )*ATAN(( -
2 2 2
SQRT(2*H*R - 2*K*R - ALPHA - EPSILON )
2 2
*SQRT( - ALPHA - EPSILON )*I*K*R -
2 2 2
SQRT(2*H*R - 2*K*R - ALPHA - EPSILON )
2 2 2
*SQRT( - ALPHA - EPSILON )*I*ALPHA -
2 2 2
SQRT(2*H*R - 2*K*R - ALPHA - EPSILON )
2 2 2 2 2
*SQRT( - ALPHA - EPSILON )*I*EPSILON )/(2*H*R *ALPHA
2 2 2 2 4
+ 2*H*R *EPSILON - 2*K*R*ALPHA - 2*K*R*EPSILON - ALPHA
2 2 4 2 2
- 2*ALPHA *EPSILON - EPSILON ))*I)/(ALPHA + EPSILON )
testint(r/sqrt(2*e*r**2-alpha**2),r);
2 2
SQRT(2*E*R - ALPHA )
-----------------------
2*E
testint(r/sqrt(2*e*r**2-alpha**2-epsilon**2),r);
2 2 2
SQRT(2*E*R - ALPHA - EPSILON )
----------------------------------
2*E
testint(r/sqrt(2*e*r**2-alpha**2-2*k*r**4),r);
(SQRT(K)*SQRT(2)*(ATAN((
2 4 2 2
- 2*SQRT(K)*SQRT(2*E*R - 2*K*R - ALPHA )*SQRT(2)*E*R
2 4 2 4
+ 4*SQRT(K)*SQRT(2*E*R - 2*K*R - ALPHA )*SQRT(2)*K*R
2 4 2 2
+ SQRT(K)*SQRT(2*E*R - 2*K*R - ALPHA )*SQRT(2)*ALPHA )/(
4 2 6 2 2
8*E*K*R - 8*K *R - 4*K*R *ALPHA )) + ATAN((
2 4 2 2 2
2*SQRT(K)*SQRT(2*E*R - 2*K*R - ALPHA )*SQRT(2)*E *R
2 4 2 2
- SQRT(K)*SQRT(2*E*R - 2*K*R - ALPHA )*SQRT(2)*E*ALPHA
2 4 2 2
- 2*SQRT(K)*SQRT(2*E*R - 2*K*R - ALPHA )*SQRT(2)*K*R
2 2 2 2 4 2 4
*ALPHA )/(4*E*K*R *ALPHA - 4*K *R *ALPHA - 2*K*ALPHA ))))
/(4*K)
testint(r/sqrt(2*e*r**2-alpha**2-2*k*r),r);
2 2
(2*SQRT(2*E*R - 2*K*R - ALPHA )*E + SQRT(E)*SQRT(2)
2 2
*LOG(SQRT(E)*SQRT(2*E*R - 2*K*R - ALPHA )*SQRT(2) + 2*E*R - K)*K)/(
2
4*E )
testint(1/(r*sqrt(2*h*r**2-alpha**2-2*k*r**4)),r);
2 4 2 2
(ATAN(( - SQRT(2*H*R - 2*K*R - ALPHA )*H
2 4 2 2
+ SQRT(2*H*R - 2*K*R - ALPHA )*H*K*R
2 4 2 2 2
+ SQRT(2*H*R - 2*K*R - ALPHA )*K*ALPHA )/(2*H*K*R *ALPHA
2 4 3
- 2*K *R *ALPHA - K*ALPHA )) + ATAN((
2 4 2 2
SQRT(2*H*R - 2*K*R - ALPHA )*H*R
2 4 2 4
- SQRT(2*H*R - 2*K*R - ALPHA )*K*R
2 4 2 2 2
- SQRT(2*H*R - 2*K*R - ALPHA )*ALPHA )/(2*H*R *ALPHA
4 3
- 2*K*R *ALPHA - ALPHA )))/(2*ALPHA)
testint(1/(r*sqrt(2*h*r**2-alpha**2-epsilon**2-2*k*r**4)),r);
2 2
(SQRT( - ALPHA - EPSILON )*I*(ATAN(( -
2 4 2 2
SQRT(2*H*R - 2*K*R - ALPHA - EPSILON )
2 2 2
*SQRT( - ALPHA - EPSILON )*H *I +
2 4 2 2
SQRT(2*H*R - 2*K*R - ALPHA - EPSILON )
2 2 2
*SQRT( - ALPHA - EPSILON )*H*I*K*R +
2 4 2 2
SQRT(2*H*R - 2*K*R - ALPHA - EPSILON )
2 2 2
*SQRT( - ALPHA - EPSILON )*I*K*ALPHA +
2 4 2 2
SQRT(2*H*R - 2*K*R - ALPHA - EPSILON )
2 2 2 2 2
*SQRT( - ALPHA - EPSILON )*I*K*EPSILON )/(2*H*K*R *ALPHA
2 2 2 4 2 2 4 2
+ 2*H*K*R *EPSILON - 2*K *R *ALPHA - 2*K *R *EPSILON
4 2 2 4
- K*ALPHA - 2*K*ALPHA *EPSILON - K*EPSILON )) + ATAN((
2 4 2 2
SQRT(2*H*R - 2*K*R - ALPHA - EPSILON )
2 2 2
*SQRT( - ALPHA - EPSILON )*H*I*R -
2 4 2 2
SQRT(2*H*R - 2*K*R - ALPHA - EPSILON )
2 2 4
*SQRT( - ALPHA - EPSILON )*I*K*R -
2 4 2 2
SQRT(2*H*R - 2*K*R - ALPHA - EPSILON )
2 2 2
*SQRT( - ALPHA - EPSILON )*I*ALPHA -
2 4 2 2
SQRT(2*H*R - 2*K*R - ALPHA - EPSILON )
2 2 2 2 2
*SQRT( - ALPHA - EPSILON )*I*EPSILON )/(2*H*R *ALPHA
2 2 4 2 4 2
+ 2*H*R *EPSILON - 2*K*R *ALPHA - 2*K*R *EPSILON
4 2 2 4
- ALPHA - 2*ALPHA *EPSILON - EPSILON ))))/(2
2 2
*(ALPHA + EPSILON ))
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);
SIN(A + X) - COS(A + X)*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*SQRT(X)*SQRT(X + 1)*X*(2*X + 5)
-----------------------------------
3 2
15*(X + 3*X + 3*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(SQRT(X)*SQRT(2) - 1) - LOG(SQRT(X)*SQRT(2) + 1))
---------------------------------------------------------------
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);
1/3
1/3 ( - A + X)
3*( - A + X) + INT(---------------,X)*A
2
A*X - X
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)*I*EPS + A*I - X)
testint(sin(2*x+3)*cos(x)**2,x);
2
( - 4*SIN(2*X + 3)*SIN(X) *X + 2*SIN(2*X + 3)*X
2
+ 2*SIN(X) *COS(2*X + 3) - 4*SIN(X)*COS(2*X + 3)*COS(X)*X
- 3*COS(2*X + 3) + 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);
SIN(A + X)*SIN(X) + SIN(A + X)*COS(X)*X - SIN(X)*COS(A + X)*X
---------------------------------------------------------------
2
testint(cos(x+a)*sin(x),x);
SIN(A + X)*SIN(X) - SIN(A + X)*COS(X)*X + SIN(X)*COS(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) + LOG(SQRT(X + 1) + 1)
- LOG(X)
testint(x/(x**4+36)**(1/2),x);
4 2 4 2
LOG(SQRT(X + 36) + X ) - LOG(SQRT(X + 36) - X )
---------------------------------------------------
4
testint(1/(x**(1/3)+x**(1/2)),x);
1
INT(----------------,X)
1/3
X + SQRT(X)
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
INT(LOG(SQRT(X + 1) + X)*X,X)
testint(x*(e**x*sin(x)+1)**2,x);
2*X 2 2*X 2*X
(2*E *SIN(X) *X - 2*E *SIN(X)*COS(X)*X + E *SIN(X)*COS(X)
2*X 2*X X X X
+ E *X - E + 8*E *SIN(X)*X - 8*E *COS(X)*X + 8*E *COS(X)
2
+ 4*X )/8
testint(x*e**x*cos(x),x);
X
E *(SIN(X)*X - SIN(X) + COS(X)*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 2
- LOG(SQRT(X + 1) - X) + LOG(SQRT(X + 1) + X)
--------------------------------------------------
2
testint(sqrt(x**2+3),x);
2 2 2
(2*SQRT(X + 3)*X - 3*LOG(SQRT(X + 3) - X) + 3*LOG(SQRT(X + 3) + X)
)/4
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);
INT(ASIN(X),X)
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);
SIN(X)*COS(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 2
- LOG(SQRT(4*X + 9) - 2*X) + LOG(SQRT(4*X + 9) + 2*X)
----------------------------------------------------------
4
testint((x**2+4)**(-1/2),x);
2 2
- LOG(SQRT(X + 4) - X) + LOG(SQRT(X + 4) + X)
--------------------------------------------------
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*LOG(X + 1)*X - 3*LOG(X + 1)*X - 30*LOG(X - 1)*X
3 4 3 4 2
+ 30*LOG(X - 1)*X + 24*LOG(X)*X - 24*LOG(X)*X - 30*X + 12*X
3
+ 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
- INT(1/(LOG(X + 1) - 10*LOG(X + 1) + 40*LOG(X + 1)
2 2 2
- 80*LOG(X + 1) + 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
(E *LOG(X) + E *LOG(X)*X + LOG(LOG(X) + X)*LOG(X)
2 2
+ LOG(LOG(X) + X)*X - 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 4 4 4 2 2
(E *X *(SIN(PI*ZE) *X + 4*SIN(PI*ZE) *X *PI
3 3
- 4*SIN(PI*ZE) *COS(PI*ZE)*X *PI
3 3 2 2 2
- 16*SIN(PI*ZE) *COS(PI*ZE)*X*PI + 12*SIN(PI*ZE) *X *PI
3 4 4 2 2
- 24*SIN(PI*ZE)*COS(PI*ZE)*X*PI + 24*PI ))/(X + 20*X *PI
4
+ 64*PI )
% This one evaluates:
testint(erf(x),x);
2
X
E *ERF(X)*X*PI + SQRT(PI)
----------------------------
2
X
E *PI
% So why not this one?
testint(erf(x+a),x);
INT(ERF(A + X),X)
Comment some interesting integrals of algebraic functions;
% The Chebyshev integral.
testint((2*x**6+4*x**5+7*x**4-3*x**3-x*x-8*x-8)/
((2*x**2-1)**2*sqrt(x**4+4*x**3+2*x**2+1)),x);
4 3 2 4 3 2
(2*SQRT(X + 4*X + 2*X + 1)*X + SQRT(X + 4*X + 2*X + 1)
2 2 2
- 4*LOG(X + 4*X + 2)*X + 2*LOG(X + 4*X + 2) + 2*LOG(
4 3 2 4 3 2
SQRT(X + 4*X + 2*X + 1)*SQRT(2) + 4*SQRT(X + 4*X + 2*X + 1)
2
- 2*SQRT(2)*X - 4*SQRT(2) - X - 2)*X - LOG(
4 3 2 4 3 2
SQRT(X + 4*X + 2*X + 1)*SQRT(2) + 4*SQRT(X + 4*X + 2*X + 1)
- 2*SQRT(2)*X - 4*SQRT(2) - X - 2) + 2*LOG(
4 3 2
25*SQRT(X + 4*X + 2*X + 1)*SQRT(2)*X
4 3 2
+ 6*SQRT(X + 4*X + 2*X + 1)*SQRT(2)
4 3 2
+ 44*SQRT(X + 4*X + 2*X + 1)*X
4 3 2 3 2
+ 38*SQRT(X + 4*X + 2*X + 1) - 7*SQRT(2)*X - 36*SQRT(2)*X
3 2 2
- 21*SQRT(2)*X - 10*SQRT(2) - 28*X - 130*X - 84*X + 2)*X -
4 3 2
LOG(25*SQRT(X + 4*X + 2*X + 1)*SQRT(2)*X
4 3 2
+ 6*SQRT(X + 4*X + 2*X + 1)*SQRT(2)
4 3 2
+ 44*SQRT(X + 4*X + 2*X + 1)*X
4 3 2 3 2
+ 38*SQRT(X + 4*X + 2*X + 1) - 7*SQRT(2)*X - 36*SQRT(2)*X
3 2
- 21*SQRT(2)*X - 10*SQRT(2) - 28*X - 130*X - 84*X + 2) + 10*
4 3 2
LOG(16*SQRT(X + 4*X + 2*X + 1)*SQRT(2)
4 3 2 2
+ 13*SQRT(X + 4*X + 2*X + 1) - 28*SQRT(2)*X - 32*SQRT(2)*X
2 2
+ 6*SQRT(2) - 35*X - 26*X + 11)*X - 5*LOG(
4 3 2
16*SQRT(X + 4*X + 2*X + 1)*SQRT(2)
4 3 2 2
+ 13*SQRT(X + 4*X + 2*X + 1) - 28*SQRT(2)*X - 32*SQRT(2)*X
2 2 2
+ 6*SQRT(2) - 35*X - 26*X + 11) - 10*LOG(2*X - 1)*X
2 4 3 2
+ 5*LOG(2*X - 1) + 4*LOG(2*SQRT(X + 4*X + 2*X + 1)*SQRT(2)
4 3 2 2
+ SQRT(X + 4*X + 2*X + 1) - 4*SQRT(2)*X - 16*SQRT(2)*X
2 2
- 10*SQRT(2) + 5*X + 20*X + 9)*X - 2*LOG(
4 3 2 4 3 2
2*SQRT(X + 4*X + 2*X + 1)*SQRT(2) + SQRT(X + 4*X + 2*X + 1)
2 2
- 4*SQRT(2)*X - 16*SQRT(2)*X - 10*SQRT(2) + 5*X + 20*X + 9)
2 2
- 4*LOG(2*SQRT(2)*X + 4*SQRT(2) - X - 4*X - 6)*X
2
+ 2*LOG(2*SQRT(2)*X + 4*SQRT(2) - X - 4*X - 6)
2 2
- 2*LOG(SQRT(2) + 2*X)*X + LOG(SQRT(2) + 2*X) + 14*LOG(-1)*X
2
- 7*LOG(-1))/(2*(2*X - 1))
% This integral came from Dr. G.S. Joyce of Imperial College London.
testint((1+2*y)*sqrt(1-5*y-5*y**2)/(y*(1+y)*(2+y)*sqrt(1-y-y**2)),y);
2 2
( - 4*LOG(Y - 2*Y + 1) - LOG(Y + 4*Y + 4)
2 2
+ LOG(SQRT( - Y - Y + 1)*I - SQRT( - 5*Y - 5*Y + 1)*I + 2*Y) + 2*
2 2
LOG(6*SQRT( - 5*Y - 5*Y + 1)*SQRT( - Y - Y + 1)*I
2 2
+ 8*SQRT( - 5*Y - 5*Y + 1)*SQRT( - Y - Y + 1)
2 2
- 20*SQRT( - Y - Y + 1)*I*Y - 4*SQRT( - Y - Y + 1)*I
2 2
+ 15*SQRT( - Y - Y + 1)*Y + 3*SQRT( - Y - Y + 1)
2 2
- 12*SQRT( - 5*Y - 5*Y + 1)*I*Y + 4*SQRT( - 5*Y - 5*Y + 1)*I
2 2
+ 9*SQRT( - 5*Y - 5*Y + 1)*Y - 3*SQRT( - 5*Y - 5*Y + 1)
2 2
- 10*I*Y - 22*I*Y + 14*I - 30*Y + 4*Y + 2) + 2*LOG(
2 2
6*SQRT( - 5*Y - 5*Y + 1)*SQRT( - Y - Y + 1)
2 2
- 15*SQRT( - Y - Y + 1)*I*Y - 3*SQRT( - Y - Y + 1)*I
2 2
- 9*SQRT( - 5*Y - 5*Y + 1)*I*Y + 3*SQRT( - 5*Y - 5*Y + 1)*I
2
- 10*Y - 22*Y + 14) + 9
2 2
*LOG(3*SQRT( - Y - Y + 1)*I + SQRT( - 5*Y - 5*Y + 1)*I - 2*Y - 4)
2 2
+ 2*LOG(3*SQRT( - 5*Y - 5*Y + 1)*SQRT( - Y - Y + 1)*I
2 2
- 4*SQRT( - 5*Y - 5*Y + 1)*SQRT( - Y - Y + 1)
2 2
+ 15*SQRT( - Y - Y + 1)*I*Y - 3*SQRT( - Y - Y + 1)*I
2 2
+ 5*SQRT( - Y - Y + 1)*Y + 4*SQRT( - Y - Y + 1)
2 2
+ SQRT( - 5*Y - 5*Y + 1)*I*Y + 3*SQRT( - 5*Y - 5*Y + 1)*I
2 2
+ 7*SQRT( - 5*Y - 5*Y + 1)*Y - 4*SQRT( - 5*Y - 5*Y + 1)
2 2
+ 5*I*Y - 11*I*Y - 3*I + 10*Y - 2*Y + 4) + LOG(
2 2
2*SQRT( - 5*Y - 5*Y + 1)*SQRT( - Y - Y + 1)*I
2 2
+ 5*SQRT( - Y - Y + 1)*Y + 4*SQRT( - Y - Y + 1)
2 2
- 3*SQRT( - 5*Y - 5*Y + 1)*Y - 4*SQRT( - 5*Y - 5*Y + 1)
2
+ 5*I*Y + 6*I*Y - 2*I) + 2
2 2
*LOG(SQRT( - Y - Y + 1) - SQRT( - 5*Y - 5*Y + 1) - 4*I*Y - 4*I)
- 2*LOG(Y - 1) - 9*LOG(Y + 2) - 2*LOG(Y + 1) - 3*LOG(Y) + 3*LOG(-1)
)/2
% This one has a simple result.
testint(x*(sqrt(x**2-1)*x**2-4*sqrt(x**2-1)+sqrt(x**2-4)*x**2
-sqrt(x**2-4))/((1+sqrt(x**2-4)+sqrt(x**2-1))*(x**4-5*x**2+4)),x);
2 2
LOG(SQRT(X - 4) + SQRT(X - 1) + 1)
% This used to reveal bugs in the integrator which have been fixed.
% Since it takes a long time and doesn't have a closed form result,
% it has been commented out.
% testint(sqrt(-4*sqrt(2)+9)*x-sqrt(x**4+2*x**2+4*x+1)*sqrt(2),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 1 3 3
(PI *( - 36864*E *INT(--------------,S)*S*Y *MC
S/Y
384*E *S*Y
S/Y 1 2 5
- 55296*E *INT(--------------,S)*S*Y *MC
S/Y
384*E *S*Y
S/Y 1 2 4
+ 13824*E *INT(--------------,S)*S*Y *MC
S/Y
384*E *S*Y
S/Y 1 7
- 18432*E *INT(--------------,S)*S*Y*MC
S/Y
384*E *S*Y
S/Y 1 6
+ 9216*E *INT(--------------,S)*S*Y*MC
S/Y
384*E *S*Y
S/Y 1 9
- 1536*E *INT(--------------,S)*S*MC
S/Y
384*E *S*Y
S/Y 1 8
+ 1152*E *INT(--------------,S)*S*MC
S/Y
384*E *S*Y
S 2 3 S 2 3
+ 96*LOG(-----)*S *Y*MC + 96*LOG(-----)*S*Y *MC
2 2
MC MC
S 5 S 4 3
+ 144*LOG(-----)*S*Y*MC - 36*LOG(-----)*S*Y*MC - 12*S *Y*MC
2 2
MC MC
3 2 2 2 2 2 3 2 2
- 3*S *Y - 24*S *Y *MC - 6*S *Y - 176*S *Y*MC + 24*S *Y*MC
3 3 2 3 2 2
- 24*S*Y *MC - 6*S*Y - 80*S*Y *MC + 24*S*Y *MC
5 9 8 S/Y
+ 144*S*Y*MC - 4*MC + 3*MC ))/(384*E *S)
factor int;
ws;
S/Y 1 3 2 3 2 2
(384*E *INT(--------------,S)*S*MC *PI *( - 96*Y - 144*Y *MC
S/Y
384*E *S*Y
2 4 3 6 5 2
+ 36*Y *MC - 48*Y*MC + 24*Y*MC - 4*MC + 3*MC ) + PI *(
S 2 3 S 2 3
96*LOG(-----)*S *Y*MC + 96*LOG(-----)*S*Y *MC
2 2
MC MC
S 5 S 4 3
+ 144*LOG(-----)*S*Y*MC - 36*LOG(-----)*S*Y*MC - 12*S *Y*MC
2 2
MC MC
3 2 2 2 2 2 3 2 2
- 3*S *Y - 24*S *Y *MC - 6*S *Y - 176*S *Y*MC + 24*S *Y*MC
3 3 2 3 2 2 5
- 24*S*Y *MC - 6*S*Y - 80*S*Y *MC + 24*S*Y *MC + 144*S*Y*MC
9 8 S/Y
- 4*MC + 3*MC ))/(384*E *S)
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: 275
Total time taken: 132600 ms
Number of garbage collections: 58
Number of incorrect integrals: 0
Number of unevaluated integrals: 32
Integrands of unevaluated integrals are:
1
--------
LOG(X)
1
------------
LOG(X + 1)
1
------------
2 2
LOG(X) *X
LOG(LOG(LOG(LOG(X))))
P
SIN(X)
SIN(X)
--------
X
COS(X)
--------
X
SIN(X)
--------
2
X
2
SIN(X)
---------
X
4 3
TAN(X) *X
6 3
TAN(X) *X
XI
COS(--------)*COS(X)*X
SIN(X)
------------------------
2
SIN(X)
A*X
E
------
X
X
A
----
2
X
X
A *X
-------------------
2 2
B *X + 2*B*X + 1
2
X
E
X
E
E
E
E
X
E *LOG(X)
X
E *LOG(X)*X
1/3
( - A + X)
---------------
X
SQRT(SIN(X) + 1)
SQRT( - SIN(X) + 1)
SQRT(COS(X) + 1)
SQRT( - COS(X) + 1)
1
----------------
1/3
X + SQRT(X)
2
LOG(SQRT(X + 1) + X)*X
3
LOG(X)*X + LOG(X)
--------------------
4
X + 2
ASIN(X)
2
ASIN(X)*X
2 5 2 4 2 3
( - 1)/(LOG(X + 1) - 10*LOG(X + 1) + 40*LOG(X + 1)
2 2 2
- 80*LOG(X + 1) + 80*LOG(X + 1) - 32)
ERF(A + X)
SIN(2*X)
----------
COS(X)
end;
5: 5:
Time: 123352 ms plus GC time: 9520 ms
6: 6:
Quitting
Sat May 30 16:24:09 PDT 1992