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 Tanh

 http://functions.wolfram.com/01.21.21.0160.01

 Input Form

 Integrate[Sin[a z]^m Sinh[b z]^u Tanh[c z], z] == (1/c) I^u 2^(-m - u) Binomial[m, m/2] Binomial[u, u/2] Log[Cosh[c z]] (1 - Mod[m, 2]) (1 - Mod[u, 2]) - I^u 2^(-m - u) Binomial[u, u/2] (-1 + Mod[u, 2]) Sum[(-1)^k Binomial[m, k] (E^((I m Pi)/2) (-((1/(a (-2 k + m))) ((I Hypergeometric2F1[-((I a (-2 k + m))/(2 c)), 1, 1 - (I a (-2 k + m))/(2 c), -E^(2 c z)])/ E^(I a (-2 k + m) z))) + (E^((2 c - I a (-2 k + m)) z) Hypergeometric2F1[1 - (I a (-2 k + m))/(2 c), 1, 2 - (I a (-2 k + m))/(2 c), -E^(2 c z)])/ (2 c - I a (-2 k + m))) + ((1/(a (-2 k + m))) (I E^(I a (-2 k + m) z) Hypergeometric2F1[ (I a (-2 k + m))/(2 c), 1, 1 + (I a (-2 k + m))/(2 c), -E^(2 c z)]) + (E^((2 c + I a (-2 k + m)) z) Hypergeometric2F1[ 1 + (I a (-2 k + m))/(2 c), 1, 2 + (I a (-2 k + m))/(2 c), -E^(2 c z)])/(2 c + I a (-2 k + m)))/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[(-1)^k Binomial[u, k] (-((1/(b (-2 k + u))) (E^(b (-2 k + u) z) Hypergeometric2F1[ (b (-2 k + u))/(2 c), 1, 1 + (b (-2 k + u))/(2 c), -E^(2 c z)])) + (-1)^u ((1/(b (-2 k + u))) (Hypergeometric2F1[-((b (-2 k + u))/(2 c)), 1, 1 - (b (-2 k + u))/(2 c), -E^(2 c z)]/E^(b (-2 k + u) z)) + (E^((2 c - b (-2 k + u)) z) Hypergeometric2F1[ 1 - (b (-2 k + u))/(2 c), 1, 2 - (b (-2 k + u))/(2 c), -E^(2 c z)])/(2 c - b (-2 k + u))) + (E^((2 c + b (-2 k + u)) z) Hypergeometric2F1[ 1 + (b (-2 k + u))/(2 c), 1, 2 + (b (-2 k + u))/(2 c), -E^(2 c z)])/ (2 c + b (-2 k + u))), {k, 0, Floor[(1/2) (-1 + u)]}] + 2^(-m - u) Sum[(-1)^(k + s) Binomial[m, k] Binomial[u, s] ((-1)^u E^((I m Pi)/2) (-((E^(((-I) a (-2 k + m) - b (-2 s + u)) z) Hypergeometric2F1[((-I) a (-2 k + m) - b (-2 s + u))/(2 c), 1, 1 + ((-I) a (-2 k + m) - b (-2 s + u))/(2 c), -E^(2 c z)])/ ((-I) a (-2 k + m) - b (-2 s + u))) + (E^((2 c - I a (-2 k + m) - b (-2 s + u)) z) Hypergeometric2F1[ 1 + ((-I) a (-2 k + m) - b (-2 s + u))/(2 c), 1, 2 + ((-I) a (-2 k + m) - b (-2 s + u))/(2 c), -E^(2 c z)])/ (2 c - I a (-2 k + m) - b (-2 s + u))) + ((-1)^u (-((E^((I a (-2 k + m) - b (-2 s + u)) z) Hypergeometric2F1[ (I a (-2 k + m) - b (-2 s + u))/(2 c), 1, 1 + (I a (-2 k + m) - b (-2 s + u))/(2 c), -E^(2 c z)])/(I a (-2 k + m) - b (-2 s + u))) + (E^((2 c + I a (-2 k + m) - b (-2 s + u)) z) Hypergeometric2F1[1 + (I a (-2 k + m) - b (-2 s + u))/(2 c), 1, 2 + (I a (-2 k + m) - b (-2 s + u))/(2 c), -E^(2 c z)])/ (2 c + I a (-2 k + m) - b (-2 s + u))))/E^((1/2) I m Pi) + E^((I m Pi)/2) (-((E^(((-I) a (-2 k + m) + b (-2 s + u)) z) Hypergeometric2F1[((-I) a (-2 k + m) + b (-2 s + u))/(2 c), 1, 1 + ((-I) a (-2 k + m) + b (-2 s + u))/(2 c), -E^(2 c z)])/ ((-I) a (-2 k + m) + b (-2 s + u))) + (E^((2 c - I a (-2 k + m) + b (-2 s + u)) z) Hypergeometric2F1[ 1 + ((-I) a (-2 k + m) + b (-2 s + u))/(2 c), 1, 2 + ((-I) a (-2 k + m) + b (-2 s + u))/(2 c), -E^(2 c z)])/ (2 c - I a (-2 k + m) + b (-2 s + u))) + (-((E^((I a (-2 k + m) + b (-2 s + u)) z) Hypergeometric2F1[ (I a (-2 k + m) + b (-2 s + u))/(2 c), 1, 1 + (I a (-2 k + m) + b (-2 s + u))/(2 c), -E^(2 c z)])/ (I a (-2 k + m) + b (-2 s + u))) + (E^((2 c + I a (-2 k + m) + b (-2 s + u)) z) Hypergeometric2F1[ 1 + (I a (-2 k + m) + b (-2 s + u))/(2 c), 1, 2 + (I a (-2 k + m) + b (-2 s + u))/(2 c), -E^(2 c z)])/ (2 c + I a (-2 k + m) + 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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m + u + Condition z a z m b z u c z u 2 -1 m -1 u c z 1 -1 \$CellContext`m 2 1 -1 \$CellContext`u 2 c -1 Binomial m m 2 -1 Binomial u u 2 -1 -1 u 2 -1 m -1 u Binomial u u 2 -1 \$CellContext`u 2 -1 k 0 m -1 2 -1 -1 k Binomial m k -1 1 2 m a m -1 2 k z Hypergeometric2F1 a m -1 2 k 2 c -1 1 a m -1 2 k 2 c -1 1 -1 2 c z a m -1 2 k -1 2 c a m -1 2 k z Hypergeometric2F1 a m -1 2 k 2 c -1 1 1 a m -1 2 k 2 c -1 2 -1 2 c z 2 c a m -1 2 k -1 m 2 -1 2 c -1 a m -1 2 k z Hypergeometric2F1 1 -1 a m -1 2 k 2 c -1 1 2 -1 a m -1 2 k 2 c -1 -1 2 c z 2 c -1 a m -1 2 k -1 -1 -1 a m -1 2 k z Hypergeometric2F1 -1 a m -1 2 k 2 c -1 1 1 -1 a m -1 2 k 2 c -1 -1 2 c z a m -1 2 k -1 2 -1 m -1 u Binomial m m 2 -1 1 -1 \$CellContext`m 2 k 0 u -1 2 -1 -1 k Binomial u k -1 u -1 b u -1 2 k z Hypergeometric2F1 -1 b u -1 2 k 2 c -1 1 1 -1 b u -1 2 k 2 c -1 -1 2 c z b u -1 2 k -1 2 c -1 b u -1 2 k z Hypergeometric2F1 1 -1 b u -1 2 k 2 c -1 1 2 -1 b u -1 2 k 2 c -1 -1 2 c z 2 c -1 b u -1 2 k -1 -1 b u -1 2 k z Hypergeometric2F1 b u -1 2 k 2 c -1 1 b u -1 2 k 2 c -1 1 -1 2 c z b u -1 2 k -1 2 c b u -1 2 k z Hypergeometric2F1 b u -1 2 k 2 c -1 1 1 b u -1 2 k 2 c -1 2 -1 2 c z 2 c b u -1 2 k -1 2 -1 m -1 u k 0 m -1 2 -1 s 0 u -1 2 -1 -1 k s Binomial m k Binomial u s -1 1 2 m 2 c a m -1 2 k b u -1 2 s z Hypergeometric2F1 a m -1 2 k b u -1 2 s 2 c -1 1 1 a m -1 2 k b u -1 2 s 2 c -1 2 -1 2 c z 2 c a m -1 2 k b u -1 2 s -1 -1 a m -1 2 k b u -1 2 s z Hypergeometric2F1 a m -1 2 k b u -1 2 s 2 c -1 1 a m -1 2 k b u -1 2 s 2 c -1 1 -1 2 c z a m -1 2 k b u -1 2 s -1 m 2 -1 2 c -1 a m -1 2 k b u -1 2 s z Hypergeometric2F1 b u -1 2 s -1 a m -1 2 k 2 c -1 1 1 b u -1 2 s -1 a m -1 2 k 2 c -1 2 -1 2 c z 2 c -1 a m -1 2 k b u -1 2 s -1 -1 b u -1 2 s -1 a m -1 2 k z Hypergeometric2F1 b u -1 2 s -1 a m -1 2 k 2 c -1 1 b u -1 2 s -1 a m -1 2 k 2 c -1 1 -1 2 c z b u -1 2 s -1 a m -1 2 k -1 -1 u -1 1 2 m 2 c a m -1 2 k -1 b u -1 2 s z Hypergeometric2F1 a m -1 2 k -1 b u -1 2 s 2 c -1 1 1 a m -1 2 k -1 b u -1 2 s 2 c -1 2 -1 2 c z 2 c a m -1 2 k -1 b u -1 2 s -1 -1 a m -1 2 k -1 b u -1 2 s z Hypergeometric2F1 a m -1 2 k -1 b u -1 2 s 2 c -1 1 a m -1 2 k -1 b u -1 2 s 2 c -1 1 -1 2 c z a m -1 2 k -1 b u -1 2 s -1 -1 u m 2 -1 2 c -1 a m -1 2 k -1 b u -1 2 s z Hypergeometric2F1 -1 a m -1 2 k -1 b u -1 2 s 2 c -1 1 1 -1 a m -1 2 k -1 b u -1 2 s 2 c -1 2 -1 2 c z 2 c -1 a m -1 2 k -1 b u -1 2 s -1 -1 -1 a m -1 2 k -1 b u -1 2 s z Hypergeometric2F1 -1 a m -1 2 k -1 b u -1 2 s 2 c -1 1 -1 a m -1 2 k -1 b u -1 2 s 2 c -1 1 -1 2 c z -1 a m -1 2 k -1 b u -1 2 s -1 m SuperPlus u SuperPlus [/itex]

 Rule Form

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

 2002-12-18