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 Coth

 http://functions.wolfram.com/01.22.21.0202.01

 Input Form

 Integrate[E^(p z) Sin[a z]^m Sinh[c z]^\[Mu] Coth[c z], z] == ((-2^(-\[Mu] - m)) Binomial[m, m/2] (-E^((-c) z) + E^(c z))^\[Mu] ((1/(p - c \[Mu])) E^(p z) Hypergeometric2F1[(p - c \[Mu])/(2 c), 1 - \[Mu], (2 c + p - c \[Mu])/(2 c), E^(2 c z)] + (1/(2 c + p - c \[Mu])) E^((2 c + p) z) Hypergeometric2F1[ (2 c + p - c \[Mu])/(2 c), 1 - \[Mu], (4 c + p - c \[Mu])/(2 c), E^(2 c z)]) (1 - Mod[m, 2]))/(1 - E^(2 c z))^\[Mu] - (2^(-\[Mu] - m) (-E^((-c) z) + E^(c z))^\[Mu] Sum[(-1)^k Binomial[m, k] ((E^((I m Pi)/2 + ((-I) a (-2 k + m) + p) z) Hypergeometric2F1[((-I) a (-2 k + m) + p - c \[Mu])/(2 c), 1 - \[Mu], (2 c - I a (-2 k + m) + p - c \[Mu])/(2 c), E^(2 c z)])/ ((-I) a (-2 k + m) + p - c \[Mu]) + (E^((I m Pi)/2 + (2 c - I a (-2 k + m) + p) z) Hypergeometric2F1[ (2 c - I a (-2 k + m) + p - c \[Mu])/(2 c), 1 - \[Mu], (4 c - I a (-2 k + m) + p - c \[Mu])/(2 c), E^(2 c z)])/ (2 c - I a (-2 k + m) + p - c \[Mu]) + (E^((-(1/2)) I m Pi + (I a (-2 k + m) + p) z) Hypergeometric2F1[ (I a (-2 k + m) + p - c \[Mu])/(2 c), 1 - \[Mu], (2 c + I a (-2 k + m) + p - c \[Mu])/(2 c), E^(2 c z)])/ (I a (-2 k + m) + p - c \[Mu]) + (E^((-(1/2)) I m Pi + (2 c + I a (-2 k + m) + p) z) Hypergeometric2F1[(2 c + I a (-2 k + m) + p - c \[Mu])/(2 c), 1 - \[Mu], (4 c + I a (-2 k + m) + p - c \[Mu])/(2 c), E^(2 c z)])/ (2 c + I a (-2 k + m) + p - c \[Mu])), {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