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 Sech

 http://functions.wolfram.com/01.24.21.0211.01

 Input Form

 Integrate[Sin[a z]^m Tanh[c z]^u Sech[c z], z] == (1/(c (1 + u))) (I^u 2^(1 - m) E^(c (1 + u) z) Binomial[m, m/2] Binomial[u, u/2] HypergeometricPFQ[{1/2 + u/2, 1 + u}, {3/2 + u/2}, -E^(2 c z)] (1 - Mod[m, 2]) (1 - Mod[u, 2])) + I^(m + u) 2^(1 - m) E^(c (1 + u) z) Binomial[u, u/2] (1 - Mod[u, 2]) Sum[(-1)^k Binomial[m, k] (HypergeometricPFQ[{1/2 + (I a k)/c - (I a m)/(2 c) + u/2, 1 + u}, {3/2 + (I a k)/c - (I a m)/(2 c) + u/2}, -E^(2 c z)]/ E^(I a (-2 k + m) z)/(I a (2 k - m) + c (1 + u)) + ((-1)^m E^(I a (-2 k + m) z) HypergeometricPFQ[ {1/2 - (I a k)/c + (I a m)/(2 c) + u/2, 1 + u}, {3/2 - (I a k)/c + (I a m)/(2 c) + u/2}, -E^(2 c z)])/ (I a (-2 k + m) + c (1 + u))), {k, 0, Floor[(1/2) (-1 + m)]}] + 2^(1 - m) E^(c (1 + u) z) Binomial[m, m/2] (1 - Mod[m, 2]) Sum[(-1)^s Binomial[u, s] ((1/(c (1 + 2 s))) (((-1)^u HypergeometricPFQ[{1/2 + s, 1 + u}, {3/2 + s}, -E^(2 c z)])/ E^(c (-2 s + u) z)) + (E^(c (-2 s + u) z) HypergeometricPFQ[ {1 + u, 1/2 - s + u}, {3/2 - s + u}, -E^(2 c z)])/ (c (1 - 2 s + 2 u))), {s, 0, Floor[(1/2) (-1 + u)]}] + 2^(1 - m) E^(c (1 + u) z) Sum[(-1)^k Binomial[m, k] Sum[(-1)^s Binomial[u, s] (((-1)^u E^((I m Pi)/2 + ((-I) a (-2 k + m) - c (-2 s + u)) z) HypergeometricPFQ[{1/2 + (I a k)/c - (I a m)/(2 c) + s, 1 + u}, {3/2 + (I a k)/c - (I a m)/(2 c) + s}, -E^(2 c z)])/ (c + 2 I a k - I a m + 2 c s) + ((-1)^u E^((-(1/2)) I m Pi + (I a (-2 k + m) - c (-2 s + u)) z) HypergeometricPFQ[{1/2 - (I a k)/c + (I a m)/(2 c) + s, 1 + u}, {3/2 - (I a k)/c + (I a m)/(2 c) + s}, -E^(2 c z)])/ (c - 2 I a k + I a m + 2 c s) + (E^((I m Pi)/2 + ((-I) a (-2 k + m) + c (-2 s + u)) z) HypergeometricPFQ[{1 + u, 1/2 + (I a k)/c - (I a m)/(2 c) - s + u}, {3/2 + (I a k)/c - (I a m)/(2 c) - s + u}, -E^(2 c z)])/ (I a (2 k - m) + c (1 - 2 s + 2 u)) + (E^((-(1/2)) I m Pi + (I a (-2 k + m) + c (-2 s + u)) z) HypergeometricPFQ[{1 + u, 1/2 - (I a k)/c + (I a m)/(2 c) - s + u}, {3/2 - (I a k)/c + (I a m)/(2 c) - s + u}, -E^(2 c z)])/ (I a (-2 k + m) + c (1 - 2 s + 2 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