>>13719754I don't think there's a nice way to compute this type of sum except knowing some Complex Analysis and having a good eye. For example, I don't believe there are any known closed forms if you replace the exponent in the numerator by a positive integer not of the form , or if you extend the sum to all integers instead of just odd ones, or if you replace the by pretty much anything else.
In this specific case one can recognize that the sum is related to the Dirichlet Eta function (
https://en.wikipedia.org/wiki/Dirichlet_eta_function#Integral_representations), so that you can write the summand as a Mellin inversion integral (
https://en.wikipedia.org/wiki/Mellin_inversion_theorem). Then you exchange sum and integral, do the summation, and get an ugly expression with Zeta functions. The remaining integral can then be calculated because the integrand has a pretty specific symmetry (which is where all the conditions I mentioned come in).
