This link has discussion on finding partial fraction decomposition of $1/(e^x-1)$, so I experimented with Mathematica to see if M can do it, but looks like not. Similar is the case with one more CAS I tried.
I can clearly see that its not working with M but question is why ?
What I think is that probably CAS gives algebraically closed solutions without extending domain hence it returns
Apart[1/(x^2+1)] without success though it has roots in complex numbers(but its just a thought).
Apart[1/(a*x^2-1)] is not evaluated. Looks like symbolic computation on Apart is not supported or may be no such algorithm exists. Don't know !!
Can someone suggest why complex roots are not considered part of decomposition on CASes ?
Quite frankly, as a user of Mathematica, I would expect the CAS to return results based on choice whether its algebraically closed or complex roots.