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REDUCE 3.6, 15-Jul-95, patched to 6 Mar 96 ... 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; !*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); 1/4 sqrt(5)*x 1/4 1/4 sqrt(5)*5 *( - 2*atan(-----------) + log(5 *x - 1) - log(5 *x + 1)) 1/4 5 --------------------------------------------------------------------------- 20 testint(1/(3*x**4+7),x); 1/4 1/4 sqrt(2)*21 - 2*sqrt(3)*x (sqrt(6)*21 *( - 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 (sqrt(2)*(2*sqrt(3)*atan(-----------------------)*i sqrt(6)*i - sqrt(2)*i 2*x + 6*atan(-----------------------)*i sqrt(6)*i - sqrt(2)*i - 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) ------------------- 2*i 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); 1 int(------------------------,x) 4 2 sqrt( - x + x - 1)*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); r int(-----------------------------------,r) 2 2 4 sqrt( - alpha + 2*e*r - 2*k*r ) testint(r/sqrt(2*e*r**2-alpha**2-2*k*r),r); 2 2 (2*sqrt( - alpha + 2*e*r - 2*k*r)*e + sqrt(e)*sqrt(2) 2 2 sqrt(e)*sqrt( - alpha + 2*e*r - 2*k*r)*sqrt(2) + 2*e*r - k 2 *log(--------------------------------------------------------------)*k)/(4*e ) 2 2 sqrt(2*alpha *e + k ) testint(1/(r*sqrt(2*h*r**2-alpha**2-2*k*r**4)),r); 1 int(-------------------------------------,r) 2 2 4 sqrt( - alpha + 2*h*r - 2*k*r )*r testint(1/(r*sqrt(2*h*r**2-alpha**2-epsilon**2-2*k*r**4)),r); 1 int(------------------------------------------------,r) 2 2 2 4 sqrt( - alpha - epsilon + 2*h*r - 2*k*r )*r Comment many of these integrals used to require Steve Harrington's code to evaluate. They originated in Novosibirsk as examples of using Analytik. There are still a few examples that could be evaluated using better heuristics; testint(a*sin(3*x+5)**2*cos(3*x+5),x); 3 sin(3*x + 5) *a ----------------- 9 testint(log(x**2)/x**3,x); 2 - (log(x ) + 1) ------------------ 2 2*x testint(x*sin(x+a),x); - cos(a + x)*x + sin(a + x) testint((log(x)*(1-x)-1)/(e**x*log(x)**2),x); x ----------- x e *log(x) testint(x**3*(a*x**2+b)**(-1),x); 2 2 - log(a*x + b)*b + a*x --------------------------- 2 2*a testint(x**(1/2)*(x+1)**(-7/2),x); 2 2 (2*( - 2*sqrt(x + 1)*x - 4*sqrt(x + 1)*x - 2*sqrt(x + 1) + 2*sqrt(x)*x 2 + 5*sqrt(x)*x))/(15*sqrt(x + 1)*(x + 2*x + 1)) testint(x**(-1)*(x+1)**(-1),x); - log(x + 1) + log(x) testint(x**(-1/2)*(2*x-1)**(-1),x); sqrt(2)*(log(2*sqrt(x) - sqrt(2)) - log(2*sqrt(x) + sqrt(2))) --------------------------------------------------------------- 2 testint((x**2+1)*x**(1/2),x); 2 2*sqrt(x)*x*(3*x + 7) ------------------------ 21 testint(x**(-1)*(x-a)**(1/3),x); 1/6 1/6 2*( - a + x) - a *sqrt(3) ( - 2*sqrt(3)*atan(--------------------------------)*a 1/6 a 1/6 1/6 2*( - a + x) + a *sqrt(3) 2/3 1/3 + 2*sqrt(3)*atan(--------------------------------)*a + 6*a *( - a + x) 1/6 a 1/3 1/3 - 2*log(( - a + x) + a )*a 1/6 1/6 1/3 1/3 + log( - a *( - a + x) *sqrt(3) + ( - a + x) + a )*a 1/6 1/6 1/3 1/3 2/3 + log(a *( - a + x) *sqrt(3) + ( - a + x) + a )*a)/(2*a ) testint(x*sinh(x),x); cosh(x)*x - sinh(x) testint(x*cosh(x),x); - cosh(x) + sinh(x)*x testint(sinh(2*x)/cosh(2*x),x); log(cosh(2*x)) ---------------- 2 testint((i*eps*sinh x-1)/(eps*i*cosh x+i*a-x),x); log(cosh(x)*eps*i + a*i - x) testint(sin(2*x+3)*cos(x)**2,x); 2 ( - 4*cos(2*x + 3)*cos(x)*sin(x)*x + 2*cos(2*x + 3)*sin(x) - 3*cos(2*x + 3) 2 - 4*sin(2*x + 3)*sin(x) *x + 2*sin(2*x + 3)*x + 3)/8 testint(x*atan(x),x); 2 atan(x)*x + atan(x) - x -------------------------- 2 testint(x*acot(x),x); 2 acot(x)*x + acot(x) + x -------------------------- 2 testint(x*log(x**2+a),x); 2 2 2 2 log(a + x )*a + log(a + x )*x - x ------------------------------------- 2 testint(sin(x+a)*cos(x),x); - cos(a + x)*cos(x) - cos(a + x)*sin(x)*x + cos(x)*sin(a + x)*x ------------------------------------------------------------------ 2 testint(cos(x+a)*sin(x),x); - cos(a + x)*cos(x) + cos(a + x)*sin(x)*x - cos(x)*sin(a + x)*x ------------------------------------------------------------------ 2 testint((1+sin(x))**(1/2),x); int(sqrt(sin(x) + 1),x) testint((1-sin(x))**(1/2),x); int(sqrt( - sin(x) + 1),x) testint((1+cos(x))**(1/2),x); int(sqrt(cos(x) + 1),x) testint((1-cos(x))**(1/2),x); int(sqrt( - cos(x) + 1),x) testint(1/(x**(1/2)-(x-1)**(1/2)),x); 2*(sqrt(x - 1)*x - sqrt(x - 1) + sqrt(x)*x) --------------------------------------------- 3 testint(1/(1-(x+1)**(1/2)),x); - 2*(sqrt(x + 1) + log(sqrt(x + 1) - 1)) testint(x/(x**4+36)**(1/2),x); 4 2 sqrt(x + 36) + x log(--------------------) 6 --------------------------- 2 testint(1/(x**(1/3)+x**(1/2)),x); 1/6 1/3 1/6 6*x - 3*x + 2*sqrt(x) - 6*log(x + 1) testint(log(2+3*x**2),x); 3*x 2 2*sqrt(6)*atan(---------) + 3*log(3*x + 2)*x - 6*x sqrt(6) ----------------------------------------------------- 3 testint(cot(x),x); x 2 x - log(tan(---) + 1) + log(tan(---)) 2 2 testint(cot x**4,x); 3 - cot(x) + 3*cot(x) + 3*x ----------------------------- 3 testint(tanh(x),x); 2*x log(e + 1) - x testint(coth(x),x); x x log(e - 1) + log(e + 1) - x testint(b**x,x); x b -------- log(b) testint((x**4+x**(-4)+2)**(1/2),x); 4 x - 3 -------- 3*x testint((2*x+1)/(3*x+2),x); - log(3*x + 2) + 6*x ----------------------- 9 testint(x*log(x+(x**2+1)**(1/2)),x); 2 2 2 2 - sqrt(x + 1)*x + 2*log(sqrt(x + 1) + x)*x + log(sqrt(x + 1) + x) ------------------------------------------------------------------------ 4 testint(x*(e**x*sin(x)+1)**2,x); 2*x 2*x x x ( - 2*e *cos(x)*sin(x)*x + e *cos(x)*sin(x) - 8*e *cos(x)*x + 8*e *cos(x) 2*x 2 2*x 2*x x 2 + 2*e *sin(x) *x + e *x - e + 8*e *sin(x)*x + 4*x )/8 testint(x*e**x*cos(x),x); x e *(cos(x)*x + sin(x)*x - sin(x)) ----------------------------------- 2 Comment the following set came from Herbert Stoyan; testint(1/(x-3)**4,x); - 1 --------------------------- 3 2 3*(x - 9*x + 27*x - 27) testint(x/(x**3-1),x); 2*x + 1 2 2*sqrt(3)*atan(---------) - log(x + x + 1) + 2*log(x - 1) sqrt(3) ------------------------------------------------------------ 6 testint(x/(x**4-1),x); 2 - log(x + 1) + log(x - 1) + log(x + 1) ------------------------------------------ 4 testint(log(x)*(x**3+1)/(x**4+2),x); log(x) log(x) 2 - 4*int(----------,x) + 2*int(--------,x) + log(x) 5 4 x + 2*x x + 2 ------------------------------------------------------ 2 testint(log(x)+log(x+1)+log(x+2),x); log(x + 2)*x + 2*log(x + 2) + log(x + 1)*x + log(x + 1) + log(x)*x - 3*x testint(1/(x**3+5),x); 1/3 1/3 5 - 2*x 2/3 1/3 2 (5 *( - 2*sqrt(3)*atan(--------------) - log(5 - 5 *x + x ) 1/3 sqrt(3)*5 1/3 + 2*log(5 + x)))/30 testint(1/sqrt(1+x**2),x); 2 log(sqrt(x + 1) + x) testint(sqrt(x**2+3),x); 2 2 sqrt(x + 3) + x sqrt(x + 3)*x + 3*log(------------------) sqrt(3) -------------------------------------------- 2 testint(x/(x+1)**2,x); log(x + 1)*x + log(x + 1) - x ------------------------------- x + 1 COMMENT The following integrals were used among others as a test of Moses' SIN program; testint(asin x,x); 2 asin(x)*x + sqrt( - x + 1) testint(x**2*asin x,x); 2 int(asin(x)*x ,x) testint(sec x**2/(1+sec x**2-3*tan x),x); x x log( - sqrt(5) + 2*tan(---) + 1) - log( - sqrt(2) + tan(---) + 1) 2 2 x x + log(sqrt(5) + 2*tan(---) + 1) - log(sqrt(2) + tan(---) + 1) 2 2 testint(1/sec x**2,x); cos(x)*sin(x) + x ------------------- 2 testint((5*x**2-3*x-2)/(x**2*(x-2)),x); 3*log(x - 2)*x + 2*log(x)*x - 1 --------------------------------- x testint(1/(4*x**2+9)**(1/2),x); 2 sqrt(4*x + 9) + 2*x log(----------------------) 3 ----------------------------- 2 testint((x**2+4)**(-1/2),x); 2 sqrt(x + 4) + x log(------------------) 2 testint(1/(9*x**2-12*x+10),x); 3*x - 2 sqrt(6)*atan(---------) sqrt(6) ------------------------- 18 testint(1/(x**8-2*x**7+2*x**6-2*x**5+x**4),x); 2 4 2 3 4 3 (3*log(x + 1)*x - 3*log(x + 1)*x - 30*log(x - 1)*x + 30*log(x - 1)*x 4 3 4 2 3 + 24*log(x)*x - 24*log(x)*x - 30*x + 12*x + 8*x + 4)/(12*x *(x - 1)) testint((a*x**3+b*x**2+c*x+d)/((x+1)*x*(x-3)),x); (27*log(x - 3)*a + 9*log(x - 3)*b + 3*log(x - 3)*c + log(x - 3)*d - 3*log(x + 1)*a + 3*log(x + 1)*b - 3*log(x + 1)*c + 3*log(x + 1)*d - 4*log(x)*d + 12*a*x)/12 testint(1/(2-log(x**2+1))**5,x); 2 5 2 4 2 3 2 2 - int(1/(log(x + 1) - 10*log(x + 1) + 40*log(x + 1) - 80*log(x + 1) 2 + 80*log(x + 1) - 32),x) % The next integral appeared in Risch's 1968 paper. testint(2*x*e**(x**2)*log(x)+e**(x**2)/x+(log(x)-2)/(log(x)**2+x)**2+ ((2/x)*log(x)+(1/x)+1)/(log(x)**2+x),x); 2 2 x 3 x 2 2 2 (e *log(x) + e *log(x)*x + log(log(x) + x)*log(x) + log(log(x) + x)*x 2 - log(x))/(log(x) + x) % The following integral would not evaluate in REDUCE 3.3. testint(exp(x*ze+x/2)*sin(pi*ze)**4*x**4,ze); (2*x*ze + x)/2 3 3 3 (e *x *( - 16*cos(pi*ze)*sin(pi*ze) *pi *x 3 3 3 - 4*cos(pi*ze)*sin(pi*ze) *pi*x - 24*cos(pi*ze)*sin(pi*ze)*pi *x 4 2 2 4 4 2 2 2 4 + 4*sin(pi*ze) *pi *x + sin(pi*ze) *x + 12*sin(pi*ze) *pi *x + 24*pi ))/ 4 2 2 4 (64*pi + 20*pi *x + x ) % This one evaluates: testint(erf(x),x); 2 x e *erf(x)*pi*x + sqrt(pi) ---------------------------- 2 x e *pi % So why not this one? testint(erf(x+a),x); int(erf(a + x),x) Comment here is an example of using the integrator with pattern matching; for all m,n let int(k1**m*log(k1)**n/(p**2-k1**2),k1)=foo(m,n), int(k1*log(k1)**n/(p**2-k1**2),k1)=foo(1,n), int(k1**m*log(k1)/(p**2-k1**2),k1)=foo(m,1), int(k1*log(k1)/(p**2-k1**2),k1)=foo(1,1), int(log(k1)**n/(k1*(p**2-k1**2)),k1)=foo(-1,n); int(k1**2*log(k1)/(p**2-k1**2),k1); *** foo declared operator foo(2,1) COMMENT It is interesting to see how much of this one can be done; let f1s= (12*log(s/mc**2)*s**2*pi**2*mc**3*(-8*s-12*mc**2+3*mc) + pi**2*(12*s**4*mc+3*s**4+176*s**3*mc**3-24*s**3*mc**2 -144*s**2*mc**5-48*s*mc**7+24*s*mc**6+4*mc**9-3*mc**8)) /(384*e**(s/y)*s**2); int(f1s,s); 2 s/y - s 9 s/y - s 8 (pi *( - 4*e *ei(------)*mc *s + 3*e *ei(------)*mc *s y y s/y - s 7 s/y - s 6 - 48*e *ei(------)*mc *s*y + 24*e *ei(------)*mc *s*y y y s/y - s 5 2 s/y - s 4 2 - 144*e *ei(------)*mc *s*y + 36*e *ei(------)*mc *s*y y y s/y - s 3 3 s 5 2 - 96*e *ei(------)*mc *s*y + 144*log(-----)*mc *s*y y 2 mc s 4 2 s 3 2 2 - 36*log(-----)*mc *s*y + 96*log(-----)*mc *s *y 2 2 mc mc s 3 3 9 8 5 2 + 96*log(-----)*mc *s*y - 4*mc *y + 3*mc *y + 144*mc *s*y 2 mc 3 2 2 3 3 2 2 2 2 3 3 2 - 176*mc *s *y - 80*mc *s*y + 24*mc *s *y + 24*mc *s*y - 12*mc*s *y 2 3 4 3 2 2 3 4 s/y - 24*mc*s *y - 24*mc*s*y - 3*s *y - 6*s *y - 6*s*y ))/(384*e *s*y) factor ei,log; ws; s/y - s 3 2 (e *ei(------)*mc *pi *s y 6 5 4 3 2 2 2 3 *( - 4*mc + 3*mc - 48*mc *y + 24*mc *y - 144*mc *y + 36*mc*y - 96*y ) s 3 2 2 2 2 9 + 12*log(-----)*mc *pi *s*y *(12*mc - 3*mc + 8*s + 8*y) + pi *y*( - 4*mc 2 mc 8 5 3 2 3 2 2 2 + 3*mc + 144*mc *s*y - 176*mc *s *y - 80*mc *s*y + 24*mc *s *y 2 2 3 2 2 3 3 2 2 + 24*mc *s*y - 12*mc*s *y - 24*mc*s *y - 24*mc*s*y - 3*s *y - 6*s *y 3 s/y - 6*s*y ))/(384*e *s*y) Comment the following integrals reveal deficiencies in the current integrator; %high degree denominator; %testint(1/(2-log(x**2+1))**5,x); %this example should evaluate; testint(sin(2*x)/cos(x),x); sin(2*x) int(----------,x) cos(x) %this example, which appeared in Tobey's thesis, needs factorization %over algebraic fields. It currently gives an ugly answer and so has %been suppressed; % testint((7*x**13+10*x**8+4*x**7-7*x**6-4*x**3-4*x**2+3*x+3)/ % (x**14-2*x**8-2*x**7-2*x**4-4*x**3-x**2+2*x+1),x); symbolic summarize!-integral!-test(); ***** SUMMARY OF INTEGRAL TESTS ***** Number of integrals tested: 278 Total time taken: 49440 ms Number of incorrect integrals: 0 Number of unevaluated integrals: 22 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 3 3 - 3*sqrt(x + 1) + 3*x + 3 ------------------------------ 4 2*x + 2*x r ----------------------------------- 2 2 4 sqrt( - alpha + 2*e*r - 2*k*r ) 1 ------------------------------------- 2 2 4 sqrt( - alpha + 2*h*r - 2*k*r )*r 1 ------------------------------------------------ 2 2 2 4 sqrt( - alpha - epsilon + 2*h*r - 2*k*r )*r sqrt(sin(x) + 1) sqrt( - sin(x) + 1) sqrt(cos(x) + 1) sqrt( - cos(x) + 1) 3 log(x)*x + log(x) -------------------- 4 x + 2 2 asin(x)*x 2 5 2 4 2 3 2 2 ( - 1)/(log(x + 1) - 10*log(x + 1) + 40*log(x + 1) - 80*log(x + 1) 2 + 80*log(x + 1) - 32) erf(a + x) sin(2*x) ---------- cos(x) end; (TIME: int 49530 52429)