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http://functions.wolfram.com/01.23.21.0203.01
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Integrate[Sin[a z]^m Sinh[c z]^\[Mu] Csch[c z], z] ==
((-(1/(c (1 - \[Mu])))) 2^(1 - m) E^(c z) Sinh[c z]^\[Mu] Binomial[m, m/2]
Hypergeometric2F1[(c - c \[Mu])/(2 c), 1 - \[Mu], (3 - \[Mu])/2,
E^(2 c z)] (1 - Mod[m, 2]))/(1 - E^(2 c z))^\[Mu] -
(2^(1 - m) E^(c z) Sinh[c z]^\[Mu] Sum[(-1)^k Binomial[m, k]
((E^((I m Pi)/2 + I a (2 k - m) z) Hypergeometric2F1[
(c + I a (2 k - m) - c \[Mu])/(2 c), 1 - \[Mu],
(I a (2 k - m) + c (3 - \[Mu]))/(2 c), E^(2 c z)])/
(I a (2 k - m) + c (1 - \[Mu])) +
(E^((-(1/2)) I m Pi + I a (-2 k + m) z) Hypergeometric2F1[
(c + I a (-2 k + m) - c \[Mu])/(2 c), 1 - \[Mu],
(I a (-2 k + m) + c (3 - \[Mu]))/(2 c), E^(2 c z)])/
(I a (-2 k + m) + c (1 - \[Mu]))), {k, 0, Floor[(1/2) (-1 + m)]}])/
(1 - E^(2 c z))^\[Mu] /; Element[m, Integers] && m > 0
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