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 Cosh

 http://functions.wolfram.com/01.20.21.4390.01

 Input Form

 Integrate[Sqrt[a Sinh[e z] + b Cosh[e z]], z] == (a (a^2 - b^2) HypergeometricPFQ[{-(1/2), -(1/4)}, {3/4}, Cosh[e z + ArcTanh[a/b]]^2] Sinh[e z + ArcTanh[a/b]] + Sqrt[-Sinh[e z + ArcTanh[a/b]]^2] (2 Sqrt[1 - a^2/b^2] b^3 Cosh[e z] + 2 b (a^2 - b^2) Cosh[e z + ArcTanh[a/b]] + a (2 Sqrt[1 - a^2/b^2] b^2 Sinh[e z] + (-a^2 + b^2) Sinh[e z + ArcTanh[a/b]])))/(a Sqrt[1 - a^2/b^2] b e Sqrt[b Cosh[e z] + a Sinh[e z]] Sqrt[-Sinh[e z + ArcTanh[a/b]]^2])

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List["\[Integral]", RowBox[List[SqrtBox[RowBox[List[RowBox[List["a", " ", RowBox[List["Sinh", "[", RowBox[List["e", " ", "z"]], "]"]]]], "+", RowBox[List["b", " ", RowBox[List["Cosh", "[", RowBox[List["e", " ", "z"]], "]"]]]]]]], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["a", " ", RowBox[List["(", RowBox[List[SuperscriptBox["a", "2"], "-", SuperscriptBox["b", "2"]]], ")"]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], ",", RowBox[List["-", FractionBox["1", "4"]]]]], "}"]], ",", RowBox[List["{", FractionBox["3", "4"], "}"]], ",", SuperscriptBox[RowBox[List["Cosh", "[", RowBox[List[RowBox[List["e", " ", "z"]], "+", RowBox[List["ArcTanh", "[", FractionBox["a", "b"], "]"]]]], "]"]], "2"]]], "]"]], " ", RowBox[List["Sinh", "[", RowBox[List[RowBox[List["e", " ", "z"]], "+", RowBox[List["ArcTanh", "[", FractionBox["a", "b"], "]"]]]], "]"]]]], "+", RowBox[List[SqrtBox[RowBox[List["-", SuperscriptBox[RowBox[List["Sinh", "[", RowBox[List[RowBox[List["e", " ", "z"]], "+", RowBox[List["ArcTanh", "[", FractionBox["a", "b"], "]"]]]], "]"]], "2"]]]], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", SqrtBox[RowBox[List["1", "-", FractionBox[SuperscriptBox["a", "2"], SuperscriptBox["b", "2"]]]]], " ", SuperscriptBox["b", "3"], " ", RowBox[List["Cosh", "[", RowBox[List["e", " ", "z"]], "]"]]]], "+", RowBox[List["2", " ", "b", " ", RowBox[List["(", RowBox[List[SuperscriptBox["a", "2"], "-", SuperscriptBox["b", "2"]]], ")"]], " ", RowBox[List["Cosh", "[", RowBox[List[RowBox[List["e", " ", "z"]], "+", RowBox[List["ArcTanh", "[", FractionBox["a", "b"], "]"]]]], "]"]]]], "+", RowBox[List["a", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", SqrtBox[RowBox[List["1", "-", FractionBox[SuperscriptBox["a", "2"], SuperscriptBox["b", "2"]]]]], " ", SuperscriptBox["b", "2"], " ", RowBox[List["Sinh", "[", RowBox[List["e", " ", "z"]], "]"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", SuperscriptBox["a", "2"]]], "+", SuperscriptBox["b", "2"]]], ")"]], " ", RowBox[List["Sinh", "[", RowBox[List[RowBox[List["e", " ", "z"]], "+", RowBox[List["ArcTanh", "[", FractionBox["a", "b"], "]"]]]], "]"]]]]]], ")"]]]]]], ")"]]]]]], ")"]], "/", RowBox[List["(", RowBox[List["a", " ", SqrtBox[RowBox[List["1", "-", FractionBox[SuperscriptBox["a", "2"], SuperscriptBox["b", "2"]]]]], " ", "b", " ", "e", " ", SqrtBox[RowBox[List[RowBox[List["b", " ", RowBox[List["Cosh", "[", RowBox[List["e", " ", "z"]], "]"]]]], "+", RowBox[List["a", " ", RowBox[List["Sinh", "[", RowBox[List["e", " ", "z"]], "]"]]]]]]], " ", SqrtBox[RowBox[List["-", SuperscriptBox[RowBox[List["Sinh", "[", RowBox[List[RowBox[List["e", " ", "z"]], "+", RowBox[List["ArcTanh", "[", FractionBox["a", "b"], "]"]]]], "]"]], "2"]]]]]], ")"]]]]]]]]

 MathML Form

 a sinh ( e z ) + b cosh ( e z ) z ( a ( a 2 - b 2 ) 2 F 1 ( - 1 2 , - 1 4 ; 3 4 ; cosh 2 ( e z + tanh - 1 ( a b ) ) ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["2", TraditionalForm]], SubscriptBox["F", FormBox["1", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[RowBox[List["-", FractionBox["1", "2"]]], HypergeometricPFQ], ",", TagBox[RowBox[List["-", FractionBox["1", "4"]]], HypergeometricPFQ]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ], ";", TagBox[TagBox[TagBox[FractionBox["3", "4"], HypergeometricPFQ], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ], ";", TagBox[RowBox[List[SuperscriptBox["cosh", "2"], "(", RowBox[List[RowBox[List["e", " ", "z"]], "+", RowBox[List[SuperscriptBox["tanh", RowBox[List["-", "1"]]], "(", FractionBox["a", "b"], ")"]]]], ")"]], HypergeometricPFQ]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]]], HypergeometricPFQ] sinh ( e z + tanh - 1 ( a b ) ) + - sinh 2 ( e z + tanh - 1 ( a b ) ) ( 2 1 - a 2 b 2 cosh ( e z ) b 3 + 2 ( a 2 - b 2 ) cosh ( e z + tanh - 1 ( a b ) ) b + a ( 2 1 - a 2 b 2 sinh ( e z ) b 2 + ( b 2 - a 2 ) sinh ( e z + tanh - 1 ( a b ) ) ) ) ) / ( a 1 - a 2 b 2 b e b cosh ( e z ) + a sinh ( e z ) - sinh 2 ( e z + tanh - 1 ( a b ) ) ) z a e z b e z 1 2 a a 2 -1 b 2 HypergeometricPFQ -1 1 2 -1 1 4 3 4 e z a b -1 2 e z a b -1 -1 e z a b -1 2 1 2 2 1 -1 a 2 b 2 -1 1 2 e z b 3 2 a 2 -1 b 2 e z a b -1 b a 2 1 -1 a 2 b 2 -1 1 2 e z b 2 b 2 -1 a 2 e z a b -1 a 1 -1 a 2 b 2 -1 1 2 b e b e z a e z 1 2 -1 e z a b -1 2 1 2 -1 [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["\[Integral]", RowBox[List[SqrtBox[RowBox[List[RowBox[List["a_", " ", RowBox[List["Sinh", "[", RowBox[List["e_", " ", "z_"]], "]"]]]], "+", RowBox[List["b_", " ", RowBox[List["Cosh", "[", RowBox[List["e_", " ", "z_"]], "]"]]]]]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List[RowBox[List["a", " ", RowBox[List["(", RowBox[List[SuperscriptBox["a", "2"], "-", SuperscriptBox["b", "2"]]], ")"]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], ",", RowBox[List["-", FractionBox["1", "4"]]]]], "}"]], ",", RowBox[List["{", FractionBox["3", "4"], "}"]], ",", SuperscriptBox[RowBox[List["Cosh", "[", RowBox[List[RowBox[List["e", " ", "z"]], "+", RowBox[List["ArcTanh", "[", FractionBox["a", "b"], "]"]]]], "]"]], "2"]]], "]"]], " ", RowBox[List["Sinh", "[", RowBox[List[RowBox[List["e", " ", "z"]], "+", RowBox[List["ArcTanh", "[", FractionBox["a", "b"], "]"]]]], "]"]]]], "+", RowBox[List[SqrtBox[RowBox[List["-", SuperscriptBox[RowBox[List["Sinh", "[", RowBox[List[RowBox[List["e", " ", "z"]], "+", RowBox[List["ArcTanh", "[", FractionBox["a", "b"], "]"]]]], "]"]], "2"]]]], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", SqrtBox[RowBox[List["1", "-", FractionBox[SuperscriptBox["a", "2"], SuperscriptBox["b", "2"]]]]], " ", SuperscriptBox["b", "3"], " ", RowBox[List["Cosh", "[", RowBox[List["e", " ", "z"]], "]"]]]], "+", RowBox[List["2", " ", "b", " ", RowBox[List["(", RowBox[List[SuperscriptBox["a", "2"], "-", SuperscriptBox["b", "2"]]], ")"]], " ", RowBox[List["Cosh", "[", RowBox[List[RowBox[List["e", " ", "z"]], "+", RowBox[List["ArcTanh", "[", FractionBox["a", "b"], "]"]]]], "]"]]]], "+", RowBox[List["a", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", SqrtBox[RowBox[List["1", "-", FractionBox[SuperscriptBox["a", "2"], SuperscriptBox["b", "2"]]]]], " ", SuperscriptBox["b", "2"], " ", RowBox[List["Sinh", "[", RowBox[List["e", " ", "z"]], "]"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", SuperscriptBox["a", "2"]]], "+", SuperscriptBox["b", "2"]]], ")"]], " ", RowBox[List["Sinh", "[", RowBox[List[RowBox[List["e", " ", "z"]], "+", RowBox[List["ArcTanh", "[", FractionBox["a", "b"], "]"]]]], "]"]]]]]], ")"]]]]]], ")"]]]]]], RowBox[List["a", " ", SqrtBox[RowBox[List["1", "-", FractionBox[SuperscriptBox["a", "2"], SuperscriptBox["b", "2"]]]]], " ", "b", " ", "e", " ", SqrtBox[RowBox[List[RowBox[List["b", " ", RowBox[List["Cosh", "[", RowBox[List["e", " ", "z"]], "]"]]]], "+", RowBox[List["a", " ", RowBox[List["Sinh", "[", RowBox[List["e", " ", "z"]], "]"]]]]]]], " ", SqrtBox[RowBox[List["-", SuperscriptBox[RowBox[List["Sinh", "[", RowBox[List[RowBox[List["e", " ", "z"]], "+", RowBox[List["ArcTanh", "[", FractionBox["a", "b"], "]"]]]], "]"]], "2"]]]]]]]]]]]

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

 2002-12-18