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 Coth

 http://functions.wolfram.com/01.22.21.0275.01

 Input Form

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

 Standard Form

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"b"]], ")"]], " ", RowBox[List["Cosh", "[", RowBox[List["2", " ", "c", " ", "z"]], "]"]]]]]], ")"]], "2"]]], ")"]]]]]], ")"]]]], ")"]]]]]]]]

 MathML Form

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

 Rule Form

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

 2002-12-18