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 Cosh

 http://functions.wolfram.com/01.20.21.4417.01

 Input Form

 Integrate[1/Sqrt[a Sinh[e z]^2 + b Cosh[e z]^2], z] == -2 I (1 + Cosh[e z]) Sqrt[(-a + b + (a + b) Cosh[2 e z])/(1 + Cosh[e z])^2] 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[1 + (b Tanh[(e z)/2]^2)/(2 a + b - 2 Sqrt[a] Sqrt[a + b])] (Sqrt[1 + (b Tanh[(e z)/2]^2)/(2 a + b + 2 Sqrt[a] Sqrt[a + b])]/ (Sqrt[(2 a + b - 2 Sqrt[a] Sqrt[a + b])/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[FractionBox["1", 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["-", "2"]], " ", "\[ImaginaryI]", " ", 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["EllipticF", "[", RowBox[List[RowBox[List["\[ImaginaryI]", " ", RowBox[List["ArcSinh", "[", RowBox[List[SqrtBox[FractionBox[RowBox[List[RowBox[List["2", " ", "a"]], "+", "b", "-", RowBox[List["2", " ", SqrtBox["a"], " ", SqrtBox[RowBox[List["a", "+", "b"]]]]]]], "b"]], " ", RowBox[List["Tanh", "[", FractionBox[RowBox[List["e", " ", "z"]], "2"], "]"]]]], "]"]]]], ",", FractionBox[RowBox[List[RowBox[List["2", " ", "a"]], "+", "b", "+", RowBox[List["2", " ", SqrtBox["a"], " ", SqrtBox[RowBox[List["a", "+", "b"]]]]]]], RowBox[List[RowBox[List["2", " ", "a"]], "+", "b", "-", RowBox[List["2", " ", SqrtBox["a"], " ", SqrtBox[RowBox[List["a", "+", "b"]]]]]]]]]], "]"]], " ", SqrtBox[RowBox[List["1", "+", FractionBox[RowBox[List["b", " ", SuperscriptBox[RowBox[List["Tanh", "[", FractionBox[RowBox[List["e", " ", "z"]], "2"], "]"]], "2"]]], RowBox[List[RowBox[List["2", " ", "a"]], "+", "b", "-", RowBox[List["2", " ", SqrtBox["a"], " ", SqrtBox[RowBox[List["a", "+", "b"]]]]]]]]]]], " ", RowBox[List[SqrtBox[RowBox[List["1", "+", FractionBox[RowBox[List["b", " ", SuperscriptBox[RowBox[List["Tanh", "[", FractionBox[RowBox[List["e", " ", "z"]], "2"], "]"]], "2"]]], RowBox[List[RowBox[List["2", " ", "a"]], "+", "b", "+", RowBox[List["2", " ", SqrtBox["a"], " ", SqrtBox[RowBox[List["a", "+", "b"]]]]]]]]]]], "/", RowBox[List["(", RowBox[List[SqrtBox[FractionBox[RowBox[List[RowBox[List["2", " ", "a"]], "+", "b", "-", RowBox[List["2", " ", SqrtBox["a"], " ", SqrtBox[RowBox[List["a", "+", "b"]]]]]]], "b"]], " ", "e", " ", SqrtBox[RowBox[List[RowBox[List["-", "a"]], "+", "b", "+", RowBox[List[RowBox[List["(", RowBox[List["a", "+", "b"]], ")"]], " ", RowBox[List["Cosh", "[", RowBox[List["2", " ", "e", " ", "z"]], "]"]]]]]]], " ", SqrtBox[RowBox[List[RowBox[List["4", " ", "a", " ", SuperscriptBox[RowBox[List["Tanh", "[", FractionBox[RowBox[List["e", " ", "z"]], "2"], "]"]], "2"]]], "+", RowBox[List["b", " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", SuperscriptBox[RowBox[List["Tanh", "[", FractionBox[RowBox[List["e", " ", "z"]], "2"], "]"]], "2"]]], ")"]], "2"]]]]]]]], ")"]]]]]]]]]]

 MathML Form

 1 a sinh 2 ( e z ) + b cosh 2 ( e z ) z - 2 ( cosh ( e z ) + 1 ) - a + b + ( a + b ) cosh ( 2 e z ) ( cosh ( e z ) + 1 ) 2 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 tanh 2 ( e z 2 ) 2 a - 2 a + b a + b + 1 b tanh 2 ( e z 2 ) 2 a + 2 a + b a + b + 1 / ( 2 a - 2 a + b a + b 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 1 a e z 2 b e z 2 1 2 -1 -2 e z 1 -1 a b a b 2 e z e z 1 2 -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 e z 2 -1 2 2 a -1 2 a b 1 2 a 1 2 b -1 1 1 2 b e z 2 -1 2 2 a 2 a b 1 2 a 1 2 b -1 1 1 2 2 a -1 2 a b 1 2 a 1 2 b b -1 1 2 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

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["\[Integral]", RowBox[List[FractionBox["1", 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_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List["-", FractionBox[RowBox[List["2", " ", "\[ImaginaryI]", " ", 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["EllipticF", "[", RowBox[List[RowBox[List["\[ImaginaryI]", " ", RowBox[List["ArcSinh", "[", RowBox[List[SqrtBox[FractionBox[RowBox[List[RowBox[List["2", " ", "a"]], "+", "b", "-", RowBox[List["2", " ", SqrtBox["a"], " ", SqrtBox[RowBox[List["a", "+", "b"]]]]]]], "b"]], " ", RowBox[List["Tanh", "[", FractionBox[RowBox[List["e", " ", "z"]], "2"], "]"]]]], "]"]]]], ",", FractionBox[RowBox[List[RowBox[List["2", " ", "a"]], "+", "b", "+", RowBox[List["2", " ", SqrtBox["a"], " ", SqrtBox[RowBox[List["a", "+", "b"]]]]]]], RowBox[List[RowBox[List["2", " ", "a"]], "+", "b", "-", RowBox[List["2", " ", SqrtBox["a"], " ", SqrtBox[RowBox[List["a", "+", "b"]]]]]]]]]], "]"]], " ", SqrtBox[RowBox[List["1", "+", FractionBox[RowBox[List["b", " ", SuperscriptBox[RowBox[List["Tanh", "[", FractionBox[RowBox[List["e", " ", "z"]], "2"], "]"]], "2"]]], RowBox[List[RowBox[List["2", " ", "a"]], "+", "b", "-", RowBox[List["2", " ", SqrtBox["a"], " ", SqrtBox[RowBox[List["a", "+", "b"]]]]]]]]]]], " ", SqrtBox[RowBox[List["1", "+", FractionBox[RowBox[List["b", " ", SuperscriptBox[RowBox[List["Tanh", "[", FractionBox[RowBox[List["e", " ", "z"]], "2"], "]"]], "2"]]], RowBox[List[RowBox[List["2", " ", "a"]], "+", "b", "+", RowBox[List["2", " ", SqrtBox["a"], " ", SqrtBox[RowBox[List["a", "+", "b"]]]]]]]]]]]]], RowBox[List[SqrtBox[FractionBox[RowBox[List[RowBox[List["2", " ", "a"]], "+", "b", "-", RowBox[List["2", " ", SqrtBox["a"], " ", SqrtBox[RowBox[List["a", "+", "b"]]]]]]], "b"]], " ", "e", " ", SqrtBox[RowBox[List[RowBox[List["-", "a"]], "+", "b", "+", RowBox[List[RowBox[List["(", RowBox[List["a", "+", "b"]], ")"]], " ", RowBox[List["Cosh", "[", RowBox[List["2", " ", "e", " ", "z"]], "]"]]]]]]], " ", SqrtBox[RowBox[List[RowBox[List["4", " ", "a", " ", SuperscriptBox[RowBox[List["Tanh", "[", FractionBox[RowBox[List["e", " ", "z"]], "2"], "]"]], "2"]]], "+", RowBox[List["b", " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", SuperscriptBox[RowBox[List["Tanh", "[", FractionBox[RowBox[List["e", " ", "z"]], "2"], "]"]], "2"]]], ")"]], "2"]]]]]]]]]]]]]]]

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

 2002-12-18