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There are very similar integrals (coming from the same source, shockingly) in arXiv:1002.1905 (Bridgeman/Kahn), where they seem to be evaluate evaluated in closed form.

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There are very similar integrals (coming from the same source, shockingly) in arXiv:1002.1905 (Bridgeman/Kahn), where they seem to be evaluate in closed form.