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 Tanh

 http://functions.wolfram.com/01.21.21.0187.01

 Input Form

 Integrate[E^(p z) Sin[a z]^m Cosh[b z]^u Tanh[c z], z] == 2^(-m - u) Binomial[m, m/2] Binomial[u, u/2] (-((E^(p z) Hypergeometric2F1[p/(2 c), 1, 1 + p/(2 c), -E^(2 c z)])/p) + (E^((2 c + p) z) Hypergeometric2F1[1 + p/(2 c), 1, 2 + p/(2 c), -E^(2 c z)])/(2 c + p)) (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) (-((E^(((-I) a (-2 k + m) + p) z) Hypergeometric2F1[ ((-I) a (-2 k + m) + p)/(2 c), 1, 1 + ((-I) a (-2 k + m) + p)/(2 c), -E^(2 c z)])/((-I) a (-2 k + m) + p)) + (E^((2 c - I a (-2 k + m) + p) z) Hypergeometric2F1[ 1 + ((-I) a (-2 k + m) + p)/(2 c), 1, 2 + ((-I) a (-2 k + m) + p)/ (2 c), -E^(2 c z)])/(2 c - I a (-2 k + m) + p)) + (-((E^((I a (-2 k + m) + p) z) Hypergeometric2F1[(I a (-2 k + m) + p)/ (2 c), 1, 1 + (I a (-2 k + m) + p)/(2 c), -E^(2 c z)])/ (I a (-2 k + m) + p)) + (E^((2 c + I a (-2 k + m) + p) z) Hypergeometric2F1[1 + (I a (-2 k + m) + p)/(2 c), 1, 2 + (I a (-2 k + m) + p)/(2 c), -E^(2 c z)])/ (2 c + I a (-2 k + m) + p))/E^((1/2) I m Pi)), {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) Hypergeometric2F1[(p - b (-2 k + u))/ (2 c), 1, 1 + (p - b (-2 k + u))/(2 c), -E^(2 c z)])/ (p - b (-2 k + u))) - (1/(p + b (-2 k + u))) (E^((p + b (-2 k + u)) z) Hypergeometric2F1[(p + b (-2 k + u))/(2 c), 1, 1 + (p + b (-2 k + u))/(2 c), -E^(2 c z)]) + (E^((2 c + p - b (-2 k + u)) z) Hypergeometric2F1[ 1 + (p - b (-2 k + u))/(2 c), 1, 2 + (p - b (-2 k + u))/(2 c), -E^(2 c z)])/(2 c + p - b (-2 k + u)) + (E^((2 c + p + b (-2 k + u)) z) Hypergeometric2F1[ 1 + (p + b (-2 k + u))/(2 c), 1, 2 + (p + b (-2 k + u))/(2 c), -E^(2 c z)])/(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) (-((E^(((-I) a (-2 k + m) + p - b (-2 s + u)) z) 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)])/((-I) a (-2 k + m) + p - b (-2 s + u))) + (E^((2 c - I a (-2 k + m) + p - b (-2 s + u)) z) 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)])/ (2 c - I a (-2 k + m) + p - b (-2 s + u))) + (-((E^((I a (-2 k + m) + p - b (-2 s + u)) z) 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)])/ (I a (-2 k + m) + p - b (-2 s + u))) + (E^((2 c + I a (-2 k + m) + p - b (-2 s + u)) z) 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)])/ (2 c + I a (-2 k + m) + p - b (-2 s + u)))/E^((1/2) I m Pi) + E^((I m Pi)/2) (-((E^(((-I) a (-2 k + m) + p + b (-2 s + u)) z) 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)])/((-I) a (-2 k + m) + p + b (-2 s + u))) + (E^((2 c - I a (-2 k + m) + p + b (-2 s + u)) z) 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)])/ (2 c - I a (-2 k + m) + p + b (-2 s + u))) + (-((E^((I a (-2 k + m) + p + b (-2 s + u)) z) 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)])/ (I a (-2 k + m) + p + b (-2 s + u))) + (E^((2 c + I a (-2 k + m) + p + b (-2 s + u)) z) 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)])/ (2 c + I a (-2 k + m) + p + b (-2 s + u)))/E^((1/2) I m Pi)), {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

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

 2002-12-18