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 Coth

 http://functions.wolfram.com/01.22.21.0207.01

 Input Form

 Integrate[E^(p z) Sin[a z]^m Cosh[b z]^u Coth[c z], z] == -((1/(p (2 c + p))) (2^(-m - u) E^(p z) Binomial[m, m/2] Binomial[u, u/2] ((2 c + p) Hypergeometric2F1[p/(2 c), 1, 1 + p/(2 c), E^(2 c z)] + E^(2 c z) p Hypergeometric2F1[1 + p/(2 c), 1, 2 + p/(2 c), E^(2 c z)]) (1 - Mod[m, 2]) (1 - Mod[u, 2]))) - 2^(-m - u) Binomial[u, u/2] (1 - Mod[u, 2]) Sum[(-1)^k Binomial[m, k] ((E^((I m Pi)/2 + ((-I) a (-2 k + m) + p) z) ((2 c - I a (-2 k + m) + p) Hypergeometric2F1[ ((-I) a (-2 k + m) + p)/(2 c), 1, 1 + ((-I) a (-2 k + m) + p)/ (2 c), E^(2 c z)] + E^(2 c z) ((-I) a (-2 k + m) + p) Hypergeometric2F1[1 + ((-I) a (-2 k + m) + p)/(2 c), 1, 2 + ((-I) a (-2 k + m) + p)/(2 c), E^(2 c z)]))/ (((-I) a (-2 k + m) + p) (2 c - I a (-2 k + m) + p)) + (E^((-(1/2)) I m Pi + (I a (-2 k + m) + p) z) ((2 c + I a (-2 k + m) + p) Hypergeometric2F1[(I a (-2 k + m) + p)/ (2 c), 1, 1 + (I a (-2 k + m) + p)/(2 c), E^(2 c z)] + E^(2 c z) (I a (-2 k + m) + p) Hypergeometric2F1[ 1 + (I a (-2 k + m) + p)/(2 c), 1, 2 + (I a (-2 k + m) + p)/ (2 c), E^(2 c z)]))/((I a (-2 k + m) + p) (2 c + I a (-2 k + m) + p))), {k, 0, Floor[(1/2) (-1 + m)]}] - 2^(-m - u) Binomial[m, m/2] (1 - Mod[m, 2]) Sum[Binomial[u, k] ((E^((p - b (-2 k + u)) z) ((2 c + p - b (-2 k + u)) Hypergeometric2F1[(p - b (-2 k + u))/ (2 c), 1, 1 + (p - b (-2 k + u))/(2 c), E^(2 c z)] + E^(2 c z) (p - b (-2 k + u)) Hypergeometric2F1[ 1 + (p - b (-2 k + u))/(2 c), 1, 2 + (p - b (-2 k + u))/(2 c), E^(2 c z)]))/((p - b (-2 k + u)) (2 c + p - b (-2 k + u))) + (E^((p + b (-2 k + u)) z) ((2 c + p + b (-2 k + u)) Hypergeometric2F1[(p + b (-2 k + u))/(2 c), 1, 1 + (p + b (-2 k + u))/(2 c), E^(2 c z)] + E^(2 c z) (p + b (-2 k + u)) Hypergeometric2F1[ 1 + (p + b (-2 k + u))/(2 c), 1, 2 + (p + b (-2 k + u))/(2 c), E^(2 c z)]))/((p + b (-2 k + u)) (2 c + p + b (-2 k + u)))), {k, 0, Floor[(1/2) (-1 + u)]}] - 2^(-m - u) Sum[(-1)^k Binomial[m, k] Binomial[u, s] ((E^((I m Pi)/2 + ((-I) a (-2 k + m) + p - b (-2 s + u)) z) ((2 c - I a (-2 k + m) + p - b (-2 s + u)) Hypergeometric2F1[ ((-I) a (-2 k + m) + p - b (-2 s + u))/(2 c), 1, 1 + ((-I) a (-2 k + m) + p - b (-2 s + u))/(2 c), E^(2 c z)] + E^(2 c z) ((-I) a (-2 k + m) + p - b (-2 s + u)) Hypergeometric2F1[1 + ((-I) a (-2 k + m) + p - b (-2 s + u))/ (2 c), 1, 2 + ((-I) a (-2 k + m) + p - b (-2 s + u))/(2 c), E^(2 c z)]))/(((-I) a (-2 k + m) + p - b (-2 s + u)) (2 c - I a (-2 k + m) + p - b (-2 s + u))) + (E^((-(1/2)) I m Pi + (I a (-2 k + m) + p - b (-2 s + u)) z) ((2 c + I a (-2 k + m) + p - b (-2 s + u)) Hypergeometric2F1[ (I a (-2 k + m) + p - b (-2 s + u))/(2 c), 1, 1 + (I a (-2 k + m) + p - b (-2 s + u))/(2 c), E^(2 c z)] + E^(2 c z) (I a (-2 k + m) + p - b (-2 s + u)) Hypergeometric2F1[ 1 + (I a (-2 k + m) + p - b (-2 s + u))/(2 c), 1, 2 + (I a (-2 k + m) + p - b (-2 s + u))/(2 c), E^(2 c z)]))/ ((I a (-2 k + m) + p - b (-2 s + u)) (2 c + I a (-2 k + m) + p - b (-2 s + u))) + (E^((I m Pi)/2 + ((-I) a (-2 k + m) + p + b (-2 s + u)) z) ((2 c - I a (-2 k + m) + p + b (-2 s + u)) Hypergeometric2F1[ ((-I) a (-2 k + m) + p + b (-2 s + u))/(2 c), 1, 1 + ((-I) a (-2 k + m) + p + b (-2 s + u))/(2 c), E^(2 c z)] + E^(2 c z) ((-I) a (-2 k + m) + p + b (-2 s + u)) Hypergeometric2F1[1 + ((-I) a (-2 k + m) + p + b (-2 s + u))/ (2 c), 1, 2 + ((-I) a (-2 k + m) + p + b (-2 s + u))/(2 c), E^(2 c z)]))/(((-I) a (-2 k + m) + p + b (-2 s + u)) (2 c - I a (-2 k + m) + p + b (-2 s + u))) + (E^((-(1/2)) I m Pi + (I a (-2 k + m) + p + b (-2 s + u)) z) ((2 c + I a (-2 k + m) + p + b (-2 s + u)) Hypergeometric2F1[ (I a (-2 k + m) + p + b (-2 s + u))/(2 c), 1, 1 + (I a (-2 k + m) + p + b (-2 s + u))/(2 c), E^(2 c z)] + E^(2 c z) (I a (-2 k + m) + p + b (-2 s + u)) Hypergeometric2F1[ 1 + (I a (-2 k + m) + p + b (-2 s + u))/(2 c), 1, 2 + (I a (-2 k + m) + p + b (-2 s + u))/(2 c), E^(2 c z)]))/ ((I a (-2 k + m) + p + b (-2 s + u)) (2 c + I a (-2 k + m) + p + b (-2 s + u)))), {s, 0, Floor[(1/2) (-1 + u)]}, {k, 0, Floor[(1/2) (-1 + m)]}] /; Element[m, Integers] && m > 0 && Element[u, Integers] && u > 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