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 Csch

 http://functions.wolfram.com/01.23.21.0316.01

 Input Form

 Integrate[Csch[c z] Sqrt[(a + b Csch[c z])^3], z] == (Cosh[c z] Sqrt[1 + I Csch[c z]] Sqrt[2 + 2 I Csch[c z]] Sqrt[(a + b Csch[c z])^3] Sinh[c z]^3 (-(((I + Csch[c z]) Sqrt[(a + b Csch[c z])/(a - I b)] (8 a (a + I b) Sqrt[-((b (-I + Csch[c z]))/(a + I b))] EllipticE[ArcSin[Sqrt[(a + b Csch[c z])/(a - I b)]], (a - I b)/(a + I b)] + (3 a^2 - b^2) Sqrt[2 + 2 I Csch[c z]] EllipticF[ArcSin[Sqrt[-((b (I + Csch[c z]))/(a - I b))]], (I a + b)/(2 b)] - 8 I a b Sqrt[-((b (-I + Csch[c z]))/(a + I b))] EllipticF[ArcSin[Sqrt[(a + b Csch[c z])/(a - I b)]], (a - I b)/(a + I b)]) Sinh[c z])/ Sqrt[-((b (I + Csch[c z]))/(a - I b))]) - 2 b Coth[c z]^2 (b + a Sinh[c z])))/(3 c Sqrt[Cosh[c z]^2] Sqrt[1 + Cosh[2 c z]] (I + Sinh[c z]) (b + a Sinh[c z])^2)

 Standard Form

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 MathML Form

 csch ( c z ) ( a + b csch ( c z ) ) 3 z ( cosh ( c z ) csch ( c z ) + 1 2 csch ( c z ) + 2 ( a + b csch ( c z ) ) 3 sinh 3 ( c z ) ( - 2 b ( b + a sinh ( c z ) ) coth 2 ( c z ) - 1 - b ( + csch ( c z ) ) a - b ( ( + csch ( c z ) ) a + b csch ( c z ) a - b ( 8 a - b ( - + csch ( c z ) ) a + b ( a + b ) E ( sin - 1 ( a + b csch ( c z ) a - b ) a - b a + b ) - 8 a b - b ( - + csch ( c z ) ) a + b F ( sin - 1 ( a + b csch ( c z ) a - b ) a - b a + b ) + ( 3 a 2 - b 2 ) 2 csch ( c z ) + 2 F ( sin - 1 ( - b ( + csch ( c z ) ) a - b ) b + a 2 b ) ) sinh ( c z ) ) ) ) / ( 3 c cosh 2 ( c z ) cosh ( 2 c z ) + 1 ( + sinh ( c z ) ) ( b + a sinh ( c z ) ) 2 ) z c z a b c z 3 1 2 c z c z 1 1 2 2 c z 2 1 2 a b c z 3 1 2 c z 3 -2 b b a c z c z 2 -1 1 -1 b c z a -1 b -1 1 2 -1 c z a b c z a -1 b -1 1 2 8 a -1 b -1 c z a b -1 1 2 a b EllipticE a b c z a -1 b -1 1 2 a -1 b a b -1 -1 8 a b -1 b -1 c z a b -1 1 2 EllipticF a b c z a -1 b -1 1 2 a -1 b a b -1 3 a 2 -1 b 2 2 c z 2 1 2 EllipticF -1 b c z a -1 b -1 1 2 b a 2 b -1 c z 3 c c z 2 1 2 2 c z 1 1 2 c z b a c z 2 -1 [/itex]

 Rule Form

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

 2002-12-18