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http://functions.wolfram.com/01.24.21.0417.01
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Integrate[(Tanh[z]^2 Sech[z]^12)^(1/3), z] ==
(3/110) Cosh[z] (13 Sinh[z] + 3 Sinh[3 z]) (Sech[z]^12 Tanh[z]^2)^(1/3)
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Cell[BoxData[RowBox[List[RowBox[List["\[Integral]", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["Tanh", "[", "z", "]"]], "2"], SuperscriptBox[RowBox[List["Sech", "[", "z", "]"]], "12"]]], " ", ")"]], RowBox[List["1", "/", "3"]]], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[FractionBox["3", "110"], " ", RowBox[List["Cosh", "[", "z", "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["13", " ", RowBox[List["Sinh", "[", "z", "]"]]]], "+", RowBox[List["3", " ", RowBox[List["Sinh", "[", RowBox[List["3", " ", "z"]], "]"]]]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["Sech", "[", "z", "]"]], "12"], " ", SuperscriptBox[RowBox[List["Tanh", "[", "z", "]"]], "2"]]], ")"]], RowBox[List["1", "/", "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> <mroot> <mrow> <mtext> </mtext> <mrow> <mrow> <msup> <mi> tanh </mi> <mn> 2 </mn> </msup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <msup> <mi> sech </mi> <mn> 12 </mn> </msup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> </mrow> <mn> 3 </mn> </mroot> <mo> ⁢ </mo> <mrow> <mo> ⅆ </mo> <mi> z </mi> </mrow> </mrow> </mrow> <mo> ⩵ </mo> <mrow> <mfrac> <mn> 3 </mn> <mn> 110 </mn> </mfrac> <mo> ⁢ </mo> <mrow> <mi> cosh </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 13 </mn> <mo> ⁢ </mo> <mrow> <mi> sinh </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mn> 3 </mn> <mo> ⁢ </mo> <mrow> <mi> sinh </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mn> 3 </mn> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mroot> <mrow> <mrow> <msup> <mi> sech </mi> <mn> 12 </mn> </msup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <msup> <mi> tanh </mi> <mn> 2 </mn> </msup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mn> 3 </mn> </mroot> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <int /> <bvar> <ci> z </ci> </bvar> <apply> <power /> <apply> <times /> <apply> <power /> <apply> <tanh /> <ci> z </ci> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <sech /> <ci> z </ci> </apply> <cn type='integer'> 12 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 3 </cn> </apply> </apply> <apply> <times /> <cn type='rational'> 3 <sep /> 110 </cn> <apply> <cosh /> <ci> z </ci> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 13 </cn> <apply> <sinh /> <ci> z </ci> </apply> </apply> <apply> <times /> <cn type='integer'> 3 </cn> <apply> <sinh /> <apply> <times /> <cn type='integer'> 3 </cn> <ci> z </ci> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <apply> <power /> <apply> <sech /> <ci> z </ci> </apply> <cn type='integer'> 12 </cn> </apply> <apply> <power /> <apply> <tanh /> <ci> z </ci> </apply> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 3 </cn> </apply> </apply> </apply> </annotation-xml> </semantics> </math>
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| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["\[Integral]", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["Tanh", "[", "z_", "]"]], "2"], " ", SuperscriptBox[RowBox[List["Sech", "[", "z_", "]"]], "12"]]], ")"]], RowBox[List["1", "/", "3"]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox["3", "110"], " ", RowBox[List["Cosh", "[", "z", "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["13", " ", RowBox[List["Sinh", "[", "z", "]"]]]], "+", RowBox[List["3", " ", RowBox[List["Sinh", "[", RowBox[List["3", " ", "z"]], "]"]]]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["Sech", "[", "z", "]"]], "12"], " ", SuperscriptBox[RowBox[List["Tanh", "[", "z", "]"]], "2"]]], ")"]], RowBox[List["1", "/", "3"]]]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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