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 Coth

 http://functions.wolfram.com/01.22.21.0386.01

 Input Form

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

 Standard Form

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 MathML Form

 1 ( a sinh ( c z ) + b coth ( c z ) ) 2 z ( csch 2 ( c z ) ( a sinh 2 ( c z ) + b cosh ( c z ) ) ( - 2 ( b + 2 a cosh ( c z ) ) sinh ( c z ) 4 a 2 + b 2 + ( 2 2 ( 4 a 2 + b ( 4 a 2 + b 2 - b ) ) tan - 1 ( ( 2 a - b + 4 a 2 + b 2 ) tanh ( c z 2 ) 2 b 4 a 2 + b 2 - b ) ( a sinh 2 ( c z ) + b cosh ( c z ) ) ) / ( b ( 4 a 2 + b 2 ) 3 / 2 4 a 2 + b 2 - b ) - ( 2 2 ( b ( b + 4 a 2 + b 2 ) - 4 a 2 ) tan - 1 ( ( - 2 a + b + 4 a 2 + b 2 ) tanh ( c z 2 ) 2 - b ( b + 4 a 2 + b 2 ) ) ( a sinh 2 ( c z ) + b cosh ( c z ) ) ) / ( ( 4 a 2 + b 2 ) 3 / 2 - b ( b + 4 a 2 + b 2 ) ) ) ) / ( 2 c ( b coth ( c z ) + a sinh ( c z ) ) 2 ) z 1 a c z b c z 2 -1 c z 2 a c z 2 b c z -1 2 b 2 a c z c z 4 a 2 b 2 -1 2 2 1 2 4 a 2 b 4 a 2 b 2 1 2 -1 b 2 a -1 b 4 a 2 b 2 1 2 c z 2 -1 2 1 2 b 1 2 4 a 2 b 2 1 2 -1 b 1 2 -1 a c z 2 b c z b 1 2 4 a 2 b 2 3 2 4 a 2 b 2 1 2 -1 b 1 2 -1 -1 2 2 1 2 b b 4 a 2 b 2 1 2 -1 4 a 2 -2 a b 4 a 2 b 2 1 2 c z 2 -1 2 1 2 -1 b b 4 a 2 b 2 1 2 1 2 -1 a c z 2 b c z 4 a 2 b 2 3 2 -1 b b 4 a 2 b 2 1 2 1 2 -1 2 c b c z a c z 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.