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 Cosh

 http://functions.wolfram.com/01.20.21.4416.01

 Input Form

 Integrate[Sqrt[a Sinh[e z]^2 + b Cosh[e z]^2], z] == (-(1 + Cosh[e z])) Sqrt[(-a + b + (a + b) Cosh[2 e z])/(1 + Cosh[e z])^2] ((I b Sqrt[(2 a + b - 2 Sqrt[a] Sqrt[a + b])/b] EllipticE[I ArcSinh[Sqrt[(2 a + b - 2 Sqrt[a] Sqrt[a + b])/b] Tanh[(e z)/2]], (2 a + b + 2 Sqrt[a] Sqrt[a + b])/ (2 a + b - 2 Sqrt[a] Sqrt[a + b])] Sqrt[(b + (2 a + b - 2 Sqrt[a] Sqrt[a + b]) Tanh[(e z)/2]^2)/b] Sqrt[(b + (2 a + b + 2 Sqrt[a] Sqrt[a + b]) Tanh[(e z)/2]^2)/b] + ((-2 a - b + 2 Sqrt[a] Sqrt[a + b]) (-a + b + (a + b) Cosh[2 e z]) Sech[(e z)/2]^2 Tanh[(e z)/2] + 4 I Sqrt[a] b (-Sqrt[a] + Sqrt[a + b]) Sqrt[(2 a + b - 2 Sqrt[a] Sqrt[a + b])/b] EllipticF[I ArcSinh[Sqrt[(2 a + b - 2 Sqrt[a] Sqrt[a + b])/b] Tanh[(e z)/2]], (2 a + b + 2 Sqrt[a] Sqrt[a + b])/ (2 a + b - 2 Sqrt[a] Sqrt[a + b])] Sqrt[(b + (2 a + b - 2 Sqrt[a] Sqrt[a + b]) Tanh[(e z)/2]^2)/b] Sqrt[(b + (2 a + b + 2 Sqrt[a] Sqrt[a + b]) Tanh[(e z)/2]^2)/b])/ (2 (2 a + b - 2 Sqrt[a] Sqrt[a + b])))/ (e Sqrt[-a + b + (a + b) Cosh[2 e z]] Sqrt[4 a Tanh[(e z)/2]^2 + b (1 + Tanh[(e z)/2]^2)^2]))

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List["\[Integral]", RowBox[List[SqrtBox[RowBox[List[RowBox[List["a", " ", SuperscriptBox[RowBox[List["Sinh", "[", RowBox[List["e", " ", "z"]], "]"]], "2"]]], "+", RowBox[List["b", " ", SuperscriptBox[RowBox[List["Cosh", "[", RowBox[List["e", " ", "z"]], "]"]], "2"]]]]]], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[RowBox[List["-", RowBox[List["(", RowBox[List["1", "+", RowBox[List["Cosh", "[", RowBox[List["e", " ", "z"]], "]"]]]], ")"]]]], " ", SqrtBox[FractionBox[RowBox[List[RowBox[List["-", "a"]], "+", "b", "+", RowBox[List[RowBox[List["(", RowBox[List["a", "+", "b"]], ")"]], " ", RowBox[List["Cosh", "[", RowBox[List["2", " ", "e", " ", "z"]], "]"]]]]]], SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", RowBox[List["Cosh", "[", RowBox[List["e", " ", "z"]], "]"]]]], ")"]], "2"]]], " ", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "b", " ", SqrtBox[FractionBox[RowBox[List[RowBox[List["2", " ", 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 MathML Form

 a sinh 2 ( e z ) + b cosh 2 ( e z ) z - ( cosh ( e z ) + 1 ) - a + b + ( a + b ) cosh ( 2 e z ) ( cosh ( e z ) + 1 ) 2 ( 1 2 ( 2 a - 2 a + b a + b ) ( ( - 2 a + 2 a + b a - b ) ( - a + b + ( a + b ) cosh ( 2 e z ) ) tanh ( e z 2 ) sech 2 ( e z 2 ) + 4 a b ( a + b - a ) 2 a - 2 a + b a + b b ( 2 a + 2 a + b a + b ) tanh 2 ( e z 2 ) + b b ( 2 a - 2 a + b a + b ) tanh 2 ( e z 2 ) + b b F ( sinh - 1 ( 2 a - 2 a + b a + b b tanh ( e z 2 ) ) 2 a + 2 a + b a + b 2 a - 2 a + b a + b ) ) + b 2 a - 2 a + b a + b b ( 2 a + 2 a + b a + b ) tanh 2 ( e z 2 ) + b b ( 2 a - 2 a + b a + b ) tanh 2 ( e z 2 ) + b b E ( sinh - 1 ( 2 a - 2 a + b a + b b tanh ( e z 2 ) ) 2 a + 2 a + b a + b 2 a - 2 a + b a + b ) ) / ( e - a + b + ( a + b ) cosh ( 2 e z ) 4 a tanh 2 ( e z 2 ) + b ( tanh 2 ( e z 2 ) + 1 ) 2 ) z a e z 2 b e z 2 1 2 -1 e z 1 -1 a b a b 2 e z e z 1 2 -1 1 2 1 2 2 a -1 2 a b 1 2 a 1 2 b -1 -2 a 2 a b 1 2 a 1 2 -1 b -1 a b a b 2 e z e z 2 -1 e z 2 -1 2 4 a 1 2 b a b 1 2 -1 a 1 2 2 a -1 2 a b 1 2 a 1 2 b b -1 1 2 2 a 2 a b 1 2 a 1 2 b e z 2 -1 2 b b -1 1 2 2 a -1 2 a b 1 2 a 1 2 b e z 2 -1 2 b b -1 1 2 EllipticF 2 a -1 2 a b 1 2 a 1 2 b b -1 1 2 e z 2 -1 2 a 2 a b 1 2 a 1 2 b 2 a -1 2 a b 1 2 a 1 2 b -1 b 2 a -1 2 a b 1 2 a 1 2 b b -1 1 2 2 a 2 a b 1 2 a 1 2 b e z 2 -1 2 b b -1 1 2 2 a -1 2 a b 1 2 a 1 2 b e z 2 -1 2 b b -1 1 2 EllipticE 2 a -1 2 a b 1 2 a 1 2 b b -1 1 2 e z 2 -1 2 a 2 a b 1 2 a 1 2 b 2 a -1 2 a b 1 2 a 1 2 b -1 e -1 a b a b 2 e z 1 2 4 a e z 2 -1 2 b e z 2 -1 2 1 2 1 2 -1 [/itex]

 Rule Form

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

 2002-12-18