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http://functions.wolfram.com/01.20.21.4399.01
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Integrate[Sinh[e z]/Sqrt[(a Sinh[e z] + b Cosh[e z])^3], z] ==
(2 Sech[e z + ArcTanh[b/a]] (b Cosh[e z] + a Sinh[e z])
(a Sqrt[Cosh[e z + ArcTanh[b/a]]^2] HypergeometricPFQ[{1/4, 1/2}, {5/4},
-Sinh[e z + ArcTanh[b/a]]^2] (b Cosh[e z] + a Sinh[e z]) +
b (a Cosh[e z] + b Sinh[e z])))/(a (a - b) (a + b) Sqrt[1 - b^2/a^2] e
Sqrt[(b Cosh[e z] + a Sinh[e z])^3])
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Cell[BoxData[RowBox[List[RowBox[List["\[Integral]", RowBox[List[FractionBox[RowBox[List["Sinh", "[", RowBox[List["e", " ", "z"]], "]"]], SqrtBox[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["a", " ", RowBox[List["Sinh", "[", RowBox[List["e", " ", "z"]], "]"]]]], "+", RowBox[List["b", " ", RowBox[List["Cosh", "[", RowBox[List["e", " ", "z"]], "]"]]]]]], ")"]], "3"]]], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[RowBox[List["(", RowBox[List["2", " ", RowBox[List["Sech", "[", RowBox[List[RowBox[List["e", " ", "z"]], "+", RowBox[List["ArcTanh", "[", FractionBox["b", "a"], "]"]]]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["b", " ", RowBox[List["Cosh", "[", RowBox[List["e", " ", "z"]], "]"]]]], "+", RowBox[List["a", " ", RowBox[List["Sinh", "[", RowBox[List["e", " ", "z"]], "]"]]]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["a", " ", SqrtBox[SuperscriptBox[RowBox[List["Cosh", "[", RowBox[List[RowBox[List["e", " ", "z"]], "+", RowBox[List["ArcTanh", "[", FractionBox["b", "a"], "]"]]]], "]"]], "2"]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[FractionBox["1", "4"], ",", FractionBox["1", "2"]]], "}"]], ",", RowBox[List["{", FractionBox["5", "4"], "}"]], ",", RowBox[List["-", SuperscriptBox[RowBox[List["Sinh", "[", RowBox[List[RowBox[List["e", " ", "z"]], "+", RowBox[List["ArcTanh", "[", FractionBox["b", "a"], "]"]]]], "]"]], "2"]]]]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["b", " ", RowBox[List["Cosh", "[", RowBox[List["e", " ", "z"]], "]"]]]], "+", RowBox[List["a", " ", RowBox[List["Sinh", "[", RowBox[List["e", " ", "z"]], "]"]]]]]], ")"]]]], "+", RowBox[List["b", " ", RowBox[List["(", RowBox[List[RowBox[List["a", " ", RowBox[List["Cosh", "[", RowBox[List["e", " ", "z"]], "]"]]]], "+", RowBox[List["b", " ", RowBox[List["Sinh", "[", RowBox[List["e", " ", "z"]], "]"]]]]]], ")"]]]]]], ")"]]]], ")"]], "/", RowBox[List["(", RowBox[List["a", " ", RowBox[List["(", RowBox[List["a", "-", "b"]], ")"]], " ", RowBox[List["(", RowBox[List["a", "+", "b"]], ")"]], " ", SqrtBox[RowBox[List["1", "-", FractionBox[SuperscriptBox["b", "2"], SuperscriptBox["a", "2"]]]]], " ", "e", " ", SqrtBox[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["b", " ", RowBox[List["Cosh", "[", RowBox[List["e", " ", "z"]], "]"]]]], "+", RowBox[List["a", " ", RowBox[List["Sinh", "[", RowBox[List["e", " ", "z"]], "]"]]]]]], ")"]], "3"]]]], ")"]]]]]]]]
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<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mo> ∫ </mo> <mrow> <mfrac> <mrow> <mi> sinh </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> e </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <msqrt> <msup> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> a </mi> <mo> ⁢ </mo> <mrow> <mi> sinh </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> e </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mi> b </mi> <mo> ⁢ </mo> <mrow> <mi> cosh </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> e </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <mn> 3 </mn> </msup> </msqrt> </mfrac> <mo> ⁢ </mo> <mrow> <mo> ⅆ </mo> <mi> z </mi> </mrow> </mrow> </mrow> <mo> ⩵ </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mrow> <mi> sech </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mrow> <mi> e </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> + </mo> <mrow> <msup> <mi> tanh </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mfrac> <mi> b </mi> <mi> a </mi> </mfrac> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> b </mi> <mo> ⁢ </mo> <mrow> <mi> cosh </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> e </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mi> a </mi> <mo> ⁢ </mo> <mrow> <mi> sinh </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> e </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> a </mi> <mo> ⁢ </mo> <msqrt> <mrow> <msup> <mi> cosh </mi> <mn> 2 </mn> </msup> <mo> ( </mo> <mrow> <mrow> <mi> e </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> + </mo> <mrow> <msup> <mi> tanh </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mfrac> <mi> b </mi> <mi> a </mi> </mfrac> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </msqrt> <mo> ⁢ </mo> <semantics> <mrow> <mrow> <msub> <mo>   </mo> <mn> 2 </mn> </msub> <msub> <mi> F </mi> <mn> 1 </mn> </msub> </mrow> <mo> ⁡ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mfrac> <mn> 1 </mn> <mn> 4 </mn> </mfrac> <mo> , </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> <mo> ; </mo> <mfrac> <mn> 5 </mn> <mn> 4 </mn> </mfrac> <mo> ; </mo> <mrow> <mo> - </mo> <mrow> <msup> <mi> sinh </mi> <mn> 2 </mn> </msup> <mo> ( </mo> <mrow> <mrow> <mi> e </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> + </mo> <mrow> <msup> <mi> tanh </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mfrac> <mi> b </mi> <mi> a </mi> </mfrac> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation encoding='Mathematica'> 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[FractionBox["1", "4"], HypergeometricPFQ], ",", TagBox[FractionBox["1", "2"], HypergeometricPFQ]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ], ";", TagBox[TagBox[TagBox[FractionBox["5", "4"], HypergeometricPFQ], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ], ";", TagBox[RowBox[List["-", RowBox[List[SuperscriptBox["sinh", "2"], "(", RowBox[List[RowBox[List["e", " ", "z"]], "+", RowBox[List[SuperscriptBox["tanh", RowBox[List["-", "1"]]], "(", FractionBox["b", "a"], ")"]]]], ")"]]]], HypergeometricPFQ]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]]], HypergeometricPFQ] </annotation> </semantics> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> b </mi> <mo> ⁢ </mo> <mrow> <mi> cosh </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> e </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mi> a </mi> <mo> ⁢ </mo> <mrow> <mi> sinh </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> e </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mi> b </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> a </mi> <mo> ⁢ </mo> <mrow> <mi> cosh </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> e </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mi> b </mi> <mo> ⁢ </mo> <mrow> <mi> sinh </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> e </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> / </mo> <mrow> <mo> ( </mo> <mrow> <mi> a </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> a </mi> <mo> - </mo> <mi> b </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> a </mi> <mo> + </mo> <mi> b </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msqrt> <mrow> <mn> 1 </mn> <mo> - </mo> <mfrac> <msup> <mi> b </mi> <mn> 2 </mn> </msup> <msup> <mi> a </mi> <mn> 2 </mn> </msup> </mfrac> </mrow> </msqrt> <mo> ⁢ </mo> <mi> e </mi> <mo> ⁢ </mo> <msqrt> <msup> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> b </mi> <mo> ⁢ </mo> <mrow> <mi> cosh </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> e </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mi> a </mi> <mo> ⁢ </mo> <mrow> <mi> sinh </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> e </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <mn> 3 </mn> </msup> </msqrt> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <int /> <bvar> <ci> z </ci> </bvar> <apply> <times /> <apply> <sinh /> <apply> <times /> <ci> e </ci> <ci> z </ci> </apply> </apply> <apply> <power /> <apply> <power /> <apply> <power /> <apply> <plus /> <apply> <times /> <ci> a </ci> <apply> <sinh /> <apply> <times /> <ci> e </ci> <ci> z </ci> </apply> </apply> </apply> <apply> <times /> <ci> b </ci> <apply> <cosh /> <apply> <times /> <ci> e </ci> <ci> z </ci> </apply> </apply> </apply> </apply> <cn type='integer'> 3 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <sech /> <apply> <plus /> <apply> <times /> <ci> e </ci> <ci> z </ci> </apply> <apply> <arctanh /> <apply> <times /> <ci> b </ci> <apply> <power /> <ci> a </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <plus /> <apply> <times /> <ci> b </ci> <apply> <cosh /> <apply> <times /> <ci> e </ci> <ci> z </ci> </apply> </apply> </apply> <apply> <times /> <ci> a </ci> <apply> <sinh /> <apply> <times /> <ci> e </ci> <ci> z </ci> </apply> </apply> </apply> </apply> <apply> <plus /> <apply> <times /> <ci> a </ci> <apply> <power /> <apply> <power /> <apply> <cosh /> <apply> <plus /> <apply> <times /> <ci> e </ci> <ci> z </ci> </apply> <apply> <arctanh /> <apply> <times /> <ci> b </ci> <apply> <power /> <ci> a </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> <cn type='integer'> 2 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <ci> HypergeometricPFQ </ci> <list> <cn type='rational'> 1 <sep /> 4 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </list> <list> <cn type='rational'> 5 <sep /> 4 </cn> </list> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <apply> <sinh /> <apply> <plus /> <apply> <times /> <ci> e </ci> <ci> z </ci> </apply> <apply> <arctanh /> <apply> <times /> <ci> b </ci> <apply> <power /> <ci> a </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <plus /> <apply> <times /> <ci> b </ci> <apply> <cosh /> <apply> <times /> <ci> e </ci> <ci> z </ci> </apply> </apply> </apply> <apply> <times /> <ci> a </ci> <apply> <sinh /> <apply> <times /> <ci> e </ci> <ci> z </ci> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <ci> b </ci> <apply> <plus /> <apply> <times /> <ci> a </ci> <apply> <cosh /> <apply> <times /> <ci> e </ci> <ci> z </ci> </apply> </apply> </apply> <apply> <times /> <ci> b </ci> <apply> <sinh /> <apply> <times /> <ci> e </ci> <ci> z </ci> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <ci> a </ci> <apply> <plus /> <ci> a </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> b </ci> </apply> </apply> <apply> <plus /> <ci> a </ci> <ci> b </ci> </apply> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <power /> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <power /> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <ci> e </ci> <apply> <power /> <apply> <power /> <apply> <plus /> <apply> <times /> <ci> b </ci> <apply> <cosh /> <apply> <times /> <ci> e </ci> <ci> z </ci> </apply> </apply> </apply> <apply> <times /> <ci> a </ci> <apply> <sinh /> <apply> <times /> <ci> e </ci> <ci> z </ci> </apply> </apply> </apply> </apply> <cn type='integer'> 3 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </annotation-xml> </semantics> </math>
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Date Added to functions.wolfram.com (modification date)
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