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 Coth

 http://functions.wolfram.com/01.22.21.0413.01

 Input Form

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

 Standard Form

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 MathML Form

 d coth 2 ( c z ) + e sinh ( c z ) ( b tanh 2 ( c z ) + a ) 3 z ( sech ( c z ) ( d coth 2 ( c z ) + e sinh ( c z ) ) ( ( a - b + ( a + b ) cosh ( 2 c z ) ) 3 / 2 ( 2 d csch 3 ( c z ) log ( a - b + ( a + b ) cosh ( 2 c z ) + 2 ( a + b ) sinh 2 ( c z ) ) ( ( a + b ) sinh 2 ( c z ) ) 3 / 2 + 2 ( a + b ) e a - b + ( a + b ) cosh ( 2 c z ) a - b + ( a + b ) cosh ( 2 c z ) ( cosh ( c z ) + 1 ) 2 ( a - b + ( a + b ) cosh ( 2 c z ) ) sech 4 ( c z 2 ) ) - 1 a 2 ( ( a + b ) ( a - b + ( a + b ) cosh ( 2 c z ) ) csch ( c z ) ( 2 b e sinh ( c z ) a 2 + ( a 3 + b a 2 - 2 b 2 a - 2 b 3 ) d + ( a 3 + 3 b a 2 + 4 b 2 a + 2 b 3 ) d cosh ( 2 c z ) ) ) ) tanh 2 ( c z ) ) / ( ( a + b ) 3 c ( 2 cosh ( 2 c z ) d + 2 d - 3 e sinh ( c z ) + e sinh ( 3 c z ) ) ( b tanh 2 ( c z ) + a ) 3 ) z d c z 2 e c z b c z 2 a 3 1 2 -1 c z d c z 2 e c z a -1 b a b 2 c z 3 2 2 1 2 d c z 3 a -1 b a b 2 c z 1 2 2 1 2 a b c z 2 1 2 a b c z 2 3 2 2 a b e a -1 b a b 2 c z 1 2 a -1 b a b 2 c z c z 1 2 -1 1 2 a -1 b a b 2 c z c z 2 -1 4 1 2 -1 -1 1 a 2 -1 a b a -1 b a b 2 c z c z 2 b e c z a 2 a 3 b a 2 -1 2 b 2 a -1 2 b 3 d a 3 3 b a 2 4 b 2 a 2 b 3 d 2 c z c z 2 a b 3 c 2 2 c z d 2 d -1 3 e c z e 3 c z b c z 2 a 3 1 2 -1 [/itex]

 Rule Form

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

 2002-12-18