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Sat Jun 29 13:41:03 PDT 1991 REDUCE 3.4, 15-Jul-91 ... 1: 1: 2: 2: 3: 3: COMMENT THE REDUCE INTEGRATION TEST PACKAGE Edited By Anthony C. Hearn The RAND Corporation This file is designed to provide a set of representative tests of the Reduce integration package. Not all examples go through, even when an integral exists, since some of the arguments are outside the domain of applicability of the current package. However, future improvements to the package will result in more closed-form evaluations in later releases. We would appreciate any additional contributions to this test file either because they illustrate some feature (good or bad) of the current package, or suggest domains which future versions should handle. Any suggestions for improved organization of this test file (e.g., in a way which corresponds more directly to the organization of a standard integration table book such as Gradshteyn and Ryznik) are welcome. Acknowledgments: The examples in this file have been contributed by the following. Any omissions to this list should be reported to the Editor. David M. Dahm James H. Davenport John P. Fitch Steven Harrington Anthony C. Hearn K. Siegfried Koelbig Ernst Krupnikov Arthur C. Norman Herbert Stoyan ; Comment we first set up a suitable testing functions; fluid '(gcknt!*); global '(faillist!* gcnumber!* inittime number!-of!-integrals unintlist!*); symbolic operator time; symbolic procedure initialize!-integral!-test; begin faillist!* := unintlist!* := nil; number!-of!-integrals := 0; gcnumber!* := gcknt!*; inittime := time() end; INITIALIZE!-INTEGRAL!-TEST symbolic procedure summarize!-integral!-test; begin scalar totaltime; totaltime := time()-inittime; prin2t " ***** SUMMARY OF INTEGRAL TESTS *****"; terpri(); prin2 "Number of integrals tested: "; prin2t number!-of!-integrals; terpri(); prin2 "Total time taken: "; prin2 totaltime; prin2t " ms"; terpri(); if gcnumber!* then <<prin2 "Number of garbage collections: "; prin2t (gcknt!* - gcnumber!*); terpri()>>; prin2 "Number of incorrect integrals: "; prin2t length faillist!*; terpri(); prin2 "Number of unevaluated integrals: "; prin2t length unintlist!*; terpri(); if faillist!* then <<prin2t "Integrands of incorrect integrals are:"; for each x in reverse faillist!* do mathprint car x>>; if unintlist!* then <<prin2t "Integrands of unevaluated integrals are:"; terpri(); for each x in reverse unintlist!* do mathprint car x>> end; SUMMARIZE!-INTEGRAL!-TEST procedure testint(a,b); begin scalar der,diffce,res,tt; tt:=time(); symbolic (number!-of!-integrals := number!-of!-integrals + 1); res:=int(a,b); % write "time for integral: ",time()-tt," ms"; der := df(res,b); diffce := der-a; if diffce neq 0 then begin for all x let cot x=cos x/sin x, sec x=1/cos x, sin x**2=1-cos x**2, tan x=sin x/cos x, tan(x/2)=sin x/(1+cos x), tanh x= (e**(x)-e**(-x))/(e**x+e**(-x)), coth x= 1/tanh x; diffce := diffce; for all x clear cot x,sec x,sin x**2,tan x,tan(x/2), tanh x,coth x end; %hopefully, difference appeared non-zero due to absence of %above transformations; if diffce neq 0 then <<write " ***** DERIVATIVE OF INTEGRAL NOT EQUAL TO INTEGRAND *****"; symbolic(faillist!* := list(a,b,res,der) . faillist!*)>>; symbolic if smemq('int,res) then unintlist!* := list(a,b,res) . unintlist!*; return res end; TESTINT symbolic initialize!-integral!-test(); % References are to Gradshteyn and Ryznik. testint(1+x+x**2,x); 2 X*(2*X + 3*X + 6) -------------------- 6 testint(x**2*(2*x**2+x)**2,x); 5 2 X *(60*X + 70*X + 21) ------------------------ 105 testint(x*(x**2+2*x+1),x); 2 2 X *(3*X + 8*X + 6) --------------------- 12 testint(1/x,x); LOG(X) % 2.01 #2; testint((x+1)**3/(x-1)**4,x); 3 2 (3*LOG(X - 1)*X - 9*LOG(X - 1)*X + 9*LOG(X - 1)*X - 3*LOG(X - 1) 3 3 2 - 6*X - 2)/(3*(X - 3*X + 3*X - 1)) testint(1/(x*(x-1)*(x+1)**2),x); (LOG(X - 1)*X + LOG(X - 1) + 3*LOG(X + 1)*X + 3*LOG(X + 1) - 4*LOG(X)*X - 4*LOG(X) + 2*X)/(4*(X + 1)) testint((a*x+b)/((x-p)*(x-q)),x); LOG(P - X)*A*P + LOG(P - X)*B - LOG(Q - X)*A*Q - LOG(Q - X)*B --------------------------------------------------------------- P - Q testint(1/(a*x**2+b*x+c),x); 2 2*A*X + B 2*SQRT(4*A*C - B )*ATAN(------------------) 2 SQRT(4*A*C - B ) --------------------------------------------- 2 4*A*C - B testint((a*x+b)/(1+x**2),x); 2 LOG(X + 1)*A + 2*ATAN(X)*B ----------------------------- 2 testint(1/(x**2-2*x+3),x); X - 1 SQRT(2)*ATAN(---------) SQRT(2) ------------------------- 2 % Rational function examples from Hardy, Pure Mathematics, p 253 et seq. testint(1/((x-1)*(x**2+1))**2,x); 2 3 2 2 2 2 (LOG(X + 1)*X - LOG(X + 1)*X + LOG(X + 1)*X - LOG(X + 1) 3 2 - 2*LOG(X - 1)*X + 2*LOG(X - 1)*X - 2*LOG(X - 1)*X + 2*LOG(X - 1) 3 2 3 + ATAN(X)*X - ATAN(X)*X + ATAN(X)*X - ATAN(X) - X - 2*X + 1)/(4 3 2 *(X - X + X - 1)) testint(x/((x-a)*(x-b)*(x-c)),x); (LOG(A - X)*A*B - LOG(A - X)*A*C - LOG(B - X)*A*B + LOG(B - X)*B*C 2 2 2 2 + LOG(C - X)*A*C - LOG(C - X)*B*C)/(A *B - A *C - A*B + A*C 2 2 + B *C - B*C ) testint(x/((x**2+a**2)*(x**2+b**2)),x); 2 2 2 2 - LOG(A + X ) + LOG(B + X ) -------------------------------- 2 2 2*(A - B ) testint(x**2/((x**2+a**2)*(x**2+b**2)),x); X X ATAN(---)*A - ATAN(---)*B A B --------------------------- 2 2 A - B testint(x/((x-1)*(x**2+1)),x); 2 - LOG(X + 1) + 2*LOG(X - 1) + 2*ATAN(X) ------------------------------------------- 4 testint(x/(1+x**3),x); 2*X - 1 2 2*SQRT(3)*ATAN(---------) + LOG(X - X + 1) - 2*LOG(X + 1) SQRT(3) ------------------------------------------------------------ 6 testint(x**3/((x-1)**2*(x**3+1)),x); 2 2 ( - 4*LOG(X - X + 1)*X + 4*LOG(X - X + 1) + 9*LOG(X - 1)*X - 9*LOG(X - 1) - LOG(X + 1)*X + LOG(X + 1) - 6*X)/(12*(X - 1)) testint(1/(1+x**4),x); 2 2 (SQRT(2)*( - LOG( - SQRT(2)*X + X + 1) + LOG(SQRT(2)*X + X + 1) SQRT(2) - 2*X SQRT(2) + 2*X - 2*ATAN(---------------) + 2*ATAN(---------------)))/8 SQRT(2) SQRT(2) testint(x**2/(1+x**4),x); 2 2 (SQRT(2)*(LOG( - SQRT(2)*X + X + 1) - LOG(SQRT(2)*X + X + 1) SQRT(2) - 2*X SQRT(2) + 2*X - 2*ATAN(---------------) + 2*ATAN(---------------)))/8 SQRT(2) SQRT(2) testint(1/(1+x**2+x**4),x); 2*X - 1 2*X + 1 (2*SQRT(3)*ATAN(---------) + 2*SQRT(3)*ATAN(---------) SQRT(3) SQRT(3) 2 2 - 3*LOG(X - X + 1) + 3*LOG(X + X + 1))/12 % Examples involving a+b*x. z := a+b*x; Z := A + B*X testint(z**p,x); P (A + B*X) *(A + B*X) ---------------------- B*(P + 1) testint(x*z**p,x); P 2 2 2 2 2 (A + B*X) *( - A + A*B*P*X + B *P*X + B *X ) ------------------------------------------------ 2 2 B *(P + 3*P + 2) testint(x**2*z**p,x); P 3 2 2 2 2 2 2 3 2 3 ((A + B*X) *(2*A - 2*A *B*P*X + A*B *P *X + A*B *P*X + B *P *X 3 3 3 3 3 3 2 + 3*B *P*X + 2*B *X ))/(B *(P + 6*P + 11*P + 6)) testint(1/z,x); LOG(A + B*X) -------------- B testint(1/z**2,x); X ------------- A*(A + B*X) testint(x/z,x); - LOG(A + B*X)*A + B*X ------------------------- 2 B testint(x**2/z,x); 2 2 2 2*LOG(A + B*X)*A - 2*A*B*X + B *X ------------------------------------- 3 2*B testint(1/(x*z),x); - LOG(A + B*X) + LOG(X) -------------------------- A testint(1/(x**2*z),x); LOG(A + B*X)*B*X - LOG(X)*B*X - A ----------------------------------- 2 A *X testint(1/(x*z)**2,x); 2 2 (2*LOG(A + B*X)*A*B*X + 2*LOG(A + B*X)*B *X - 2*LOG(X)*A*B*X 2 2 2 2 2 3 - 2*LOG(X)*B *X - A + 2*B *X )/(A *X*(A + B*X)) testint(1/(c**2+x**2),x); X ATAN(---) C ----------- C testint(1/(c**2-x**2),x); LOG( - C - X) - LOG(C - X) ---------------------------- 2*C % More complicated rational function examples, mostly contributed % by David M. Dahm, who also developed the code to integrate them. testint(1/(2*x**3-1),x); 1/3 2/3 2*2 *X + 1 2/3 2 1/3 (2 *( - 2*SQRT(3)*ATAN(--------------) - LOG(2 *X + 2 *X + 1) SQRT(3) 1/3 + 2*LOG(2 *X - 1)))/12 testint(1/(x**3-2),x); 1/3 1/3 2 + 2*X 2/3 1/3 2 (2 *( - 2*SQRT(3)*ATAN(--------------) - LOG(2 + 2 *X + X ) 1/3 2 *SQRT(3) 1/3 + 2*LOG( - 2 + X)))/12 testint(1/(a*x**3-b),x); 1/3 1/3 1/3 2/3 2*A *X + B (B *A *( - 2*SQRT(3)*ATAN(-----------------) 1/3 B *SQRT(3) 2/3 2 1/3 1/3 2/3 - LOG(A *X + B *A *X + B ) 1/3 1/3 + 2*LOG(A *X - B )))/(6*A*B) testint(1/(x**4-2),x); 1/4 1/4 1/4 X 2 *( - LOG(2 + X) + LOG( - 2 + X) - 2*ATAN(------)) 1/4 2 ------------------------------------------------------------- 8 testint(1/(5*x**4-1),x); 1/4 (SQRT(5)*5 1/4 1/4 SQRT(5)*X *(LOG(5 *X - 1) - LOG(5 *X + 1) - 2*ATAN(-----------)))/20 1/4 5 testint(1/(3*x**4+7),x); 1/4 1/4 2 (SQRT(6)*21 *( - LOG( - SQRT(2)*21 *X + SQRT(7) + SQRT(3)*X ) 1/4 2 + LOG(SQRT(2)*21 *X + SQRT(7) + SQRT(3)*X ) 1/4 SQRT(2)*21 - 2*SQRT(3)*X - 2*ATAN(-----------------------------) 1/4 SQRT(2)*21 1/4 SQRT(2)*21 + 2*SQRT(3)*X + 2*ATAN(-----------------------------)))/168 1/4 SQRT(2)*21 testint(1/(x**4+3*x**2-1),x); (SQRT(2)*(3*SQRT(SQRT(13) - 3)*SQRT(13) *LOG( - SQRT(SQRT(13) - 3) + SQRT(2)*X) - 3 *SQRT(SQRT(13) - 3)*SQRT(13) *LOG(SQRT(SQRT(13) - 3) + SQRT(2)*X) + 13 *SQRT(SQRT(13) - 3)*LOG( - SQRT(SQRT(13) - 3) + SQRT(2)*X) - 13*SQRT(SQRT(13) - 3)*LOG(SQRT(SQRT(13) - 3) + SQRT(2)*X) + 6*SQRT(SQRT(13) + 3)*SQRT(13) 2*X *ATAN(----------------------------) SQRT(SQRT(13) + 3)*SQRT(2) 2*X - 26*SQRT(SQRT(13) + 3)*ATAN(----------------------------) SQRT(SQRT(13) + 3)*SQRT(2) ))/104 testint(1/(x**4-3*x**2-1),x); (SQRT(2)*( - 6*SQRT(SQRT(13) - 3)*SQRT(13) 2*X *ATAN(----------------------------) SQRT(SQRT(13) - 3)*SQRT(2) 2*X - 26*SQRT(SQRT(13) - 3)*ATAN(----------------------------) SQRT(SQRT(13) - 3)*SQRT(2) - 3*SQRT(SQRT(13) + 3)*SQRT(13) *LOG( - SQRT(SQRT(13) + 3) + SQRT(2)*X) + 3 *SQRT(SQRT(13) + 3)*SQRT(13) *LOG(SQRT(SQRT(13) + 3) + SQRT(2)*X) + 13 *SQRT(SQRT(13) + 3)*LOG( - SQRT(SQRT(13) + 3) + SQRT(2)*X) - 13*SQRT(SQRT(13) + 3)*LOG(SQRT(SQRT(13) + 3) + SQRT(2)*X) ))/104 testint(1/(x**4-3*x**2+1),x); ( - SQRT(5)*LOG( - SQRT(5) + 2*X - 1) - SQRT(5)*LOG( - SQRT(5) + 2*X + 1) + SQRT(5)*LOG(SQRT(5) + 2*X - 1) + SQRT(5)*LOG(SQRT(5) + 2*X + 1) + 5*LOG( - SQRT(5) + 2*X - 1) - 5*LOG( - SQRT(5) + 2*X + 1) + 5*LOG(SQRT(5) + 2*X - 1) - 5*LOG(SQRT(5) + 2*X + 1))/20 testint(1/(x**4-4*x**2+1),x); X (4*SQRT(SQRT(3) - 2)*SQRT(3)*ATAN(-------------------) SQRT(SQRT(3) - 2) X + 6*SQRT(SQRT(3) - 2)*ATAN(-------------------) SQRT(SQRT(3) - 2) + 2*SQRT(SQRT(3) + 2)*SQRT(3)*LOG( - SQRT(SQRT(3) + 2) + X) - 2*SQRT(SQRT(3) + 2)*SQRT(3)*LOG(SQRT(SQRT(3) + 2) + X) - 3*SQRT(SQRT(3) + 2)*LOG( - SQRT(SQRT(3) + 2) + X) + 3*SQRT(SQRT(3) + 2)*LOG(SQRT(SQRT(3) + 2) + X))/12 testint(1/(x**4+4*x**2+1),x); ( - 2*SQRT(SQRT(3) - 2)*SQRT(3)*LOG( - SQRT(SQRT(3) - 2) + X) + 2*SQRT(SQRT(3) - 2)*SQRT(3)*LOG(SQRT(SQRT(3) - 2) + X) - 3*SQRT(SQRT(3) - 2)*LOG( - SQRT(SQRT(3) - 2) + X) + 3*SQRT(SQRT(3) - 2)*LOG(SQRT(SQRT(3) - 2) + X) X - 4*SQRT(SQRT(3) + 2)*SQRT(3)*ATAN(-------------------) SQRT(SQRT(3) + 2) X + 6*SQRT(SQRT(3) + 2)*ATAN(-------------------))/12 SQRT(SQRT(3) + 2) testint(1/(x**4+x**2+2),x); ( - SQRT(2*SQRT(2) - 1)*SQRT(2) 2 *LOG( - SQRT(2*SQRT(2) - 1)*X + SQRT(2) + X ) + SQRT(2*SQRT(2) - 1) 2 *SQRT(2)*LOG(SQRT(2*SQRT(2) - 1)*X + SQRT(2) + X ) - 4 2 *SQRT(2*SQRT(2) - 1)*LOG( - SQRT(2*SQRT(2) - 1)*X + SQRT(2) + X ) 2 + 4*SQRT(2*SQRT(2) - 1)*LOG(SQRT(2*SQRT(2) - 1)*X + SQRT(2) + X ) SQRT(2*SQRT(2) - 1) - 2*X + 2*SQRT(2*SQRT(2) + 1)*SQRT(2)*ATAN(---------------------------) SQRT(2*SQRT(2) + 1) SQRT(2*SQRT(2) - 1) + 2*X - 2*SQRT(2*SQRT(2) + 1)*SQRT(2)*ATAN(---------------------------) SQRT(2*SQRT(2) + 1) SQRT(2*SQRT(2) - 1) - 2*X - 8*SQRT(2*SQRT(2) + 1)*ATAN(---------------------------) SQRT(2*SQRT(2) + 1) SQRT(2*SQRT(2) - 1) + 2*X + 8*SQRT(2*SQRT(2) + 1)*ATAN(---------------------------))/56 SQRT(2*SQRT(2) + 1) testint(1/(x**4-x**2+2),x); SQRT(2*SQRT(2) + 1) - 2*X ( - 2*SQRT(2*SQRT(2) - 1)*SQRT(2)*ATAN(---------------------------) SQRT(2*SQRT(2) - 1) SQRT(2*SQRT(2) + 1) + 2*X + 2*SQRT(2*SQRT(2) - 1)*SQRT(2)*ATAN(---------------------------) SQRT(2*SQRT(2) - 1) SQRT(2*SQRT(2) + 1) - 2*X - 8*SQRT(2*SQRT(2) - 1)*ATAN(---------------------------) SQRT(2*SQRT(2) - 1) SQRT(2*SQRT(2) + 1) + 2*X + 8*SQRT(2*SQRT(2) - 1)*ATAN(---------------------------) + SQRT(2*SQRT(2) - 1) SQRT(2*SQRT(2) + 1)*SQRT(2) 2 *LOG( - SQRT(2*SQRT(2) + 1)*X + SQRT(2) + X ) - SQRT(2*SQRT(2) + 1) 2 *SQRT(2)*LOG(SQRT(2*SQRT(2) + 1)*X + SQRT(2) + X ) - 4 2 *SQRT(2*SQRT(2) + 1)*LOG( - SQRT(2*SQRT(2) + 1)*X + SQRT(2) + X ) 2 + 4*SQRT(2*SQRT(2) + 1)*LOG(SQRT(2*SQRT(2) + 1)*X + SQRT(2) + X ))/ 56 testint(1/(x**6-1),x); 2*X - 1 2*X + 1 ( - 2*SQRT(3)*ATAN(---------) - 2*SQRT(3)*ATAN(---------) SQRT(3) SQRT(3) 2 2 + LOG(X - X + 1) - LOG(X + X + 1) + 2*LOG(X - 1) - 2*LOG(X + 1))/ 12 testint(1/(x**6-2),x); 1/6 1/6 2 - 2*X (2 *(2*SQRT(3)*ATAN(--------------) 1/6 2 *SQRT(3) 1/6 2 + 2*X 1/6 - 2*SQRT(3)*ATAN(--------------) - 2*LOG(2 + X) 1/6 2 *SQRT(3) 1/6 1/6 1/3 2 + 2*LOG( - 2 + X) + LOG( - 2 *X + 2 + X ) 1/6 1/3 2 - LOG(2 *X + 2 + X )))/24 testint(1/(x**6+2),x); 1/6 1/6 1/3 2 (2 *( - SQRT(3)*LOG( - 2 *SQRT(3)*X + 2 + X ) 1/6 1/3 2 + SQRT(3)*LOG(2 *SQRT(3)*X + 2 + X ) 1/6 1/6 2 *SQRT(3) - 2*X 2 *SQRT(3) + 2*X - 2*ATAN(--------------------) + 2*ATAN(--------------------) 1/6 1/6 2 2 X + 4*ATAN(------)))/24 1/6 2 testint(1/(x**8+1),x); 2 ( - SQRT( - SQRT(2) + 2)*LOG( - SQRT( - SQRT(2) + 2)*X + X + 1) 2 + SQRT( - SQRT(2) + 2)*LOG(SQRT( - SQRT(2) + 2)*X + X + 1) SQRT(SQRT(2) + 2) - 2*X - 2*SQRT( - SQRT(2) + 2)*ATAN(-------------------------) SQRT( - SQRT(2) + 2) SQRT(SQRT(2) + 2) + 2*X + 2*SQRT( - SQRT(2) + 2)*ATAN(-------------------------) SQRT( - SQRT(2) + 2) 2 - SQRT(SQRT(2) + 2)*LOG( - SQRT(SQRT(2) + 2)*X + X + 1) 2 + SQRT(SQRT(2) + 2)*LOG(SQRT(SQRT(2) + 2)*X + X + 1) SQRT( - SQRT(2) + 2) - 2*X - 2*SQRT(SQRT(2) + 2)*ATAN(----------------------------) SQRT(SQRT(2) + 2) SQRT( - SQRT(2) + 2) + 2*X + 2*SQRT(SQRT(2) + 2)*ATAN(----------------------------))/16 SQRT(SQRT(2) + 2) testint(1/(x**8-1),x); 2 2 (SQRT(2)*LOG( - SQRT(2)*X + X + 1) - SQRT(2)*LOG(SQRT(2)*X + X + 1) SQRT(2) - 2*X SQRT(2) + 2*X + 2*SQRT(2)*ATAN(---------------) - 2*SQRT(2)*ATAN(---------------) SQRT(2) SQRT(2) + 2*LOG(X - 1) - 2*LOG(X + 1) - 4*ATAN(X))/16 testint(1/(x**8-x**4+1),x); ( - SQRT( - SQRT(3) + 2)*SQRT(3) 2 *LOG( - SQRT( - SQRT(3) + 2)*X + X + 1) 2 + SQRT( - SQRT(3) + 2)*SQRT(3)*LOG(SQRT( - SQRT(3) + 2)*X + X + 1) SQRT(SQRT(3) + 2) - 2*X - 2*SQRT( - SQRT(3) + 2)*SQRT(3)*ATAN(-------------------------) SQRT( - SQRT(3) + 2) SQRT(SQRT(3) + 2) + 2*X + 2*SQRT( - SQRT(3) + 2)*SQRT(3)*ATAN(-------------------------) SQRT( - SQRT(3) + 2) 2 - 3*SQRT( - SQRT(3) + 2)*LOG( - SQRT( - SQRT(3) + 2)*X + X + 1) 2 + 3*SQRT( - SQRT(3) + 2)*LOG(SQRT( - SQRT(3) + 2)*X + X + 1) SQRT(SQRT(3) + 2) - 2*X - 6*SQRT( - SQRT(3) + 2)*ATAN(-------------------------) SQRT( - SQRT(3) + 2) SQRT(SQRT(3) + 2) + 2*X + 6*SQRT( - SQRT(3) + 2)*ATAN(-------------------------) SQRT( - SQRT(3) + 2) 2 + SQRT(SQRT(3) + 2)*SQRT(3)*LOG( - SQRT(SQRT(3) + 2)*X + X + 1) 2 - SQRT(SQRT(3) + 2)*SQRT(3)*LOG(SQRT(SQRT(3) + 2)*X + X + 1) SQRT( - SQRT(3) + 2) - 2*X + 2*SQRT(SQRT(3) + 2)*SQRT(3)*ATAN(----------------------------) SQRT(SQRT(3) + 2) SQRT( - SQRT(3) + 2) + 2*X - 2*SQRT(SQRT(3) + 2)*SQRT(3)*ATAN(----------------------------) SQRT(SQRT(3) + 2) 2 - 3*SQRT(SQRT(3) + 2)*LOG( - SQRT(SQRT(3) + 2)*X + X + 1) 2 + 3*SQRT(SQRT(3) + 2)*LOG(SQRT(SQRT(3) + 2)*X + X + 1) SQRT( - SQRT(3) + 2) - 2*X - 6*SQRT(SQRT(3) + 2)*ATAN(----------------------------) SQRT(SQRT(3) + 2) SQRT( - SQRT(3) + 2) + 2*X + 6*SQRT(SQRT(3) + 2)*ATAN(----------------------------))/24 SQRT(SQRT(3) + 2) testint(x**7/(x**12+1),x); (2*SQRT(SQRT(3) + 2)*SQRT( - SQRT(3) + 2)*SQRT(3) SQRT( - SQRT(3) + 2) - 2*X *ATAN(----------------------------) + 2*SQRT(SQRT(3) + 2) SQRT(SQRT(3) + 2) SQRT( - SQRT(3) + 2) + 2*X *SQRT( - SQRT(3) + 2)*SQRT(3)*ATAN(----------------------------) - 2 SQRT(SQRT(3) + 2) *SQRT(SQRT(3) + 2)*SQRT( - SQRT(3) + 2)*SQRT(3) SQRT(SQRT(3) + 2) - 2*X *ATAN(-------------------------) - 2*SQRT(SQRT(3) + 2) SQRT( - SQRT(3) + 2) SQRT(SQRT(3) + 2) + 2*X *SQRT( - SQRT(3) + 2)*SQRT(3)*ATAN(-------------------------) SQRT( - SQRT(3) + 2) 2 + LOG( - SQRT( - SQRT(3) + 2)*X + X + 1) 2 + LOG( - SQRT(SQRT(3) + 2)*X + X + 1) 2 - 2*LOG( - SQRT(2)*X + X + 1) 2 + LOG(SQRT( - SQRT(3) + 2)*X + X + 1) 2 2 + LOG(SQRT(SQRT(3) + 2)*X + X + 1) - 2*LOG(SQRT(2)*X + X + 1))/24 % Examples involving logarithms. testint(log x,x); X*(LOG(X) - 1) testint(x*log x,x); 2 X *(2*LOG(X) - 1) ------------------- 4 testint(x**2*log x,x); 3 X *(3*LOG(X) - 1) ------------------- 9 testint(x**p*log x,x); P X *X*(LOG(X)*P + LOG(X) - 1) ------------------------------ 2 P + 2*P + 1 testint((log x)**2,x); 2 X*(LOG(X) - 2*LOG(X) + 2) testint(x**9*log x**11,x); 10 11 10 9 (X *(15625000*LOG(X) - 17187500*LOG(X) + 17187500*LOG(X) 8 7 6 - 15468750*LOG(X) + 12375000*LOG(X) - 8662500*LOG(X) 5 4 3 + 5197500*LOG(X) - 2598750*LOG(X) + 1039500*LOG(X) 2 - 311850*LOG(X) + 62370*LOG(X) - 6237))/156250000 testint(log x**2/x,x); 3 LOG(X) --------- 3 testint(1/log x,x); 1 INT(--------,X) LOG(X) testint(1/log(x+1),x); X INT(---------------------------,X) + LOG(LOG(X + 1)) LOG(X + 1)*X + LOG(X + 1) testint(1/(x*log x),x); LOG(LOG(X)) testint(1/(x*log x)**2,x); 1 - (INT(-----------,X)*LOG(X)*X + 1) 2 LOG(X)*X -------------------------------------- LOG(X)*X testint((log x)**p/x,x); P LOG(X) *LOG(X) ---------------- P + 1 testint(log x *(a*x+b),x); X*(2*LOG(X)*A*X + 4*LOG(X)*B - A*X - 4*B) ------------------------------------------- 4 testint((a*x+b)**2*log x,x); 2 2 2 2 2 (X*(6*LOG(X)*A *X + 18*LOG(X)*A*B*X + 18*LOG(X)*B - 2*A *X 2 - 9*A*B*X - 18*B ))/18 testint(log x/(a*x+b)**2,x); - LOG(A*X + B)*A*X - LOG(A*X + B)*B + LOG(X)*A*X --------------------------------------------------- A*B*(A*X + B) testint(x*log (a*x+b),x); 2 2 2 2 2 2*LOG(A*X + B)*A *X - 2*LOG(A*X + B)*B - A *X + 2*A*B*X ------------------------------------------------------------ 2 4*A testint(x**2*log(a*x+b),x); 3 3 3 3 3 2 2 (6*LOG(A*X + B)*A *X + 6*LOG(A*X + B)*B - 2*A *X + 3*A *B*X 2 3 - 6*A*B *X)/(18*A ) testint(log(x**2+a**2),x); 2 2 X LOG(A + X )*X + 2*ATAN(---)*A - 2*X A testint(x*log(x**2+a**2),x); 2 2 2 2 2 2 2 LOG(A + X )*A + LOG(A + X )*X - X ---------------------------------------- 2 testint(x**2*log(x**2+a**2),x); 2 2 3 X 3 2 3 3*LOG(A + X )*X - 6*ATAN(---)*A + 6*A *X - 2*X A ---------------------------------------------------- 9 testint(x**4*log(x**2+a**2),x); 2 2 5 X 5 4 2 3 5 15*LOG(A + X )*X + 30*ATAN(---)*A - 30*A *X + 10*A *X - 6*X A ------------------------------------------------------------------ 75 testint(log(x**2-a**2),x); 2 2 2 2 - LOG( - A + X )*A + LOG( - A + X )*X + 2*LOG( - A - X)*A - 2*X testint(log(log(log(log(x)))),x); 1 - INT(-------------------------------------,X) LOG(LOG(LOG(X)))*LOG(LOG(X))*LOG(X) + LOG(LOG(LOG(LOG(X))))*X % Examples involving circular functions. testint(sin x,x); - COS(X) % 2.01 #5; testint(cos x,x); SIN(X) % #6; testint(tan x,x); 2 LOG(TAN(X) + 1) ------------------ 2 % #11; testint(1/tan(x),x); 2 - LOG(TAN(X) + 1) + 2*LOG(TAN(X)) ------------------------------------- 2 % 2.01 #12; testint(1/(1+tan(x))**2,x); 2 2 ( - LOG(TAN(X) + 1)*TAN(X) - LOG(TAN(X) + 1) + 2*LOG(TAN(X) + 1)*TAN(X) + 2*LOG(TAN(X) + 1) + 2*TAN(X))/(4 *(TAN(X) + 1)) testint(1/cos x,x); X X - LOG(TAN(---) - 1) + LOG(TAN(---) + 1) 2 2 testint(1/sin x,x); X LOG(TAN(---)) 2 testint(sin x**2,x); - SIN(X)*COS(X) + X ---------------------- 2 testint(x**3*sin(x**2),x); 2 2 2 SIN(X ) - COS(X )*X ---------------------- 2 testint(sin x**3,x); 2 - SIN(X) *COS(X) - 2*COS(X) + 2 ---------------------------------- 3 testint(sin x**p,x); P INT(SIN(X) ,X) testint((sin x**2+1)**2*cos x,x); 4 2 SIN(X)*(3*SIN(X) + 10*SIN(X) + 15) -------------------------------------- 15 testint(cos x**2,x); SIN(X)*COS(X) + X ------------------- 2 testint(cos x**3,x); 2 SIN(X)*( - SIN(X) + 3) ------------------------- 3 testint(sin(a*x+b),x); - COS(A*X + B) ----------------- A testint(1/cos x**2,x); SIN(X) -------- COS(X) testint(sin x*sin(2*x),x); SIN(2*X)*COS(X) - 2*SIN(X)*COS(2*X) ------------------------------------- 3 testint(x*sin x,x); SIN(X) - COS(X)*X testint(x**2*sin x,x); 2 2*SIN(X)*X - COS(X)*X + 2*COS(X) testint(x*sin x**2,x); 2 2 SIN(X) - 2*SIN(X)*COS(X)*X + X - 2 -------------------------------------- 4 testint(x**2*sin x**2,x); 2 2 3 6*SIN(X) *X - 6*SIN(X)*COS(X)*X + 3*SIN(X)*COS(X) + 2*X - 3*X ----------------------------------------------------------------- 12 testint(x*sin x**3,x); 3 2 SIN(X) - 3*SIN(X) *COS(X)*X + 6*SIN(X) - 6*COS(X)*X ------------------------------------------------------ 9 testint(x*cos x,x); SIN(X)*X + COS(X) testint(x**2*cos x,x); 2 SIN(X)*X - 2*SIN(X) + 2*COS(X)*X testint(x*cos x**2,x); 2 2 - SIN(X) + 2*SIN(X)*COS(X)*X + X + 2 ----------------------------------------- 4 testint(x**2*cos x**2,x); 2 2 3 - 6*SIN(X) *X + 6*SIN(X)*COS(X)*X - 3*SIN(X)*COS(X) + 2*X + 3*X -------------------------------------------------------------------- 12 testint(x*cos x**3,x); 3 2 - 3*SIN(X) *X - SIN(X) *COS(X) + 9*SIN(X)*X + 7*COS(X) + 1 ------------------------------------------------------------- 9 testint(sin x/x,x); SIN(X) INT(--------,X) X testint(cos x/x,x); COS(X) INT(--------,X) X testint(sin x/x**2,x); SIN(X) INT(--------,X) 2 X testint(sin x**2/x,x); 2 SIN(X) INT(---------,X) X testint(tan x**3,x); 2 2 - LOG(TAN(X) + 1) + TAN(X) ------------------------------- 2 % z := a+b*x; testint(sin z,x); - COS(A + B*X) ----------------- B testint(cos z,x); SIN(A + B*X) -------------- B testint(tan z,x); 2 LOG(TAN(A + B*X) + 1) ------------------------ 2*B testint(1/tan z,x); 2 - LOG(TAN(A + B*X) + 1) + 2*LOG(TAN(A + B*X)) ------------------------------------------------- 2*B testint(1/sin z,x); A + B*X LOG(TAN(---------)) 2 --------------------- B testint(1/cos z,x); A + B*X A + B*X - LOG(TAN(---------) - 1) + LOG(TAN(---------) + 1) 2 2 ------------------------------------------------------ B testint(sin z**2,x); - SIN(A + B*X)*COS(A + B*X) + B*X ------------------------------------ 2*B testint(sin z**3,x); 2 - SIN(A + B*X) *COS(A + B*X) - 2*COS(A + B*X) + 2 ---------------------------------------------------- 3*B testint(cos z**2,x); SIN(A + B*X)*COS(A + B*X) + B*X --------------------------------- 2*B testint(cos z**3,x); 2 SIN(A + B*X)*( - SIN(A + B*X) + 3) ------------------------------------- 3*B testint(1/cos z**2,x); SIN(A + B*X) ---------------- COS(A + B*X)*B testint(1/(1+cos x),x); X TAN(---) 2 testint(1/(1-cos x),x); - 1 ---------- X TAN(---) 2 testint(1/(1+sin x),x); X 2*TAN(---) 2 -------------- X TAN(---) + 1 2 testint(1/(1-sin x),x); X - 2*TAN(---) 2 --------------- X TAN(---) - 1 2 testint(1/(a+b*sin x),x); X TAN(---)*A + B 2 2 2 2*SQRT(A - B )*ATAN(----------------) 2 2 SQRT(A - B ) ---------------------------------------- 2 2 A - B testint(1/(a+b*sin x+cos x),x); X X TAN(---)*A - TAN(---) + B 2 2 2 2 2*SQRT(A - B - 1)*ATAN(---------------------------) 2 2 SQRT(A - B - 1) ------------------------------------------------------- 2 2 A - B - 1 testint(x**2*sin z**2,x); 2 2 2 (6*SIN(A + B*X) *B*X - 6*SIN(A + B*X)*COS(A + B*X)*B *X 3 3 3 + 3*SIN(A + B*X)*COS(A + B*X) + 9*A + 2*B *X - 3*B*X)/(12*B ) testint(cos x*cos(2*x),x); 2*SIN(2*X)*COS(X) - SIN(X)*COS(2*X) ------------------------------------- 3 testint(x**2*cos z**2,x); 2 2 2 ( - 6*SIN(A + B*X) *B*X + 6*SIN(A + B*X)*COS(A + B*X)*B *X 3 3 3 - 3*SIN(A + B*X)*COS(A + B*X) + 2*B *X + 3*B*X)/(12*B ) testint(1/tan x**3,x); 2 2 2 LOG(TAN(X) + 1)*TAN(X) - 2*LOG(TAN(X))*TAN(X) - 1 ------------------------------------------------------ 2 2*TAN(X) testint(x**3*tan(x)**4,x); 2 2 3 3 (48*INT(TAN(X)*X ,X) - 6*LOG(TAN(X) + 1) + 4*TAN(X) *X 2 2 3 4 2 - 6*TAN(X) *X - 12*TAN(X)*X + 12*TAN(X)*X + 3*X - 6*X )/12 testint(x**3*tan(x)**6,x); 2 2 5 3 ( - 276*INT(TAN(X)*X ,X) + 60*LOG(TAN(X) + 1) + 12*TAN(X) *X 4 2 3 3 3 2 2 - 9*TAN(X) *X - 20*TAN(X) *X + 6*TAN(X) *X + 48*TAN(X) *X 2 3 4 2 - 3*TAN(X) + 60*TAN(X)*X - 114*TAN(X)*X - 15*X + 57*X )/60 testint(x*tan(x)**2,x); 2 2 - LOG(TAN(X) + 1) + 2*TAN(X)*X - X --------------------------------------- 2 testint(sin(2*x)*cos(3*x),x); 3*SIN(3*X)*SIN(2*X) + 2*COS(3*X)*COS(2*X) ------------------------------------------- 5 testint(sin x**2*cos x**2,x); 3 2*SIN(X) *COS(X) - SIN(X)*COS(X) + X -------------------------------------- 8 testint(1/(sin x**2*cos x**2),x); 2 2*SIN(X) - 1 --------------- SIN(X)*COS(X) testint(d**x*sin x,x); X D *(SIN(X)*LOG(D) - COS(X)) ----------------------------- 2 LOG(D) + 1 testint(d**x*cos x,x); X D *(SIN(X) + COS(X)*LOG(D)) ----------------------------- 2 LOG(D) + 1 testint(x*d**x*sin x,x); X 3 2 (D *(SIN(X)*LOG(D) *X - SIN(X)*LOG(D) + SIN(X)*LOG(D)*X + SIN(X) 2 4 - COS(X)*LOG(D) *X + 2*COS(X)*LOG(D) - COS(X)*X))/(LOG(D) 2 + 2*LOG(D) + 1) testint(x*d**x*cos x,x); X 2 3 (D *(SIN(X)*LOG(D) *X - 2*SIN(X)*LOG(D) + SIN(X)*X + COS(X)*LOG(D) *X 2 4 - COS(X)*LOG(D) + COS(X)*LOG(D)*X + COS(X)))/(LOG(D) 2 + 2*LOG(D) + 1) testint(x**2*d**x*sin x,x); X 5 2 4 3 2 (D *(SIN(X)*LOG(D) *X - 2*SIN(X)*LOG(D) *X + 2*SIN(X)*LOG(D) *X 3 2 + 2*SIN(X)*LOG(D) + SIN(X)*LOG(D)*X - 6*SIN(X)*LOG(D) 4 2 3 + 2*SIN(X)*X - COS(X)*LOG(D) *X + 4*COS(X)*LOG(D) *X 2 2 2 - 2*COS(X)*LOG(D) *X - 6*COS(X)*LOG(D) + 4*COS(X)*LOG(D)*X 2 6 4 2 - COS(X)*X + 2*COS(X)))/(LOG(D) + 3*LOG(D) + 3*LOG(D) + 1) testint(x**2*d**x*cos x,x); X 4 2 3 2 2 (D *(SIN(X)*LOG(D) *X - 4*SIN(X)*LOG(D) *X + 2*SIN(X)*LOG(D) *X 2 2 + 6*SIN(X)*LOG(D) - 4*SIN(X)*LOG(D)*X + SIN(X)*X - 2*SIN(X) 5 2 4 3 2 + COS(X)*LOG(D) *X - 2*COS(X)*LOG(D) *X + 2*COS(X)*LOG(D) *X 3 2 + 2*COS(X)*LOG(D) + COS(X)*LOG(D)*X - 6*COS(X)*LOG(D) 6 4 2 + 2*COS(X)*X))/(LOG(D) + 3*LOG(D) + 3*LOG(D) + 1) testint(x**3*d**x*sin x,x); X 7 3 6 2 5 3 (D *(SIN(X)*LOG(D) *X - 3*SIN(X)*LOG(D) *X + 3*SIN(X)*LOG(D) *X 5 4 2 4 + 6*SIN(X)*LOG(D) *X - 3*SIN(X)*LOG(D) *X - 6*SIN(X)*LOG(D) 3 3 3 + 3*SIN(X)*LOG(D) *X - 12*SIN(X)*LOG(D) *X 2 2 2 3 + 3*SIN(X)*LOG(D) *X + 36*SIN(X)*LOG(D) + SIN(X)*LOG(D)*X 2 - 18*SIN(X)*LOG(D)*X + 3*SIN(X)*X - 6*SIN(X) 6 3 5 2 4 3 - COS(X)*LOG(D) *X + 6*COS(X)*LOG(D) *X - 3*COS(X)*LOG(D) *X 4 3 2 - 18*COS(X)*LOG(D) *X + 12*COS(X)*LOG(D) *X 3 2 3 2 + 24*COS(X)*LOG(D) - 3*COS(X)*LOG(D) *X - 12*COS(X)*LOG(D) *X 2 3 + 6*COS(X)*LOG(D)*X - 24*COS(X)*LOG(D) - COS(X)*X 8 6 4 2 + 6*COS(X)*X))/(LOG(D) + 4*LOG(D) + 6*LOG(D) + 4*LOG(D) + 1 ) testint(x**3*d**x*cos x,x); X 6 3 5 2 4 3 (D *(SIN(X)*LOG(D) *X - 6*SIN(X)*LOG(D) *X + 3*SIN(X)*LOG(D) *X 4 3 2 + 18*SIN(X)*LOG(D) *X - 12*SIN(X)*LOG(D) *X 3 2 3 2 - 24*SIN(X)*LOG(D) + 3*SIN(X)*LOG(D) *X + 12*SIN(X)*LOG(D) *X 2 3 - 6*SIN(X)*LOG(D)*X + 24*SIN(X)*LOG(D) + SIN(X)*X 7 3 6 2 - 6*SIN(X)*X + COS(X)*LOG(D) *X - 3*COS(X)*LOG(D) *X 5 3 5 + 3*COS(X)*LOG(D) *X + 6*COS(X)*LOG(D) *X 4 2 4 3 3 - 3*COS(X)*LOG(D) *X - 6*COS(X)*LOG(D) + 3*COS(X)*LOG(D) *X 3 2 2 2 - 12*COS(X)*LOG(D) *X + 3*COS(X)*LOG(D) *X + 36*COS(X)*LOG(D) 3 2 + COS(X)*LOG(D)*X - 18*COS(X)*LOG(D)*X + 3*COS(X)*X 8 6 4 2 - 6*COS(X)))/(LOG(D) + 4*LOG(D) + 6*LOG(D) + 4*LOG(D) + 1) testint(sin x*sin(2*x)*sin(3*x),x); (6*SIN(3*X)*SIN(2*X)*SIN(X)*X + 3*SIN(3*X)*SIN(X)*COS(2*X) - 6*SIN(3*X)*COS(2*X)*COS(X)*X - 8*SIN(2*X)*SIN(X)*COS(3*X) + 6*SIN(2*X)*COS(3*X)*COS(X)*X + 6*SIN(X)*COS(3*X)*COS(2*X)*X - COS(3*X)*COS(2*X)*COS(X))/24 testint(cos x*cos(2*x)*cos(3*x),x); (SIN(3*X)*SIN(2*X)*SIN(X) + 6*SIN(3*X)*SIN(2*X)*COS(X)*X + 6*SIN(3*X)*SIN(X)*COS(2*X)*X + 8*SIN(3*X)*COS(2*X)*COS(X) - 6*SIN(2*X)*SIN(X)*COS(3*X)*X - 3*SIN(2*X)*COS(3*X)*COS(X) + 6*COS(3*X)*COS(2*X)*COS(X)*X)/24 testint(sin(x*kx)**3*x**2,x); 3 2 2 2 (6*SIN(X*KX) *X*KX - 9*SIN(X*KX) *COS(X*KX)*X *KX 2 2 2 + 2*SIN(X*KX) *COS(X*KX) + 36*SIN(X*KX)*X*KX - 18*COS(X*KX)*X *KX 3 + 40*COS(X*KX) + 16)/(27*KX ) testint(x*cos(xi/sin(x))*cos(x)/sin(x)**2,x); XI COS(--------)*COS(X)*X SIN(X) INT(------------------------,X) 2 SIN(X) % Mixed angles and half angles. int(cos(x)/(sin(x)*tan(x/2)),x); X - (TAN(---)*X + 1) 2 --------------------- X TAN(---) 2 % This integral produces a messy result because the code for % converting half angle tans to sin and cos is not effective enough. testint(sin(a*x)/(b+c*sin(a*x))**2,x); A*X TAN(-----)*B + C 2 2 2 2 ( - 2*SQRT(B - C )*SIN(A*X)*ATAN(------------------)*C 2 2 SQRT(B - C ) A*X TAN(-----)*B + C 2 2 2 3 - 2*SQRT(B - C )*ATAN(------------------)*B*C - COS(A*X)*B 2 2 SQRT(B - C ) 2 4 2 3 5 + COS(A*X)*B*C )/(A*(SIN(A*X)*B *C - 2*SIN(A*X)*B *C + SIN(A*X)*C 5 3 2 4 + B - 2*B *C + B*C )) % Examples involving logarithms and circular functions. testint(sin log x,x); X*(SIN(LOG(X)) - COS(LOG(X))) ------------------------------- 2 testint(cos log x,x); X*(SIN(LOG(X)) + COS(LOG(X))) ------------------------------- 2 % Examples involving exponentials. testint(e**x,x); X E % 2.01 #3; testint(a**x,x); X A -------- LOG(A) % 2.01 #4; testint(e**(a*x),x); A*X E ------ A testint(e**(a*x)/x,x); A*X E INT(------,X) X testint(1/(a+b*e**(m*x)),x); M*X - LOG(E *B + A) + M*X -------------------------- A*M testint(e**(2*x)/(1+e**x),x); X X E - LOG(E + 1) testint(e**(2*x)*e**(a*x),x); A*X + 2*X E ------------ A + 2 testint(1/(a*e**(m*x)+b*e**(-m*x)),x); M*X E *A SQRT(B)*SQRT(A)*ATAN(-----------------) SQRT(B)*SQRT(A) ----------------------------------------- A*B*M testint(x*e**(a*x),x); A*X E *(A*X - 1) ---------------- 2 A testint(x**20*e**x,x); X 20 19 18 17 16 15 E *(X - 20*X + 380*X - 6840*X + 116280*X - 1860480*X 14 13 12 + 27907200*X - 390700800*X + 5079110400*X 11 10 9 - 60949324800*X + 670442572800*X - 6704425728000*X 8 7 6 + 60339831552000*X - 482718652416000*X + 3379030566912000*X 5 4 - 20274183401472000*X + 101370917007360000*X 3 2 - 405483668029440000*X + 1216451004088320000*X - 2432902008176640000*X + 2432902008176640000) testint(a**x/b**x,x); X A ---------------------- X B *(LOG(A) - LOG(B)) testint(a**x*b**x,x); X X B *A ----------------- LOG(A) + LOG(B) testint(a**x/x**2,x); X A INT(----,X) 2 X testint(x*a**x/(1+b*x)**2,x); X X A (A - INT(-------------------,X)*LOG(A)*B*X 2 2 B *X + 2*B*X + 1 X A - INT(-------------------,X)*LOG(A) 2 2 B *X + 2*B*X + 1 X X A 2 A + INT(-------------------,X)*B *X + INT(-------------------,X)*B)/( 2 2 2 2 B *X + 2*B*X + 1 B *X + 2*B*X + 1 LOG(A)*B*(B*X + 1)) testint(x*e**(a*x)/(1+a*x)**2,x); A*X E -------------- 2 A *(A*X + 1) testint(x*k**(x**2),x); 2 X K ---------- 2*LOG(K) testint(e**(x**2),x); 2 X INT(E ,X) testint(x*e**(x**2),x); 2 X E ----- 2 testint((x+1)*e**(1/x)/x**4,x); 1/X 2 E *( - X + X - 1) ---------------------- 2 X testint((2*x**3+x)*(e**(x**2))**2*e**(1-x*e**(x**2))/(1-x*e**(x**2))**2, x); - E -------------------- 2 X 2 E *X X E *(E *X - 1) testint(e**(e**(e**(e**x))),x); X E E E INT(E ,X) % Examples involving exponentials and logarithms. testint(e**x*log x,x); X X E E *LOG(X) - INT(----,X) X testint(x*e**x*log x,x); X X X X E E *LOG(X)*X - E *LOG(X) - E + INT(----,X) X testint(e**(2*x)*log(e**x),x); 2*X E *(2*X - 1) ---------------- 4 % Examples involving square roots. testint(sqrt(2)*x**2 + 2*x,x); 2 X *(SQRT(2)*X + 3) -------------------- 3 testint(log x/sqrt(a*x+b),x); (2 SQRT(A*X + B) *(SQRT(A*X + B)*LOG(X) - 2*SQRT(A*X + B) - INT(---------------,X)*B) 2 A*X + B*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*SQRT(C + D*X)*SQRT(A + B*X)*A*D + 2*SQRT(C + D*X)*SQRT(A + B*X)*B*C + 4*SQRT(C + D*X)*SQRT(A + B*X)*B*D*X SQRT(C + D*X)*SQRT(A + B*X) 2 2 - INT(------------------------------,X)*A *D 2 A*C + A*D*X + B*C*X + B*D*X SQRT(C + D*X)*SQRT(A + B*X) + 2*INT(------------------------------,X)*A*B*C*D 2 A*C + A*D*X + B*C*X + B*D*X SQRT(C + D*X)*SQRT(A + B*X) 2 2 - INT(------------------------------,X)*B *C )/(8*B*D) 2 A*C + A*D*X + B*C*X + B*D*X 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); SQRT(A + B*X) 2*SQRT(A + B*X) + INT(---------------,X)*A 2 A*X + B*X testint(u/x**2,x); SQRT(A + B*X) - 2*SQRT(A + B*X) + INT(---------------,X)*B*X 2 A*X + B*X ------------------------------------------------- 2*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 + B*X) INT(---------------,X) 2 A*X + B*X testint(1/(x**2*u),x); SQRT(A + B*X) - 2*SQRT(A + B*X) - INT(---------------,X)*B*X 2 A*X + B*X ------------------------------------------------- 2*A*X testint(u**p,x); P/2 2*(A + B*X) *(A + B*X) -------------------------- B*(P + 2) testint(x*u**p,x); P/2 2 2 2 2 2 2*(A + B*X) *( - 2*A + A*B*P*X + B *P*X + 2*B *X ) -------------------------------------------------------- 2 2 B *(P + 6*P + 8) testint(atan((-sqrt(2)+2*x)/sqrt(2)),x); 2 ( - SQRT(2)*LOG( - SQRT(2)*X + X + 1) SQRT(2) - 2*X SQRT(2) - 2*X + 2*SQRT(2)*ATAN(---------------) - 4*ATAN(---------------)*X)/4 SQRT(2) SQRT(2) testint(1/sqrt(x**2-1),x); 2 SQRT(X - 1) INT(--------------,X) 2 X - 1 testint(sqrt(x+1)*sqrt x,x); (4*SQRT(X)*SQRT(X + 1)*X + 2*SQRT(X)*SQRT(X + 1) SQRT(X)*SQRT(X + 1) - INT(---------------------,X))/8 2 X + X % Examples from James Davenport's thesis: testint(1/sqrt(x**2-1)+10/sqrt(x**2-4),x); 2 2 2 2 SQRT(X - 4)*X SQRT(X - 1)*X 10*INT(-----------------,X) + INT(-----------------,X) 4 2 4 2 X - 5*X + 4 X - 5*X + 4 2 2 SQRT(X - 4) SQRT(X - 1) - 10*INT(---------------,X) - 4*INT(---------------,X) 4 2 4 2 X - 5*X + 4 X - 5*X + 4 % 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 SQRT(X + 1) INT(--------------,X) 2 X + 1 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(2*H*R - ALPHA ) INT(-----------------------,R) 2 2 2*H*R - ALPHA testint(1/(r*sqrt(2*h*r**2-alpha**2-epsilon**2)),r); 2 2 2 SQRT(2*H*R - ALPHA - EPSILON ) INT(----------------------------------,R) 3 2 2 2*H*R - R*ALPHA - R*EPSILON testint(1/(r*sqrt(2*h*r**2-alpha**2-2*k*r)),r); 2 2 SQRT(2*H*R - 2*K*R - ALPHA ) INT(-------------------------------,R) 3 2 2 2*H*R - 2*K*R - R*ALPHA testint(1/(r*sqrt(2*h*r**2-alpha**2-epsilon**2-2*k*r)),r); 2 2 2 SQRT(2*H*R - 2*K*R - ALPHA - EPSILON ) INT(------------------------------------------,R) 3 2 2 2 2*H*R - 2*K*R - R*ALPHA - R*EPSILON testint(r/sqrt(2*e*r**2-alpha**2),r); 2 2 SQRT(2*E*R - ALPHA ) ----------------------- 2*E testint(r/sqrt(2*e*r**2-alpha**2-epsilon**2),r); 2 2 2 SQRT(2*E*R - ALPHA - EPSILON ) ---------------------------------- 2*E testint(r/sqrt(2*e*r**2-alpha**2-2*k*r**4),r); 2 4 2 SQRT(2*E*R - 2*K*R - ALPHA )*R INT(----------------------------------,R) 2 4 2 2*E*R - 2*K*R - ALPHA testint(r/sqrt(2*e*r**2-alpha**2-2*k*r),r); 2 2 (SQRT(2*E*R - 2*K*R - ALPHA ) 2 2 SQRT(2*E*R - 2*K*R - ALPHA ) + INT(-------------------------------,R)*K)/(2*E) 2 2 2*E*R - 2*K*R - ALPHA testint(1/(r*sqrt(2*h*r**2-alpha**2-2*k*r**4)),r); 2 4 2 SQRT(2*H*R - 2*K*R - ALPHA ) INT(--------------------------------,R) 3 5 2 2*H*R - 2*K*R - R*ALPHA testint(1/(r*sqrt(2*h*r**2-alpha**2-epsilon**2-2*k*r**4)),r); 2 4 2 2 SQRT(2*H*R - 2*K*R - ALPHA - EPSILON ) INT(-------------------------------------------,R) 3 5 2 2 2*H*R - 2*K*R - R*ALPHA - R*EPSILON Comment many of these integrals used to require Steve Harrington's code to evaluate. They originated in Novosibirsk as examples of using Analytik. There are still a few examples that could be evaluated using better heuristics; testint(a*sin(3*x+5)**2*cos(3*x+5),x); 3 SIN(3*X + 5) *A ----------------- 9 testint(log(x**2)/x**3,x); 2 - (LOG(X ) + 1) ------------------ 2 2*X testint(x*sin(x+a),x); SIN(A + X) - COS(A + X)*X testint((log(x)*(1-x)-1)/(e**x*log(x)**2),x); X ----------- X E *LOG(X) testint(x**3*(a*x**2+b)**(-1),x); 2 2 - LOG(A*X + B)*B + A*X --------------------------- 2 2*A testint(x**(1/2)*(x+1)**(-7/2),x); 2*SQRT(X)*SQRT(X + 1)*X*(2*X + 5) ----------------------------------- 3 2 15*(X + 3*X + 3*X + 1) testint(x**(-1)*(x+1)**(-1),x); - LOG(X + 1) + LOG(X) testint(x**(-1/2)*(2*x-1)**(-1),x); SQRT(X) INT(----------,X) 2 2*X - X testint((x**2+1)*x**(1/2),x); 2 2*SQRT(X)*X*(3*X + 7) ------------------------ 21 testint(x**(-1)*(x-a)**(1/3),x); 1/3 1/3 ( - A + X) 3*( - A + X) + INT(---------------,X)*A 2 A*X - X testint(x*sinh(x),x); COSH(X)*X - SINH(X) testint(x*cosh(x),x); - COSH(X) + SINH(X)*X testint(sinh(2*x)/cosh(2*x),x); LOG(COSH(2*X)) ---------------- 2 testint((i*eps*sinh x-1)/(eps*i*cosh x+i*a-x),x); LOG(COSH(X)*I*EPS + A*I - X) testint(sin(2*x+3)*cos(x)**2,x); 2 ( - 4*SIN(2*X + 3)*SIN(X) *X + 2*SIN(2*X + 3)*X 2 + 2*SIN(X) *COS(2*X + 3) - 4*SIN(X)*COS(2*X + 3)*COS(X)*X - 3*COS(2*X + 3) + 3)/8 testint(x*atan(x),x); 2 ATAN(X)*X + ATAN(X) - X -------------------------- 2 testint(x*acot(x),x); 2 ACOT(X)*X + ACOT(X) + X -------------------------- 2 testint(x*log(x**2+a),x); 2 2 2 2 LOG(A + X )*A + LOG(A + X )*X - X ------------------------------------- 2 testint(sin(x+a)*cos(x),x); SIN(A + X)*COS(X)*X - SIN(X)*COS(A + X)*X - COS(A + X)*COS(X) --------------------------------------------------------------- 2 testint(cos(x+a)*sin(x),x); - SIN(A + X)*COS(X)*X + SIN(X)*COS(A + X)*X - COS(A + X)*COS(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); SQRT(X + 1) - 2*SQRT(X + 1) - INT(-------------,X) - LOG(X) 2 X + X testint(x/(x**4+36)**(1/2),x); 4 SQRT(X + 36)*X INT(-----------------,X) 4 X + 36 testint(1/(x**(1/3)+x**(1/2)),x); 1 INT(----------------,X) 1/3 X + SQRT(X) testint(log(2+3*x**2),x); 3*X 2 2*SQRT(6)*ATAN(---------) + 3*LOG(3*X + 2)*X - 6*X SQRT(6) ----------------------------------------------------- 3 testint(cot(x),x); X 2 X - LOG(TAN(---) + 1) + LOG(TAN(---)) 2 2 testint(cot x**4,x); 3 - COT(X) + 3*COT(X) + 3*X ----------------------------- 3 testint(tanh(x),x); 2*X LOG(E + 1) - X testint(coth(x),x); X X LOG(E - 1) + LOG(E + 1) - X testint(b**x,x); X B -------- LOG(B) testint((x**4+x**(-4)+2)**(1/2),x); 4 X - 3 -------- 3*X testint((2*x+1)/(3*x+2),x); - LOG(3*X + 2) + 6*X ----------------------- 9 testint(x*log(x+(x**2+1)**(1/2)),x); 2 INT(LOG(SQRT(X + 1) + X)*X,X) testint(x*(e**x*sin(x)+1)**2,x); 2*X 2 2*X 2*X (2*E *SIN(X) *X - 2*E *SIN(X)*COS(X)*X + E *SIN(X)*COS(X) 2*X 2*X X X X + E *X - E + 8*E *SIN(X)*X - 8*E *COS(X)*X + 8*E *COS(X) 2 + 4*X )/8 testint(x*e**x*cos(x),x); X E *(SIN(X)*X - SIN(X) + COS(X)*X) ----------------------------------- 2 Comment the following set came from Herbert Stoyan; testint(1/(x-3)**4,x); - 1 --------------------------- 3 2 3*(X - 9*X + 27*X - 27) testint(x/(x**3-1),x); 2*X + 1 2 2*SQRT(3)*ATAN(---------) - LOG(X + X + 1) + 2*LOG(X - 1) SQRT(3) ------------------------------------------------------------ 6 testint(x/(x**4-1),x); 2 - LOG(X + 1) + LOG(X - 1) + LOG(X + 1) ------------------------------------------ 4 testint(log(x)*(x**3+1)/(x**4+2),x); LOG(X) LOG(X) 2 - 4*INT(----------,X) + 2*INT(--------,X) + LOG(X) 5 4 X + 2*X X + 2 ------------------------------------------------------ 2 testint(log(x)+log(x+1)+log(x+2),x); LOG(X + 2)*X + 2*LOG(X + 2) + LOG(X + 1)*X + LOG(X + 1) + LOG(X)*X - 3*X testint(1/(x**3+5),x); 1/3 1/3 5 - 2*X 2/3 1/3 2 (5 *( - 2*SQRT(3)*ATAN(--------------) - LOG(5 - 5 *X + X ) 1/3 SQRT(3)*5 1/3 + 2*LOG(5 + X)))/30 testint(1/sqrt(1+x**2),x); 2 SQRT(X + 1) INT(--------------,X) 2 X + 1 testint(sqrt(x**2+3),x); 2 2 SQRT(X + 3) SQRT(X + 3)*X + 3*INT(--------------,X) 2 X + 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); INT(ASIN(X),X) testint(x**2*asin x,x); 2 INT(ASIN(X)*X ,X) testint(sec x**2/(1+sec x**2-3*tan x),x); X X LOG( - SQRT(5) + 2*TAN(---) + 1) - LOG( - SQRT(2) + TAN(---) + 1) 2 2 X X + LOG(SQRT(5) + 2*TAN(---) + 1) - LOG(SQRT(2) + TAN(---) + 1) 2 2 testint(1/sec x**2,x); SIN(X)*COS(X) + X ------------------- 2 testint((5*x**2-3*x-2)/(x**2*(x-2)),x); 3*LOG(X - 2)*X + 2*LOG(X)*X - 1 --------------------------------- X testint(1/(4*x**2+9)**(1/2),x); 2 SQRT(4*X + 9) INT(----------------,X) 2 4*X + 9 testint((x**2+4)**(-1/2),x); 2 SQRT(X + 4) INT(--------------,X) 2 X + 4 testint(1/(9*x**2-12*x+10),x); 3*X - 2 SQRT(6)*ATAN(---------) SQRT(6) ------------------------- 18 testint(1/(x**8-2*x**7+2*x**6-2*x**5+x**4),x); 2 4 2 3 4 (3*LOG(X + 1)*X - 3*LOG(X + 1)*X - 30*LOG(X - 1)*X 3 4 3 4 2 + 30*LOG(X - 1)*X + 24*LOG(X)*X - 24*LOG(X)*X - 30*X + 12*X 3 + 8*X + 4)/(12*X *(X - 1)) testint((a*x**3+b*x**2+c*x+d)/((x+1)*x*(x-3)),x); (27*LOG(X - 3)*A + 9*LOG(X - 3)*B + 3*LOG(X - 3)*C + LOG(X - 3)*D - 3*LOG(X + 1)*A + 3*LOG(X + 1)*B - 3*LOG(X + 1)*C + 3*LOG(X + 1)*D - 4*LOG(X)*D + 12*A*X)/12 testint(1/(2-log(x**2+1))**5,x); 2 5 2 4 2 3 - INT(1/(LOG(X + 1) - 10*LOG(X + 1) + 40*LOG(X + 1) 2 2 2 - 80*LOG(X + 1) + 80*LOG(X + 1) - 32),X) % The next integral appeared in Risch's 1968 paper. testint(2*x*e**(x**2)*log(x)+e**(x**2)/x+(log(x)-2)/(log(x)**2+x)**2+ ((2/x)*log(x)+(1/x)+1)/(log(x)**2+x),x); 2 2 X 3 X 2 2 (E *LOG(X) + E *LOG(X)*X + LOG(LOG(X) + X)*LOG(X) 2 2 + LOG(LOG(X) + X)*X - LOG(X))/(LOG(X) + X) % The following integral would not evaluate in REDUCE 3.3. testint(exp(x*ze+x/2)*sin(pi*ze)**4*x**4,ze); (2*X*ZE + X)/2 3 4 4 4 2 2 (E *X *(SIN(PI*ZE) *X + 4*SIN(PI*ZE) *X *PI 3 3 - 4*SIN(PI*ZE) *COS(PI*ZE)*X *PI 3 3 2 2 2 - 16*SIN(PI*ZE) *COS(PI*ZE)*X*PI + 12*SIN(PI*ZE) *X *PI 3 4 4 2 2 - 24*SIN(PI*ZE)*COS(PI*ZE)*X*PI + 24*PI ))/(X + 20*X *PI 4 + 64*PI ) % This one evaluates: testint(erf(x),x); 2 X E *ERF(X)*X*PI + SQRT(PI) ---------------------------- 2 X E *PI % So why not this one? testint(erf(x+a),x); INT(ERF(A + X),X) Comment here is an example of using the integrator with pattern matching; for all m,n let int(k1**m*log(k1)**n/(p**2-k1**2),k1)=foo(m,n), int(k1*log(k1)**n/(p**2-k1**2),k1)=foo(1,n), int(k1**m*log(k1)/(p**2-k1**2),k1)=foo(m,1), int(k1*log(k1)/(p**2-k1**2),k1)=foo(1,1), int(log(k1)**n/(k1*(p**2-k1**2)),k1)=foo(-1,n); int(k1**2*log(k1)/(p**2-k1**2),k1); *** FOO declared operator FOO(2,1) COMMENT It is interesting to see how much of this one can be done; let f1s= (12*log(s/mc**2)*s**2*pi**2*mc**3*(-8*s-12*mc**2+3*mc) + pi**2*(12*s**4*mc+3*s**4+176*s**3*mc**3-24*s**3*mc**2 -144*s**2*mc**5-48*s*mc**7+24*s*mc**6+4*mc**9-3*mc**8)) /(384*e**(s/y)*s**2); int(f1s,s); 2 S/Y 1 3 3 (PI *( - 36864*E *INT(--------------,S)*S*Y *MC S/Y 384*E *S*Y S/Y 1 2 5 - 55296*E *INT(--------------,S)*S*Y *MC S/Y 384*E *S*Y S/Y 1 2 4 + 13824*E *INT(--------------,S)*S*Y *MC S/Y 384*E *S*Y S/Y 1 7 - 18432*E *INT(--------------,S)*S*Y*MC S/Y 384*E *S*Y S/Y 1 6 + 9216*E *INT(--------------,S)*S*Y*MC S/Y 384*E *S*Y S/Y 1 9 - 1536*E *INT(--------------,S)*S*MC S/Y 384*E *S*Y S/Y 1 8 + 1152*E *INT(--------------,S)*S*MC S/Y 384*E *S*Y S 2 3 S 2 3 + 96*LOG(-----)*S *Y*MC + 96*LOG(-----)*S*Y *MC 2 2 MC MC S 5 S 4 3 + 144*LOG(-----)*S*Y*MC - 36*LOG(-----)*S*Y*MC - 12*S *Y*MC 2 2 MC MC 3 2 2 2 2 2 3 2 2 - 3*S *Y - 24*S *Y *MC - 6*S *Y - 176*S *Y*MC + 24*S *Y*MC 3 3 2 3 2 2 - 24*S*Y *MC - 6*S*Y - 80*S*Y *MC + 24*S*Y *MC 5 9 8 S/Y + 144*S*Y*MC - 4*MC + 3*MC ))/(384*E *S) factor int; ws; S/Y 1 2 3 3 2 2 (384*E *INT(--------------,S)*S*PI *MC *( - 96*Y - 144*Y *MC S/Y 384*E *S*Y 2 4 3 6 5 2 + 36*Y *MC - 48*Y*MC + 24*Y*MC - 4*MC + 3*MC ) + PI *( S 2 3 S 2 3 96*LOG(-----)*S *Y*MC + 96*LOG(-----)*S*Y *MC 2 2 MC MC S 5 S 4 3 + 144*LOG(-----)*S*Y*MC - 36*LOG(-----)*S*Y*MC - 12*S *Y*MC 2 2 MC MC 3 2 2 2 2 2 3 2 2 - 3*S *Y - 24*S *Y *MC - 6*S *Y - 176*S *Y*MC + 24*S *Y*MC 3 3 2 3 2 2 5 - 24*S*Y *MC - 6*S*Y - 80*S*Y *MC + 24*S*Y *MC + 144*S*Y*MC 9 8 S/Y - 4*MC + 3*MC ))/(384*E *S) Comment the following integrals reveal deficiencies in the current integrator; %high degree denominator; %testint(1/(2-log(x**2+1))**5,x); %this example should evaluate; testint(sin(2*x)/cos(x),x); SIN(2*X) INT(----------,X) COS(X) %this example, which appeared in Tobey's thesis, needs factorization %over algebraic fields. It currently gives an ugly answer and so has %been suppressed; % testint((7*x**13+10*x**8+4*x**7-7*x**6-4*x**3-4*x**2+3*x+3)/ % (x**14-2*x**8-2*x**7-2*x**4-4*x**3-x**2+2*x+1),x); symbolic summarize!-integral!-test(); ***** SUMMARY OF INTEGRAL TESTS ***** Number of integrals tested: 272 Total time taken: 175865 ms Number of garbage collections: 168 Number of incorrect integrals: 0 Number of unevaluated integrals: 59 Integrands of unevaluated integrals are: 1 -------- LOG(X) 1 ------------ LOG(X + 1) 1 ------------ 2 2 LOG(X) *X LOG(LOG(LOG(LOG(X)))) P SIN(X) SIN(X) -------- X COS(X) -------- X SIN(X) -------- 2 X 2 SIN(X) --------- X 4 3 TAN(X) *X 6 3 TAN(X) *X XI COS(--------)*COS(X)*X SIN(X) ------------------------ 2 SIN(X) A*X E ------ X X A ---- 2 X X A *X ------------------- 2 2 B *X + 2*B*X + 1 2 X E X E E E E X E *LOG(X) X E *LOG(X)*X LOG(X) --------------- SQRT(A*X + B) SQRT(C + D*X)*SQRT(A + B*X) SQRT(A + B*X) --------------- X SQRT(A + B*X) --------------- 2 X 1 ----------------- SQRT(A + B*X)*X 1 ------------------ 2 SQRT(A + B*X)*X 1 -------------- 2 SQRT(X - 1) SQRT(X)*SQRT(X + 1) 2 2 SQRT(X - 4) + 10*SQRT(X - 1) -------------------------------- 2 2 SQRT(X - 1)*SQRT(X - 4) 2 2 SQRT(SQRT(A + X ) + X) ------------------------- X 2 SQRT(X + 1) -------------- 2 X + 1 3 3 - 3*SQRT(X + 1) + 3*X + 3 ------------------------------ 4 2*X + 2*X 1 ----------------------- 2 2 SQRT(2*H*R - ALPHA ) 1 ------------------------------------ 2 2 2 SQRT(2*H*R - ALPHA - EPSILON )*R 1 --------------------------------- 2 2 SQRT(2*H*R - 2*K*R - ALPHA )*R 1 -------------------------------------------- 2 2 2 SQRT(2*H*R - 2*K*R - ALPHA - EPSILON )*R R -------------------------------- 2 4 2 SQRT(2*E*R - 2*K*R - ALPHA ) R ------------------------------- 2 2 SQRT(2*E*R - 2*K*R - ALPHA ) 1 ---------------------------------- 2 4 2 SQRT(2*H*R - 2*K*R - ALPHA )*R 1 --------------------------------------------- 2 4 2 2 SQRT(2*H*R - 2*K*R - ALPHA - EPSILON )*R 1 ----------------------- 2*SQRT(X)*X - SQRT(X) 1/3 ( - A + X) --------------- X SQRT(SIN(X) + 1) SQRT( - SIN(X) + 1) SQRT(COS(X) + 1) SQRT( - COS(X) + 1) - 1 ----------------- SQRT(X + 1) - 1 X --------------- 4 SQRT(X + 36) 1 ---------------- 1/3 X + SQRT(X) 2 LOG(SQRT(X + 1) + X)*X 3 LOG(X)*X + LOG(X) -------------------- 4 X + 2 1 -------------- 2 SQRT(X + 1) 2 SQRT(X + 3) ASIN(X) 2 ASIN(X)*X 1 ---------------- 2 SQRT(4*X + 9) 1 -------------- 2 SQRT(X + 4) 2 5 2 4 2 3 ( - 1)/(LOG(X + 1) - 10*LOG(X + 1) + 40*LOG(X + 1) 2 2 2 - 80*LOG(X + 1) + 80*LOG(X + 1) - 32) ERF(A + X) SIN(2*X) ---------- COS(X) end; 4: 4: Quitting Sat Jun 29 13:44:51 PDT 1991