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http://functions.wolfram.com/01.24.21.0393.01
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Integrate[Sqrt[Cosh[2 z]^3] Sech[z]^3, z] ==
-((1/(2 Cosh[2 z]^(3/2))) (Sqrt[Cosh[2 z]^3]
(-4 Sqrt[2] ArcSinh[Sqrt[2] Sinh[z]] +
5 ArcTanh[Sinh[z]/Sqrt[Cosh[2 z]]] + Sqrt[Cosh[2 z]] Sech[z] Tanh[z])))
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Cell[BoxData[RowBox[List[RowBox[List["\[Integral]", RowBox[List[RowBox[List[SqrtBox[SuperscriptBox[RowBox[List["Cosh", "[", RowBox[List["2", " ", "z"]], "]"]], "3"]], " ", SuperscriptBox[RowBox[List["Sech", "[", "z", "]"]], "3"]]], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List["-", RowBox[List[FractionBox["1", RowBox[List["2", " ", SuperscriptBox[RowBox[List["Cosh", "[", RowBox[List["2", " ", "z"]], "]"]], RowBox[List["3", "/", "2"]]]]]], RowBox[List["(", RowBox[List[SqrtBox[SuperscriptBox[RowBox[List["Cosh", "[", RowBox[List["2", " ", "z"]], "]"]], "3"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "4"]], " ", SqrtBox["2"], " ", RowBox[List["ArcSinh", "[", RowBox[List[SqrtBox["2"], " ", RowBox[List["Sinh", "[", "z", "]"]]]], "]"]]]], "+", RowBox[List["5", " ", RowBox[List["ArcTanh", "[", FractionBox[RowBox[List["Sinh", "[", "z", "]"]], SqrtBox[RowBox[List["Cosh", "[", RowBox[List["2", " ", "z"]], "]"]]]], "]"]]]], "+", RowBox[List[SqrtBox[RowBox[List["Cosh", "[", RowBox[List["2", " ", "z"]], "]"]]], " ", RowBox[List["Sech", "[", "z", "]"]], " ", RowBox[List["Tanh", "[", "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> <mrow> <msqrt> <mrow> <msup> <mi> cosh </mi> <mn> 3 </mn> </msup> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </msqrt> <mo> ⁢ </mo> <mrow> <msup> <mi> sech </mi> <mn> 3 </mn> </msup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mo> ⁢ </mo> <mrow> <mo> ⅆ </mo> <mi> z </mi> </mrow> </mrow> </mrow> <mo> ⩵ </mo> <mrow> <mo> - </mo> <mrow> <mfrac> <mn> 1 </mn> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mrow> <msup> <mi> cosh </mi> <mfrac> <mn> 3 </mn> <mn> 2 </mn> </mfrac> </msup> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mfrac> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <msqrt> <mrow> <msup> <mi> cosh </mi> <mn> 3 </mn> </msup> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </msqrt> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mo> - </mo> <mn> 4 </mn> </mrow> <mo> ⁢ </mo> <msqrt> <mn> 2 </mn> </msqrt> <mo> ⁢ </mo> <mrow> <msup> <mi> sinh </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mrow> <msqrt> <mn> 2 </mn> </msqrt> <mo> ⁢ </mo> <mrow> <mi> sinh </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mn> 5 </mn> <mo> ⁢ </mo> <mrow> <msup> <mi> tanh </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mfrac> <mrow> <mi> sinh </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mrow> <msup> <mi> cosh </mi> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </msup> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mfrac> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mrow> <msup> <mi> cosh </mi> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </msup> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> sech </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> tanh </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <int /> <bvar> <ci> z </ci> </bvar> <apply> <times /> <apply> <power /> <apply> <power /> <apply> <cosh /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> z </ci> </apply> </apply> <cn type='integer'> 3 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <sech /> <ci> z </ci> </apply> <cn type='integer'> 3 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <apply> <cosh /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> z </ci> </apply> </apply> <cn type='rational'> 3 <sep /> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <apply> <power /> <apply> <power /> <apply> <cosh /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> z </ci> </apply> </apply> <cn type='integer'> 3 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> -4 </cn> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <arcsinh /> <apply> <times /> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <sinh /> <ci> z </ci> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 5 </cn> <apply> <arctanh /> <apply> <times /> <apply> <sinh /> <ci> z </ci> </apply> <apply> <power /> <apply> <power /> <apply> <cosh /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> z </ci> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <power /> <apply> <cosh /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> z </ci> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <sech /> <ci> z </ci> </apply> <apply> <tanh /> <ci> z </ci> </apply> </apply> </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[RowBox[List[SqrtBox[SuperscriptBox[RowBox[List["Cosh", "[", RowBox[List["2", " ", "z_"]], "]"]], "3"]], " ", SuperscriptBox[RowBox[List["Sech", "[", "z_", "]"]], "3"]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List["-", FractionBox[RowBox[List[SqrtBox[SuperscriptBox[RowBox[List["Cosh", "[", RowBox[List["2", " ", "z"]], "]"]], "3"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "4"]], " ", SqrtBox["2"], " ", RowBox[List["ArcSinh", "[", RowBox[List[SqrtBox["2"], " ", RowBox[List["Sinh", "[", "z", "]"]]]], "]"]]]], "+", RowBox[List["5", " ", RowBox[List["ArcTanh", "[", FractionBox[RowBox[List["Sinh", "[", "z", "]"]], SqrtBox[RowBox[List["Cosh", "[", RowBox[List["2", " ", "z"]], "]"]]]], "]"]]]], "+", RowBox[List[SqrtBox[RowBox[List["Cosh", "[", RowBox[List["2", " ", "z"]], "]"]]], " ", RowBox[List["Sech", "[", "z", "]"]], " ", RowBox[List["Tanh", "[", "z", "]"]]]]]], ")"]]]], RowBox[List["2", " ", SuperscriptBox[RowBox[List["Cosh", "[", RowBox[List["2", " ", "z"]], "]"]], RowBox[List["3", "/", "2"]]]]]]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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