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 Tanh

 http://functions.wolfram.com/01.21.21.0354.01

 Input Form

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

 Standard Form

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 MathML Form

 1 ( a cosh ( c z ) + b tanh ( c z ) ) 2 z ( sech 2 ( c z ) ( cosh ( 2 c z ) a + a + 2 b sinh ( c z ) ) ( 2 cosh ( c z ) ( b + 2 a sinh ( c z ) ) 4 a 2 - b 2 + ( 2 ( b ( b 2 - 4 a 2 - b ) - 4 a 2 ) tan - 1 ( 2 a + ( b 2 - 4 a 2 - b ) tanh ( c z 2 ) 2 b b 2 - 4 a 2 - b ) ( cosh ( 2 c z ) a + a + 2 b sinh ( c z ) ) ) / ( b ( b 2 - 4 a 2 ) 3 / 2 b 2 - 4 a 2 - b ) + ( 2 ( 4 a 2 + b ( b + b 2 - 4 a 2 ) ) tan - 1 ( 2 a - ( b + b 2 - 4 a 2 ) tanh ( c z 2 ) 2 - b ( b + b 2 - 4 a 2 ) ) ( cosh ( 2 c z ) a + a + 2 b sinh ( c z ) ) ) / ( ( b 2 - 4 a 2 ) 3 / 2 - b ( b + b 2 - 4 a 2 ) ) ) ) / ( 4 c ( a cosh ( c z ) + b tanh ( c z ) ) 2 ) z 1 a c z b c z 2 -1 c z 2 2 c z a a 2 b c z 2 c z b 2 a c z 4 a 2 -1 b 2 -1 2 1 2 b b 2 -1 4 a 2 1 2 -1 b -1 4 a 2 2 a b 2 -1 4 a 2 1 2 -1 b c z 2 -1 2 1 2 b 1 2 b 2 -1 4 a 2 1 2 -1 b 1 2 -1 2 c z a a 2 b c z b 1 2 b 2 -1 4 a 2 3 2 b 2 -1 4 a 2 1 2 -1 b 1 2 -1 2 1 2 4 a 2 b b b 2 -1 4 a 2 1 2 2 a -1 b b 2 -1 4 a 2 1 2 c z 2 -1 2 1 2 -1 b b b 2 -1 4 a 2 1 2 1 2 -1 2 c z a a 2 b c z b 2 -1 4 a 2 3 2 -1 b b b 2 -1 4 a 2 1 2 1 2 -1 4 c a c z b c z 2 -1 [/itex]

 Rule Form

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

 2002-12-18