# Fourier series expansion of Tanh(cos(x))

I need the coefficients of the Fourier series expansion of $\tanh(\alpha cos(x))$, for every real positive number $\alpha >0$ namely, since it is an even function, I need the computation of the following integral:

$\int_{-\pi}^{\pi} dx \cos(nx) \tanh(\alpha cos(x))$

Can anyone help me?

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What techniques have you tried? Are you able to do the integral in any special cases? Have you looked at similar but simpler forms for insight? E.g., $\int \exp(\exp(x)) dx$ can be evaluated as an exponential integral. –  Ben Crowell Feb 19 at 20:18