How to integrate x^5/((e^x-1)(1-e^(-x))) and have a simplified result within the limit from 0 to (a/b)?

Question

I have tried it with mathematica but the result is too complicated to be simplified. I need a simplified expression. By the way a/b is a finite quantity.

I have tried it with mathematica but the result is too complicated to be simplified. I need a simplified expression. By the way a/b is a finite quantity.
I need a solution of the integral in terms of (a/b).

Answer 1

Well, the denominator (e^x-1)(1-e^(-x))=2(cosh(x)-1). Try a trig substitution cosh(x)=>y. Not really sure what sort of simplification you're looking for, though.

Edit: Actually, things get simpler. 2(cosh-1)=4sinh^2, and 1/(4sinh^2)=(coth^2-1)/4

so now you have int{x^5coth^2(x)}/4-int{x^5}/4

which can be done by parts a few times.

Answer 2

My friend suggested partial fraction decomposition, so that you get two integrals. Maybe this would help simplify a bit.