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 Coth

 http://functions.wolfram.com/01.22.21.0179.01

 Input Form

 Integrate[Cos[a z]^m Sinh[c z]^\[Mu] Coth[c z], z] == ((-(1/(c (-2 + \[Mu]) \[Mu]))) 2^(-m - \[Mu]) (-E^((-c) z) + E^(c z))^\[Mu] Binomial[m, m/2] (E^(2 c z) \[Mu] Hypergeometric2F1[1 - \[Mu]/2, 1 - \[Mu], 2 - \[Mu]/2, E^(2 c z)] + (-2 + \[Mu]) Hypergeometric2F1[-(\[Mu]/2), 1 - \[Mu], 1 - \[Mu]/2, E^(2 c z)]) (-1 + Mod[m, 2]))/(1 - E^(2 c z))^\[Mu] - (2^(-m - \[Mu]) (-E^((-c) z) + E^(c z))^\[Mu] Sum[Binomial[m, k] (Hypergeometric2F1[((-I) a (-2 k + m) - c \[Mu])/ (2 c), 1 - \[Mu], ((-I) a (-2 k + m) - c (\[Mu] - 2))/(2 c), E^(2 c z)]/E^(I a (-2 k + m) z)/((-I) a (-2 k + m) - c \[Mu]) + (E^((2 c - I a (-2 k + m)) z) Hypergeometric2F1[ ((-I) a (-2 k + m) - c (\[Mu] - 2))/(2 c), 1 - \[Mu], ((-I) a (-2 k + m) - c (\[Mu] - 4))/(2 c), E^(2 c z)])/ (I a (2 k - m) - c (\[Mu] - 2)) + (E^(I a (-2 k + m) z) Hypergeometric2F1[(I a (-2 k + m) - c \[Mu])/(2 c), 1 - \[Mu], (I a (-2 k + m) - c (\[Mu] - 2))/(2 c), E^(2 c z)])/ (I a (-2 k + m) - c \[Mu]) + (E^((2 c + I a (-2 k + m)) z) Hypergeometric2F1[(I a (-2 k + m) - c (\[Mu] - 2))/(2 c), 1 - \[Mu], (I a (-2 k + m) - c (\[Mu] - 4))/(2 c), E^(2 c z)])/ (I a (-2 k + m) - c (\[Mu] - 2))), {k, 0, Floor[(1/2) (-1 + m)]}])/ (1 - E^(2 c z))^\[Mu] /; Element[m, Integers] && m > 0

 Standard Form

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 MathML Form

 Rule Form

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 Date Added to functions.wolfram.com (modification date)

 2002-12-18