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 Csch

 http://functions.wolfram.com/01.23.21.0442.01

 Input Form

 Integrate[1/(a Tanh[z] + b Csch[z])^2, z] == (1/a^2) (((-Sqrt[2]) Sqrt[(-b) (b + Sqrt[4 a^2 + b^2])] (-4 a^4 + b^3 (-b + Sqrt[4 a^2 + b^2]) + a^2 b (-7 b + 5 Sqrt[4 a^2 + b^2])) ArcTan[((2 a - b + Sqrt[4 a^2 + b^2]) Tanh[z/2])/ (Sqrt[2] Sqrt[b] Sqrt[-b + Sqrt[4 a^2 + b^2]])] + Sqrt[b] Sqrt[-b + Sqrt[4 a^2 + b^2]] ((4 a^2 + b^2)^(3/2) Sqrt[(-b) (b + Sqrt[4 a^2 + b^2])] z + Sqrt[2] (4 a^4 + b^3 (b + Sqrt[4 a^2 + b^2]) + a^2 b (7 b + 5 Sqrt[4 a^2 + b^2])) ArcTan[((-2 a + b + Sqrt[4 a^2 + b^2]) Tanh[z/2])/ (Sqrt[2] Sqrt[(-b) (b + Sqrt[4 a^2 + b^2])])]))/ (Sqrt[b] (4 a^2 + b^2)^(3/2) Sqrt[-b + Sqrt[4 a^2 + b^2]] Sqrt[(-b) (b + Sqrt[4 a^2 + b^2])]) - (a ((-a) b + (2 a^2 + b^2) Cosh[z]) Sinh[z])/ ((4 a^2 + b^2) (b Cosh[z] + a Sinh[z]^2)))

 Standard Form

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 MathML Form

 1 ( a tanh ( z ) + b csch ( z ) ) 2 z 1 a 2 ( ( b 4 a 2 + b 2 - b ( - b ( b + 4 a 2 + b 2 ) z ( 4 a 2 + b 2 ) 3 / 2 + 2 ( 4 a 4 + b ( 7 b + 5 4 a 2 + b 2 ) a 2 + b 3 ( b + 4 a 2 + b 2 ) ) tan - 1 ( ( - 2 a + b + 4 a 2 + b 2 ) tanh ( z 2 ) 2 - b ( b + 4 a 2 + b 2 ) ) ) - 2 - b ( b + 4 a 2 + b 2 ) ( - 4 a 4 + b ( 5 4 a 2 + b 2 - 7 b ) a 2 + b 3 ( 4 a 2 + b 2 - b ) ) tan - 1 ( ( 2 a - b + 4 a 2 + b 2 ) tanh ( z 2 ) 2 b 4 a 2 + b 2 - b ) ) / ( b ( 4 a 2 + b 2 ) 3 / 2 4 a 2 + b 2 - b - b ( b + 4 a 2 + b 2 ) ) - a ( ( 2 a 2 + b 2 ) cosh ( z ) - a b ) sinh ( z ) ( 4 a 2 + b 2 ) ( a sinh 2 ( z ) + b cosh ( z ) ) ) z 1 a z b z 2 -1 1 a 2 -1 b 1 2 4 a 2 b 2 1 2 -1 b 1 2 -1 b b 4 a 2 b 2 1 2 1 2 z 4 a 2 b 2 3 2 2 1 2 4 a 4 b 7 b 5 4 a 2 b 2 1 2 a 2 b 3 b 4 a 2 b 2 1 2 -2 a b 4 a 2 b 2 1 2 z 2 -1 2 1 2 -1 b b 4 a 2 b 2 1 2 1 2 -1 -1 2 1 2 -1 b b 4 a 2 b 2 1 2 1 2 -4 a 4 b 5 4 a 2 b 2 1 2 -1 7 b a 2 b 3 4 a 2 b 2 1 2 -1 b 2 a -1 b 4 a 2 b 2 1 2 z 2 -1 2 1 2 b 1 2 4 a 2 b 2 1 2 -1 b 1 2 -1 b 1 2 4 a 2 b 2 3 2 4 a 2 b 2 1 2 -1 b 1 2 -1 b b 4 a 2 b 2 1 2 1 2 -1 -1 a 2 a 2 b 2 z -1 a b z 4 a 2 b 2 a z 2 b z -1 [/itex]

 Rule Form

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

 2002-12-18