Some integrals with problematic evaluation

[parisse]

Your explanations seems to be essentially what I call table lookup, very different from what I implemented inside giac : because table lookup is not general enough, it's boring and it does not give much pedagogical added value.
For example for all rational fractions, I do Hermite reduction to have a square-free denominator, then if the denominator can be factored on the field of it's coefficients, partial fraction decomposition and integration. This explains why 1/(x^5-a^5) is currently not fully integrated, since the factorization of x^5-a^5 over Q[a,x] is (x-a)*(x^4+a*x^3+a^2*x^2+a^3*x+a^4) where the second factor is irreducible: for a=1 there is code to recognize a cyclotomic polynomial and factor it, I can extend it to a symbolic a with a little code change:
http://www-fourier.ujf-grenoble.fr/%7epa...x^5-a^5))&
For assume(x>0); int(1/(x*(1-ln(x))),x), Xcas displays in step by step mode

Code:


===== Step/step primitive of 1/(x*(1-ln(x))) with respect to x =====
Integration of 1/((1-ln(x))*x): f(u)*u' where f=x->-1/(x-1) and u=ln(x)
Integrate rational fraction 1/(t_nostep-1)
Partial fraction integration of 1/(t_nostep-1)
Hence primitive of 1/(x*(1-ln(x))) with respect to x is -ln(abs(ln(x)-1))