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 Sech

 http://functions.wolfram.com/01.24.21.0462.01

 Input Form

 Integrate[1/(a Csch[z]^2 + b Sech[z]^2)^2, z] == ((a - b + (a + b) Cosh[2 z]) Csch[z]^4 Sech[z]^4 (24 a^3 z - 168 a^2 b z + 168 a b^2 z - 24 b^3 z + 96 a^(5/2) Sqrt[b] ArcTan[(Sqrt[b] Tanh[z])/Sqrt[a]] - 192 a^(3/2) b^(3/2) ArcTan[(Sqrt[b] Tanh[z])/Sqrt[a]] + 96 Sqrt[a] b^(5/2) ArcTan[(Sqrt[b] Tanh[z])/Sqrt[a]] + 24 (a + b) ((a^2 - 6 a b + b^2) z + 4 Sqrt[a] (a - b) Sqrt[b] ArcTan[(Sqrt[b] Tanh[z])/Sqrt[a]]) Cosh[2 z] - 3 (5 a^3 - 17 a^2 b - 17 a b^2 + 5 b^3) Sinh[2 z] - 6 a^3 Sinh[4 z] - 6 a^2 b Sinh[4 z] + 6 a b^2 Sinh[4 z] + 6 b^3 Sinh[4 z] + a^3 Sinh[6 z] + 3 a^2 b Sinh[6 z] + 3 a b^2 Sinh[6 z] + b^3 Sinh[6 z]))/ (256 (a + b)^4 (a Csch[z]^2 + b Sech[z]^2)^2)

 Standard Form

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 MathML Form

 1 ( a csch 2 ( z ) + b sech 2 ( z ) ) 2 z ( ( a - b + ( a + b ) cosh ( 2 z ) ) csch 4 ( z ) sech 4 ( z ) ( - 192 a 3 / 2 tan - 1 ( b tanh ( z ) a ) b 3 / 2 + 96 a tan - 1 ( b tanh ( z ) a ) b 5 / 2 - 24 z b 3 + 6 sinh ( 4 z ) b 3 + sinh ( 6 z ) b 3 + 168 a z b 2 + 6 a sinh ( 4 z ) b 2 + 3 a sinh ( 6 z ) b 2 - 168 a 2 z b - 6 a 2 sinh ( 4 z ) b + 3 a 2 sinh ( 6 z ) b + 96 a 5 / 2 tan - 1 ( b tanh ( z ) a ) b + 24 a 3 z + 24 ( a + b ) ( ( a 2 - 6 b a + b 2 ) z + 4 a ( a - b ) b tan - 1 ( b tanh ( z ) a ) ) cosh ( 2 z ) - 3 ( 5 a 3 - 17 b a 2 - 17 b 2 a + 5 b 3 ) sinh ( 2 z ) - 6 a 3 sinh ( 4 z ) + a 3 sinh ( 6 z ) ) ) / ( 256 ( a + b ) 4 ( a csch 2 ( z ) + b sech 2 ( z ) ) 2 ) z 1 a z 2 b z 2 2 -1 a -1 b a b 2 z z 4 z 4 -192 a 3 2 b 1 2 z a 1 2 -1 b 3 2 96 a 1 2 b 1 2 z a 1 2 -1 b 5 2 -1 24 z b 3 6 4 z b 3 6 z b 3 168 a z b 2 6 a 4 z b 2 3 a 6 z b 2 -1 168 a 2 z b -1 6 a 2 4 z b 3 a 2 6 z b 96 a 5 2 b 1 2 z a 1 2 -1 b 1 2 24 a 3 z 24 a b a 2 -1 6 b a b 2 z 4 a 1 2 a -1 b b 1 2 b 1 2 z a 1 2 -1 2 z -1 3 5 a 3 -1 17 b a 2 -1 17 b 2 a 5 b 3 2 z -1 6 a 3 4 z a 3 6 z 256 a b 4 a z 2 b z 2 2 -1 [/itex]

 Rule Form

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

 2002-12-18

© 1998-2013 Wolfram Research, Inc.