On Tuesday, August 6, 2013 1:51:41 PM UTC-10, clicl...@freenet.de wrote:
> I thought it would be instructive to list those integrals Rubi 4.1 fails on: > > Timofeev's example 29 from Chapter 1: > INT(TAN(x)*TAN(x - a), x) > = 1/TAN(a)*LN(1 + TAN(a)*TAN(x)) - x > > Timofeev's example 64 from Chapter 1: > INT(ASIN(x/a)^(3/2)/SQRT(a^2 - x^2), x) > = 2/5*ASIN(x/a)^(3/2)*ATAN(x/SQRT(a^2 - x^2)) > = 2/5*(a/SQRT(a^2 - x^2))*SQRT(1 - (x/a)^2)*ASIN(x/a)^(5/2) > > Timofeev's example 6 from Chapter 7: > INT(x*SIN(x)^3/COS(x)^4, x) > = x*(1/(3*COS(x)^3) - 1/COS(x)) - SIN(x)/(6*COS(x)^2) + 5/6*ATANH(SIN(x)) > > Timofeev's example 10 from Chapter 7: > INT((2*x + SIN(2*x))/(x*SIN(x) + COS(x))^2, x) > = - 2*COS(x)/(x*SIN(x) + COS(x)) > > Timofeev's example 11 from Chapter 7: > INT((x/(x*COS(x) - SIN(x)))^2, x) > = (x*SIN(x) + COS(x))/(x*COS(x) - SIN(x)) > > In my view, the first two failures are deplorable but understandable, > while the third one must be counted as a bug. The last two are simply to > be expected. > > Timofeev's book can be found at: > > <http://www.math-life.com/jdownloads/view.download/27/279.html>
You probably thought I forgot or ignored your above post. Actually, the first three examples inspired me to make significant enhancements to the just released version 4.2 of Rubi. In particular, Timofeev's example 64 from Chapter 1 resulted in the replacement of an ad hoc collection of inverse trig integration rules with a coherent set capable of integrating ANY integrable expression of the form
x^m (d+e x^2)^p (a+b arcsin(c x))^n
for any integer (including 0), fraction or symbolic m, n and p; and when c^2 d+e=0 (as is the case in example 64) and when not. Rules were also defined for analogous expressions involving arccos, arctan and arccot; as well as their inverse hyperbolic counterparts. The major computer algebra systems are woefully inadequate at integrating this difficult class of problems.
Rubi 4.2 gets all the Timofeev examples from Chapters 1, 3 and 7 with the exception of #10 and #11 from Chapter 7. So your table of test results can be revised to read as follows:
I would also like to add the Timofeev examples from the other chapters to the Rubi test-suite (currently 44,656 problems). Unfortunately when I attempt to follow the above link, I am confronted with the unfriendly message ?Forbidden: You don?t have permission to access? the html file. Any suggestions?