Simplify result of this definite integral
It is well known that for n∈N and n>0 (an maybe even for more than these restrictions): In=∫∞0xnex−1dx=ζ(n+1)n! which, analytically can be shown easily by expanding 1/(1−e−x) into a geometric series, which leads to trivial integrals, and by using ζ(n+1)=∑∞l=1l−(n+1). So, eg.: I1=π2/6 I2=2ζ(3) a.s.o...
Now, if I try even the simplest case with sage, I get this 'nifty' little results
sage: integrate(x/(exp(x)-1),x,0,oo)
-1/6*pi^2 + limit(-1/2*x^2 + x*log(-e^x + 1) + polylog(2, e^x), x,
+Infinity, minus)
Is there any trick to simplify this down to the final result, or is this about as far as I can get with sage alone?
PS.: it is probably needless to say that (once again ... :( ...)
In[1]:= Integrate[x/(Exp[x] - 1), {x, 0, Infinity}]
Out[1]:= Pi^2/6