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I have this function ## v(x) = -\sum_{i=1} x^i \sqrt{2}^{i-2} \int_{-\infty}^{\infty} m^{i-1} \cosh(m)^{-4} dm## which I can not seem to figure out how to simplify.I tried looking at some partial integration but repeated integration of ## \cosh ## gives polylogarithms which seemed to only complicate matters. Does anyone have some good tips?

Some observations:

Only uneven ## i ## will contribute.

For uneven ##i \geq 3## the integral (## I_i = \int_{-\infty}^{\infty} m^{i-1} \cosh(m)^{-4} dm ##) seems to follow the pattern ## I_i = a_i\pi^{(i-1)} + b_i\pi^{(i-3)} ##, where ## a_i ## is going towards ## \infty ## with ## i ## and the converse is true for ## b_i ##

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# I Help with simplifying series of hyperbolic integrals

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