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- Tue Feb 10 12:26:19 2004 run on Linux
- 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 )
- % These two integrals will evaluate, but they take a very long time
- % and the results are messy (compared with the algint results).
- % testint(1/(r*sqrt(2*h*r**2-alpha**2-2*k*r**4)),r);
- % testint(1/(r*sqrt(2*h*r**2-alpha**2-epsilon**2-2*k*r**4)),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 is an example of integrals that used to loop
- forever. They were first revealed by problems with Bessel
- function integration when specfn was loaded,
- e.g., int(x*besseli(2,x),x) or int(besselj(n,x),x);
- operator f;
- let {df(f(~x),x) => x*f(x-1)};
- int(f x,x);
- int(f(x),x)
- 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: 276
- Total time taken: 1449 ms
- Number of incorrect integrals: 0
- Number of unevaluated integrals: 19
- 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
- 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;
- Time for test: 1449 ms, plus GC time: 80 ms
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