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Sech




Mathematica Notation

Traditional Notation









Elementary Functions > Sech[z] > Integration > Indefinite integration > Involving functions of the direct function and hyperbolic functions > Involving powers of the direct function and hyperbolic functions > Involving algebraic functions of tanh





http://functions.wolfram.com/01.24.21.0419.01









  


  








Input Form





Integrate[(Tanh[z] (1 + (1 - 8 Tanh[z]^2)^(1/3)) Sech[z]^2)/ ((1 - 8 Tanh[z]^2)^2)^(1/3), z] == (3 (-9 + 7 Cosh[2 z]) Sech[z]^2 (4 + 2^(2/3) ((9 - 7 Cosh[2 z]) Sech[z]^2)^(1/3)))/ (128 ((1 - 8 Tanh[z]^2)^2)^(1/3))








Standard Form





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MathML Form







<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mo> &#8747; </mo> <mrow> <mfrac> <mrow> <mrow> <mi> tanh </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mroot> <mrow> <mn> 1 </mn> <mo> - </mo> <mrow> <mn> 8 </mn> <mo> &#8290; </mo> <mrow> <msup> <mi> tanh </mi> <mn> 2 </mn> </msup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> </mrow> <mn> 3 </mn> </mroot> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <msup> <mi> sech </mi> <mn> 2 </mn> </msup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mtext> </mtext> </mrow> <mroot> <msup> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mrow> <mn> 8 </mn> <mo> &#8290; </mo> <mrow> <msup> <mi> tanh </mi> <mn> 2 </mn> </msup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> <mn> 3 </mn> </mroot> </mfrac> <mo> &#8290; </mo> <mrow> <mo> &#8518; </mo> <mi> z </mi> </mrow> </mrow> </mrow> <mo> &#10869; </mo> <mfrac> <mrow> <mn> 3 </mn> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 7 </mn> <mo> &#8290; </mo> <mrow> <mi> cosh </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mn> 9 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <msup> <mi> sech </mi> <mn> 2 </mn> </msup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <msup> <mn> 2 </mn> <mrow> <mn> 2 </mn> <mo> / </mo> <mn> 3 </mn> </mrow> </msup> <mo> &#8290; </mo> <mroot> <mrow> <mrow> <mo> ( </mo> <mrow> <mn> 9 </mn> <mo> - </mo> <mrow> <mn> 7 </mn> <mo> &#8290; </mo> <mrow> <mi> cosh </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <msup> <mi> sech </mi> <mn> 2 </mn> </msup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mn> 3 </mn> </mroot> </mrow> <mo> + </mo> <mn> 4 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mrow> <mn> 128 </mn> <mo> &#8290; </mo> <mroot> <msup> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mrow> <mn> 8 </mn> <mo> &#8290; </mo> <mrow> <msup> <mi> tanh </mi> <mn> 2 </mn> </msup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> <mn> 3 </mn> </mroot> </mrow> </mfrac> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <int /> <bvar> <ci> z </ci> </bvar> <apply> <times /> <apply> <tanh /> <ci> z </ci> </apply> <apply> <plus /> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 8 </cn> <apply> <power /> <apply> <tanh /> <ci> z </ci> </apply> <cn type='integer'> 2 </cn> </apply> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 3 </cn> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <power /> <apply> <sech /> <ci> z </ci> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <power /> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 8 </cn> <apply> <power /> <apply> <tanh /> <ci> z </ci> </apply> <cn type='integer'> 2 </cn> </apply> </apply> </apply> </apply> <cn type='integer'> 2 </cn> </apply> <cn type='rational'> 1 <sep /> 3 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 3 </cn> <apply> <plus /> <apply> <times /> <cn type='integer'> 7 </cn> <apply> <cosh /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> z </ci> </apply> </apply> </apply> <cn type='integer'> -9 </cn> </apply> <apply> <power /> <apply> <sech /> <ci> z </ci> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <plus /> <apply> <times /> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='rational'> 2 <sep /> 3 </cn> </apply> <apply> <power /> <apply> <times /> <apply> <plus /> <cn type='integer'> 9 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 7 </cn> <apply> <cosh /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> z </ci> </apply> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <sech /> <ci> z </ci> </apply> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 3 </cn> </apply> </apply> <cn type='integer'> 4 </cn> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 128 </cn> <apply> <power /> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 8 </cn> <apply> <power /> <apply> <tanh /> <ci> z </ci> </apply> <cn type='integer'> 2 </cn> </apply> </apply> </apply> </apply> <cn type='integer'> 2 </cn> </apply> <cn type='rational'> 1 <sep /> 3 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["\[Integral]", RowBox[List[FractionBox[RowBox[List[RowBox[List["Tanh", "[", "z_", "]"]], " ", RowBox[List["(", RowBox[List["1", "+", SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", RowBox[List["8", " ", SuperscriptBox[RowBox[List["Tanh", "[", "z_", "]"]], "2"]]]]], ")"]], RowBox[List["1", "/", "3"]]]]], ")"]], " ", SuperscriptBox[RowBox[List["Sech", "[", "z_", "]"]], "2"]]], SuperscriptBox[RowBox[List["(", SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", RowBox[List["8", " ", SuperscriptBox[RowBox[List["Tanh", "[", "z_", "]"]], "2"]]]]], ")"]], "2"], ")"]], RowBox[List["1", "/", "3"]]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List["3", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "9"]], "+", RowBox[List["7", " ", RowBox[List["Cosh", "[", RowBox[List["2", " ", "z"]], "]"]]]]]], ")"]], " ", SuperscriptBox[RowBox[List["Sech", "[", "z", "]"]], "2"], " ", RowBox[List["(", RowBox[List["4", "+", RowBox[List[SuperscriptBox["2", RowBox[List["2", "/", "3"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["(", RowBox[List["9", "-", RowBox[List["7", " ", RowBox[List["Cosh", "[", RowBox[List["2", " ", "z"]], "]"]]]]]], ")"]], " ", SuperscriptBox[RowBox[List["Sech", "[", "z", "]"]], "2"]]], ")"]], RowBox[List["1", "/", "3"]]]]]]], ")"]]]], RowBox[List["128", " ", SuperscriptBox[RowBox[List["(", SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", RowBox[List["8", " ", SuperscriptBox[RowBox[List["Tanh", "[", "z", "]"]], "2"]]]]], ")"]], "2"], ")"]], RowBox[List["1", "/", "3"]]]]]]]]]]


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





2002-12-18