Axel Vogt schrieb: > > Ok, for this 86 given integrals (translated to Maple syntax) Maple 17 solves all, > BUT ONE, Int(exp(1/2*x)/(exp(x)-1)^(1/2),x) = 2*ln((exp(x)-1)^(1/2)+exp(1/2*x)). > > That can be done manually: > > Int(exp(1/2*x)/(exp(x)-1)^(1/2),x); > Change(%, exp(1/2*x) = y, y); > value(%); > subs(y = exp(1/2*x), %); # re-substitute > combine(%); # re-write, giving the asserted > > For the remaining 85 solutions it can automatically prove that lhs = rhs after > differentiation w.r.t. x and it fails for no. 50, 52, 53: > > #Int(x/(a^4-x^4)^(1/2),x) > 1/2*arctan(x^2/(a^4-x^4)^(1/2)) = > 1/2*arcsin(x^2/a^2), > > #Int(1/(x*(a^2-x^2)^(1/2)),x) > -1/(a^2)^(1/2)*ln((2*a^2+2*(a^2)^(1/2)*(a^2-x^2)^(1/2))/x) = > -1/a*arctanh((a^2-x^2)^(1/2)/a), > > #Int(1/(x*(a^2+x^2)^(1/2)),x) > -1/(a^2)^(1/2)*ln((2*a^2+2*(a^2)^(1/2)*(a^2+x^2)^(1/2))/x) = > -1/a*arctanh(a/(a^2+x^2)^(1/2)) > > Assuming a<>0 resolves the last two ones. > > For the first one I think that is false for a= exp(I*Pi/4), > as already said in a previous post (20.04.2013)
I have repaired the evaluation of INT(x/SQRT(a^4-x^4)), x), it now applies to complex 'a' as well. I also saw that some other evaluations (i.e. 3, 4, 5, 55, 57, 71) could be shortened by converting from LN to ATANH, and have made the substitutions. The full set of corrected integrals from chapter 1 of Timofeev's book is appended below.
I am still counting integrals involving subcases as a single integral, because splitting them into separate integrals would destroy the correspondence with Timofeev's numbering. In order to allow easy reference, the number would then have to be included explicitely, e.g. as
and vectors would still have to be parsed. I am therefore undecided on this issue.
The following table summarizes the performance data made available so far. I am no longer counting integral 29 (requiring a non-default setting of a simplification flag) among the successes of Derive. I am assuming that results produced by FriCAS never consist of error messages or involve unresolved integrals or non-elementary functions. And I am counting INT(EXP(x/2)/SQRT(EXP(x)-1), x) as a failure for Maple. Yes.