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http://functions.wolfram.com/01.20.21.4137.01
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Integrate[Sqrt[Sinh[z]^4 Cosh[z]], z] ==
(2 Csch[z]^2 Sqrt[Cosh[z] Sinh[z]^4] (2 I EllipticE[(I z)/2, 2] +
Cosh[z]^(3/2) Sinh[z]))/(5 Sqrt[Cosh[z]])
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Cell[BoxData[RowBox[List[RowBox[List["\[Integral]", RowBox[List[SqrtBox[RowBox[List[SuperscriptBox[RowBox[List["Sinh", "[", "z", "]"]], "4"], " ", RowBox[List["Cosh", "[", "z", "]"]]]]], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", FractionBox[RowBox[List["2", " ", SuperscriptBox[RowBox[List["Csch", "[", "z", "]"]], "2"], " ", SqrtBox[RowBox[List[RowBox[List["Cosh", "[", "z", "]"]], " ", SuperscriptBox[RowBox[List["Sinh", "[", "z", "]"]], "4"]]]], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "\[ImaginaryI]", " ", RowBox[List["EllipticE", "[", RowBox[List[FractionBox[RowBox[List["\[ImaginaryI]", " ", "z"]], "2"], ",", "2"]], "]"]]]], "+", RowBox[List[SuperscriptBox[RowBox[List["Cosh", "[", "z", "]"]], RowBox[List["3", "/", "2"]]], " ", RowBox[List["Sinh", "[", "z", "]"]]]]]], ")"]]]], RowBox[List["5", " ", SqrtBox[RowBox[List["Cosh", "[", "z", "]"]]]]]]]]]]
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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> <msqrt> <mrow> <mrow> <msup> <mi> sinh </mi> <mn> 4 </mn> </msup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> cosh </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> </msqrt> <mo> ⁢ </mo> <mrow> <mo> ⅆ </mo> <mi> z </mi> </mrow> </mrow> </mrow> <mo> ⩵ </mo> <mrow> <mfrac> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mrow> <msup> <mi> csch </mi> <mn> 2 </mn> </msup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msqrt> <mrow> <mrow> <mi> cosh </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <msup> <mi> sinh </mi> <mn> 4 </mn> </msup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> </msqrt> </mrow> <mrow> <mn> 5 </mn> <mo> ⁢ </mo> <mrow> <msup> <mi> cosh </mi> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </msup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> </mfrac> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mi> sinh </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <msup> <mi> cosh </mi> <mfrac> <mn> 3 </mn> <mn> 2 </mn> </mfrac> </msup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> ⅈ </mi> <mo> ⁢ </mo> <mrow> <mi> E </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mfrac> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mn> 2 </mn> </mfrac> <mo> ❘ </mo> <mn> 2 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <int /> <bvar> <ci> z </ci> </bvar> <apply> <power /> <apply> <times /> <apply> <power /> <apply> <sinh /> <ci> z </ci> </apply> <cn type='integer'> 4 </cn> </apply> <apply> <cosh /> <ci> z </ci> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <apply> <csch /> <ci> z </ci> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <times /> <apply> <cosh /> <ci> z </ci> </apply> <apply> <power /> <apply> <sinh /> <ci> z </ci> </apply> <cn type='integer'> 4 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 5 </cn> <apply> <power /> <apply> <cosh /> <ci> z </ci> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <sinh /> <ci> z </ci> </apply> <apply> <power /> <apply> <cosh /> <ci> z </ci> </apply> <cn type='rational'> 3 <sep /> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <imaginaryi /> <apply> <ci> EllipticE </ci> <apply> <times /> <imaginaryi /> <ci> z </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='integer'> 2 </cn> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>
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| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["\[Integral]", RowBox[List[SqrtBox[RowBox[List[SuperscriptBox[RowBox[List["Sinh", "[", "z_", "]"]], "4"], " ", RowBox[List["Cosh", "[", "z_", "]"]]]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List["2", " ", SuperscriptBox[RowBox[List["Csch", "[", "z", "]"]], "2"], " ", SqrtBox[RowBox[List[RowBox[List["Cosh", "[", "z", "]"]], " ", SuperscriptBox[RowBox[List["Sinh", "[", "z", "]"]], "4"]]]], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "\[ImaginaryI]", " ", RowBox[List["EllipticE", "[", RowBox[List[FractionBox[RowBox[List["\[ImaginaryI]", " ", "z"]], "2"], ",", "2"]], "]"]]]], "+", RowBox[List[SuperscriptBox[RowBox[List["Cosh", "[", "z", "]"]], RowBox[List["3", "/", "2"]]], " ", RowBox[List["Sinh", "[", "z", "]"]]]]]], ")"]]]], RowBox[List["5", " ", SqrtBox[RowBox[List["Cosh", "[", "z", "]"]]]]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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