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 Cosh

 http://functions.wolfram.com/01.20.21.4407.01

 Input Form

 Integrate[1/Sqrt[a + b Sinh[e z] + c Cosh[e z]], z] == (1/(b Sqrt[1 - c^2/b^2] e)) (2 AppellF1[1/2, 1/2, 1/2, 3/2, (a + c Cosh[e z] + b Sinh[e z])/(a + I b Sqrt[1 - c^2/b^2]), (a + c Cosh[e z] + b Sinh[e z])/(a - I b Sqrt[1 - c^2/b^2])] Sech[e z + ArcTanh[c/b]] Sqrt[a + c Cosh[e z] + b Sinh[e z]] Sqrt[-((c Cosh[e z] + b ((-I) Sqrt[1 - c^2/b^2] + Sinh[e z]))/ (a + I b Sqrt[1 - c^2/b^2]))] Sqrt[-((b Sqrt[1 - c^2/b^2] (I + Sinh[e z + ArcTanh[c/b]]))/ (a - I b Sqrt[1 - c^2/b^2]))])

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List["\[Integral]", RowBox[List[FractionBox["1", SqrtBox[RowBox[List["a", "+", RowBox[List["b", " ", RowBox[List["Sinh", "[", RowBox[List["e", " ", "z"]], "]"]]]], "+", RowBox[List["c", " ", RowBox[List["Cosh", "[", RowBox[List["e", " ", "z"]], "]"]]]]]]]], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[FractionBox["1", RowBox[List["b", " ", SqrtBox[RowBox[List["1", "-", FractionBox[SuperscriptBox["c", "2"], SuperscriptBox["b", "2"]]]]], " ", "e"]]], RowBox[List["(", RowBox[List["2", " ", RowBox[List["AppellF1", "[", RowBox[List[FractionBox["1", "2"], ",", FractionBox["1", "2"], ",", FractionBox["1", "2"], ",", FractionBox["3", "2"], ",", FractionBox[RowBox[List["a", "+", RowBox[List["c", " ", RowBox[List["Cosh", "[", RowBox[List["e", " ", "z"]], "]"]]]], "+", RowBox[List["b", " ", RowBox[List["Sinh", "[", RowBox[List["e", " ", "z"]], "]"]]]]]], RowBox[List["a", "+", RowBox[List["\[ImaginaryI]", " ", "b", " ", SqrtBox[RowBox[List["1", "-", FractionBox[SuperscriptBox["c", "2"], SuperscriptBox["b", "2"]]]]]]]]]], ",", FractionBox[RowBox[List["a", "+", RowBox[List["c", " ", RowBox[List["Cosh", "[", RowBox[List["e", " ", "z"]], "]"]]]], "+", RowBox[List["b", " ", RowBox[List["Sinh", "[", RowBox[List["e", " ", "z"]], "]"]]]]]], RowBox[List["a", "-", RowBox[List["\[ImaginaryI]", " ", "b", " ", SqrtBox[RowBox[List["1", "-", FractionBox[SuperscriptBox["c", "2"], SuperscriptBox["b", "2"]]]]]]]]]]]], "]"]], " ", RowBox[List["Sech", "[", RowBox[List[RowBox[List["e", " ", "z"]], "+", RowBox[List["ArcTanh", "[", FractionBox["c", "b"], "]"]]]], "]"]], " ", SqrtBox[RowBox[List["a", "+", RowBox[List["c", " ", RowBox[List["Cosh", "[", RowBox[List["e", " ", "z"]], "]"]]]], "+", RowBox[List["b", " ", RowBox[List["Sinh", "[", RowBox[List["e", " ", "z"]], "]"]]]]]]], " ", SqrtBox[RowBox[List["-", FractionBox[RowBox[List[RowBox[List["c", " ", RowBox[List["Cosh", "[", RowBox[List["e", " ", "z"]], "]"]]]], "+", RowBox[List["b", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", SqrtBox[RowBox[List["1", "-", FractionBox[SuperscriptBox["c", "2"], SuperscriptBox["b", "2"]]]]]]], "+", RowBox[List["Sinh", "[", RowBox[List["e", " ", "z"]], "]"]]]], ")"]]]]]], RowBox[List["a", "+", RowBox[List["\[ImaginaryI]", " ", "b", " ", SqrtBox[RowBox[List["1", "-", FractionBox[SuperscriptBox["c", "2"], SuperscriptBox["b", "2"]]]]]]]]]]]]], " ", SqrtBox[RowBox[List["-", FractionBox[RowBox[List["b", " ", SqrtBox[RowBox[List["1", "-", FractionBox[SuperscriptBox["c", "2"], SuperscriptBox["b", "2"]]]]], " ", RowBox[List["(", RowBox[List["\[ImaginaryI]", "+", RowBox[List["Sinh", "[", RowBox[List[RowBox[List["e", " ", "z"]], "+", RowBox[List["ArcTanh", "[", FractionBox["c", "b"], "]"]]]], "]"]]]], ")"]]]], RowBox[List["a", "-", RowBox[List["\[ImaginaryI]", " ", "b", " ", SqrtBox[RowBox[List["1", "-", FractionBox[SuperscriptBox["c", "2"], SuperscriptBox["b", "2"]]]]]]]]]]]]]]], ")"]]]]]]]]

 MathML Form

 1 a + b sinh ( e z ) + c cosh ( e z ) z 1 b 1 - c 2 b 2 e ( 2 F 1 AppellF1 ( 1 2 ; 1 2 , 1 2 ; 3 2 ; a + c cosh ( e z ) + b sinh ( e z ) a + b 1 - c 2 b 2 , a + c cosh ( e z ) + b sinh ( e z ) a - b 1 - c 2 b 2 ) sech ( e z + tanh - 1 ( c b ) ) a + c cosh ( e z ) + b sinh ( e z ) - c cosh ( e z ) + b ( sinh ( e z ) - 1 - c 2 b 2 ) a + b 1 - c 2 b 2 - b 1 - c 2 b 2 ( sinh ( e z + tanh - 1 ( c b ) ) + ) a - b 1 - c 2 b 2 ) z 1 a b e z c e z 1 2 -1 1 b 1 -1 c 2 b 2 -1 1 2 e -1 2 AppellF1 1 2 1 2 1 2 3 2 a c e z b e z a b 1 -1 c 2 b 2 -1 1 2 -1 a c e z b e z a -1 b 1 -1 c 2 b 2 -1 1 2 -1 e z c b -1 a c e z b e z 1 2 -1 c e z b e z -1 1 -1 c 2 b 2 -1 1 2 a b 1 -1 c 2 b 2 -1 1 2 -1 1 2 -1 b 1 -1 c 2 b 2 -1 1 2 e z c b -1 a -1 b 1 -1 c 2 b 2 -1 1 2 -1 1 2 [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["\[Integral]", RowBox[List[FractionBox["1", SqrtBox[RowBox[List["a_", "+", RowBox[List["b_", " ", RowBox[List["Sinh", "[", RowBox[List["e_", " ", "z_"]], "]"]]]], "+", RowBox[List["c_", " ", RowBox[List["Cosh", "[", RowBox[List["e_", " ", "z_"]], "]"]]]]]]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List["2", " ", RowBox[List["AppellF1", "[", RowBox[List[FractionBox["1", "2"], ",", FractionBox["1", "2"], ",", FractionBox["1", "2"], ",", FractionBox["3", "2"], ",", FractionBox[RowBox[List["a", "+", RowBox[List["c", " ", RowBox[List["Cosh", "[", RowBox[List["e", " ", "z"]], "]"]]]], "+", RowBox[List["b", " ", RowBox[List["Sinh", "[", RowBox[List["e", " ", "z"]], "]"]]]]]], RowBox[List["a", "+", RowBox[List["\[ImaginaryI]", " ", "b", " ", SqrtBox[RowBox[List["1", "-", FractionBox[SuperscriptBox["c", "2"], SuperscriptBox["b", "2"]]]]]]]]]], ",", FractionBox[RowBox[List["a", "+", RowBox[List["c", " ", RowBox[List["Cosh", "[", RowBox[List["e", " ", "z"]], "]"]]]], "+", RowBox[List["b", " ", RowBox[List["Sinh", "[", RowBox[List["e", " ", "z"]], "]"]]]]]], RowBox[List["a", "-", RowBox[List["\[ImaginaryI]", " ", "b", " ", SqrtBox[RowBox[List["1", "-", FractionBox[SuperscriptBox["c", "2"], SuperscriptBox["b", "2"]]]]]]]]]]]], "]"]], " ", RowBox[List["Sech", "[", RowBox[List[RowBox[List["e", " ", "z"]], "+", RowBox[List["ArcTanh", "[", FractionBox["c", "b"], "]"]]]], "]"]], " ", SqrtBox[RowBox[List["a", "+", RowBox[List["c", " ", RowBox[List["Cosh", "[", RowBox[List["e", " ", "z"]], "]"]]]], "+", RowBox[List["b", " ", RowBox[List["Sinh", "[", RowBox[List["e", " ", "z"]], "]"]]]]]]], " ", SqrtBox[RowBox[List["-", FractionBox[RowBox[List[RowBox[List["c", " ", RowBox[List["Cosh", "[", RowBox[List["e", " ", "z"]], "]"]]]], "+", RowBox[List["b", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", SqrtBox[RowBox[List["1", "-", FractionBox[SuperscriptBox["c", "2"], SuperscriptBox["b", "2"]]]]]]], "+", RowBox[List["Sinh", "[", RowBox[List["e", " ", "z"]], "]"]]]], ")"]]]]]], RowBox[List["a", "+", RowBox[List["\[ImaginaryI]", " ", "b", " ", SqrtBox[RowBox[List["1", "-", FractionBox[SuperscriptBox["c", "2"], SuperscriptBox["b", "2"]]]]]]]]]]]]], " ", SqrtBox[RowBox[List["-", FractionBox[RowBox[List["b", " ", SqrtBox[RowBox[List["1", "-", FractionBox[SuperscriptBox["c", "2"], SuperscriptBox["b", "2"]]]]], " ", RowBox[List["(", RowBox[List["\[ImaginaryI]", "+", RowBox[List["Sinh", "[", RowBox[List[RowBox[List["e", " ", "z"]], "+", RowBox[List["ArcTanh", "[", FractionBox["c", "b"], "]"]]]], "]"]]]], ")"]]]], RowBox[List["a", "-", RowBox[List["\[ImaginaryI]", " ", "b", " ", SqrtBox[RowBox[List["1", "-", FractionBox[SuperscriptBox["c", "2"], SuperscriptBox["b", "2"]]]]]]]]]]]]]]], RowBox[List["b", " ", SqrtBox[RowBox[List["1", "-", FractionBox[SuperscriptBox["c", "2"], SuperscriptBox["b", "2"]]]]], " ", "e"]]]]]]]

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

 2002-12-18