integrate(x/(exp(2*pi*x)-1), (x, 0, oo)) can not be calculated

[ghost]

The result is 1/24

>>> integrate(x/(exp(2*pi*x)-1), (x, 0, oo))
Integral(x/((exp(pi*x) - 1)*(exp(pi*x) + 1)), (x, 0, oo))

Wolfram alpha can't do it either

[ghost]

Closely related to #8169

[tverticalis]

My Wolfram Mathematica 11.2 calculated it in 0.1 sec. If "-1" is replaced with "+1", Mathematica gives 1/48. I posted a similar issue a few days ago (#14035). My feeling is SymPy's integration is not yet ready for prime time.

[ghost]

@tverticalis, if SymPy was successful with integrate(x/(exp(2*pi*x)-1), (x, 0, oo)) then I was going to test it with integrate(sin(a*x)/(exp(2*pi*x)-1), (x, 0, oo))
And if SymPy was successful with integrate(x**2/cosh(x), (x, 0, oo)), then I was going to test it with integrate(x**(2*n)/cosh(x), (x, 0, oo))

[tverticalis]

The first integral yields ConditionalExpression[(-2 + a Coth[a/2])/(4 a), Abs[Im[a]] < 2 Pi].
The second one ConditionalExpression[
2^(-4 n) n Gamma[2 n] (Zeta[1 + 2 n, 1/4] - Zeta[1 + 2 n, 3/4]),
Re[n] > -(1/2)].

[ghost]

The 2nd integral can be expressed more cleanly, it is a rational multiple of pi**(2n+1), and the rational number can be written explicitly in terms of Euler's numbers (I was implicitly assuming that n is a nonnegative integer)
A similar, but perhaps more attractive integral is integrate(cosh(a*x)/cosh(x), (x, 0, oo)), which evaluates to pi/(2*cos((pi*a)/2))

[tverticalis]

This level of analytical depth/elegance may be too much for automated systems.
Assuming[n > 0 && n [Element] Integers, Integrate[x^(2*n)/Cosh[x], {x, 0, Infinity}]] // FullSimplify
2^(-1 - 4 n) (-PolyGamma[2 n, 1/4] + PolyGamma[2 n, 3/4])

[ghost]

I'm amazed that symbolic algebra systems can compute so many sums and integrals in terms of polylogs and polygammas, completely different to the way I do it