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 Coth

 http://functions.wolfram.com/01.22.21.0394.01

 Input Form

 Integrate[1/(a Tanh[c z]^2 + b Coth[c z]^2)^2, z] == ((-4 (a - b) Cosh[2 c z] + (a + b) (3 + Cosh[4 c z]))^2 Csch[c z]^4 Sech[c z]^4 ((1024 c z)/(a + b)^2 + ((-1)^(1/4) ((-I) Sqrt[a] + Sqrt[b]) (3 a^2 + 20 I a^(3/2) Sqrt[b] + 174 a b - 20 I Sqrt[a] b^(3/2) + 3 b^2) ArcTan[((-1)^(1/4) a^(1/4) Tanh[c z])/b^(1/4)])/ (a^(7/4) (Sqrt[a] - I Sqrt[b])^2 b^(7/4)) + ((-1)^(3/4) (I Sqrt[a] + Sqrt[b]) (3 a^2 - 20 I a^(3/2) Sqrt[b] + 174 a b + 20 I Sqrt[a] b^(3/2) + 3 b^2) ArcTan[((-1)^(3/4) a^(1/4) Tanh[c z])/b^(1/4)])/ (a^(7/4) (Sqrt[a] + I Sqrt[b])^2 b^(7/4)) + (8 (14 (-a^2 + b^2) + (15 a^2 - 98 a b + 15 b^2) Cosh[2 c z]) Sinh[2 c z])/(a b (a + b) (-4 (a - b) Cosh[2 c z] + (a + b) (3 + Cosh[4 c z]))) + (1/(a^(7/4) b^(7/4))) (3 (-((3 (-1)^(3/4) (a + b)^2 ArcTan[((-1)^(1/4) a^(1/4) Tanh[c z])/ b^(1/4)])/(Sqrt[a] + I Sqrt[b])) - (3 (-1)^(1/4) (a + b)^2 ArcTan[((-1)^(3/4) a^(1/4) Tanh[c z])/ b^(1/4)])/(Sqrt[a] - I Sqrt[b]) + (8 a^(3/4) b^(3/4) (-2 a + 2 b + (a + b) Cosh[2 c z]) Sinh[2 c z])/ (4 (a - b) Cosh[2 c z] - (a + b) (3 + Cosh[4 c z])))) + (1/(a^(7/4) b^(7/4))) (4 ((-1)^(3/4) (Sqrt[a] + I Sqrt[b]) (3 a + 2 I Sqrt[a] Sqrt[b] - 3 b) ArcTan[((-1)^(1/4) a^(1/4) Tanh[c z])/b^(1/4)] + (-1)^(1/4) (Sqrt[a] - I Sqrt[b]) (3 a - 2 I Sqrt[a] Sqrt[b] - 3 b) ArcTan[((-1)^(3/4) a^(1/4) Tanh[c z])/b^(1/4)] + (8 a^(3/4) b^(3/4) (-2 a + 2 b + 3 (a + b) Cosh[2 c z]) Sinh[2 c z])/ (4 (a - b) Cosh[2 c z] - (a + b) (3 + Cosh[4 c z]))))))/ (65536 c (b Coth[c z]^2 + a Tanh[c z]^2)^2)

 Standard Form

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 MathML Form

 1 ( a tanh 2 ( c z ) + b coth 2 ( c z ) ) 2 z ( ( ( a + b ) ( cosh ( 4 c z ) + 3 ) - 4 ( a - b ) cosh ( 2 c z ) ) 2 csch 4 ( c z ) sech 4 ( c z ) ( 1024 c z ( a + b ) 2 + 1 a 7 / 4 ( a - b ) 2 b 7 / 4 ( - 1 4 ( - a + b ) ( 20 b a 3 / 2 + 3 a 2 + 174 b a - 20 b 3 / 2 a + 3 b 2 ) tan - 1 ( - 1 4 a 4 tanh ( c z ) b 4 ) ) + 1 a 7 / 4 ( a + b ) 2 b 7 / 4 ( ( - 1 ) 3 / 4 ( a + b ) ( - 20 b a 3 / 2 + 3 a 2 + 174 b a + 20 b 3 / 2 a + 3 b 2 ) tan - 1 ( ( - 1 ) 3 / 4 a 4 tanh ( c z ) b 4 ) ) + 1 a 7 / 4 b 7 / 4 ( 3 ( 8 a 3 / 4 ( - 2 a + 2 b + ( a + b ) cosh ( 2 c z ) ) sinh ( 2 c z ) b 3 / 4 4 ( a - b ) cosh ( 2 c z ) - ( a + b ) ( cosh ( 4 c z ) + 3 ) - 3 ( - 1 ) 3 / 4 ( a + b ) 2 tan - 1 ( - 1 4 a 4 tanh ( c z ) b 4 ) a + b - 3 - 1 4 ( a + b ) 2 tan - 1 ( ( - 1 ) 3 / 4 a 4 tanh ( c z ) b 4 ) a - b ) ) + 1 a 7 / 4 b 7 / 4 ( 4 ( ( - 1 ) 3 / 4 ( a + b ) ( 3 a + 2 b a - 3 b ) tan - 1 ( - 1 4 a 4 tanh ( c z ) b 4 ) + - 1 4 ( a - b ) ( 3 a - 2 b a - 3 b ) tan - 1 ( ( - 1 ) 3 / 4 a 4 tanh ( c z ) b 4 ) + 8 a 3 / 4 b 3 / 4 ( - 2 a + 2 b + 3 ( a + b ) cosh ( 2 c z ) ) sinh ( 2 c z ) 4 ( a - b ) cosh ( 2 c z ) - ( a + b ) ( cosh ( 4 c z ) + 3 ) ) ) + 8 ( 14 ( b 2 - a 2 ) + ( 15 a 2 - 98 b a + 15 b 2 ) cosh ( 2 c z ) ) sinh ( 2 c z ) a b ( a + b ) ( ( a + b ) ( cosh ( 4 c z ) + 3 ) - 4 ( a - b ) cosh ( 2 c z ) ) ) ) / ( 65536 c ( b coth 2 ( c z ) + a tanh 2 ( c z ) ) 2 ) z 1 a c z 2 b c z 2 2 -1 a b 4 c z 3 -1 4 a -1 b 2 c z 2 c z 4 c z 4 1024 c z a b 2 -1 1 a 7 4 a 1 2 -1 b 1 2 2 b 7 4 -1 -1 1 4 -1 a 1 2 b 1 2 20 b 1 2 a 3 2 3 a 2 174 b a -1 20 b 3 2 a 1 2 3 b 2 -1 1 4 a 1 4 c z b 1 4 -1 1 a 7 4 a 1 2 b 1 2 2 b 7 4 -1 -1 3 4 a 1 2 b 1 2 -20 b 1 2 a 3 2 3 a 2 174 b a 20 b 3 2 a 1 2 3 b 2 -1 3 4 a 1 4 c z b 1 4 -1 1 a 7 4 b 7 4 -1 3 8 a 3 4 -2 a 2 b a b 2 c z 2 c z b 3 4 4 a -1 b 2 c z -1 a b 4 c z 3 -1 -1 3 -1 3 4 a b 2 -1 1 4 a 1 4 c z b 1 4 -1 a 1 2 b 1 2 -1 -1 3 -1 1 4 a b 2 -1 3 4 a 1 4 c z b 1 4 -1 a 1 2 -1 b 1 2 -1 1 a 7 4 b 7 4 -1 4 -1 3 4 a 1 2 b 1 2 3 a 2 b 1 2 a 1 2 -1 3 b -1 1 4 a 1 4 c z b 1 4 -1 -1 1 4 a 1 2 -1 b 1 2 3 a -1 2 b 1 2 a 1 2 -1 3 b -1 3 4 a 1 4 c z b 1 4 -1 8 a 3 4 b 3 4 -2 a 2 b 3 a b 2 c z 2 c z 4 a -1 b 2 c z -1 a b 4 c z 3 -1 8 14 b 2 -1 a 2 15 a 2 -1 98 b a 15 b 2 2 c z 2 c z a b a b a b 4 c z 3 -1 4 a -1 b 2 c z -1 65536 c b c z 2 a c z 2 2 -1 [/itex]

 Rule Form

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

 2002-12-18