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Putting aside a weird technical issue with Maxima engine that you observed, it is generally not a good idea to use Cauchy's formula for computing coefficients of the series expansion as the resulting integrals are tough for evaluation. In particular, evaluating the one in question is well failed by Maple as well. Mathematica does evaluate it, but I'm not sure if you'd be satisfied with its result.

A more feasible approach is to compute a few coefficients numerically and then try to guess their formula - e.g. see https://ask.sagemath.org/question/68486/