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 Cosh

 http://functions.wolfram.com/01.20.21.4410.01

 Input Form

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

 Standard Form

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

 MathML Form

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

 Rule Form

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

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

 2002-12-18