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ArcSech




Mathematica Notation

Traditional Notation









Elementary Functions > ArcSech[z] > Integration > Indefinite integration > Involving one direct function and elementary functions > Arguments involving hyperbolic functions > Involving cosh





http://functions.wolfram.com/01.30.21.0023.01









  


  








Input Form





Integrate[ArcSech[Cosh[z]], z] == z ArcSech[Cosh[z]] + I Coth[z/2] (z (Log[1 - I/E^z] - Log[1 + I/E^z]) + PolyLog[2, -I/E^z] - PolyLog[2, I/E^z]) Sqrt[-Tanh[z/2]^2]








Standard Form





Cell[BoxData[RowBox[List[RowBox[List["\[Integral]", RowBox[List[RowBox[List["ArcSech", "[", RowBox[List["Cosh", "[", "z", "]"]], "]"]], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[RowBox[List["z", " ", RowBox[List["ArcSech", "[", RowBox[List["Cosh", "[", "z", "]"]], "]"]]]], "+", RowBox[List["\[ImaginaryI]", " ", RowBox[List["Coth", "[", FractionBox["z", "2"], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["z", " ", RowBox[List["(", RowBox[List[RowBox[List["Log", "[", RowBox[List["1", "-", RowBox[List["\[ImaginaryI]", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["-", "z"]]]]]]], "]"]], "-", RowBox[List["Log", "[", RowBox[List["1", "+", RowBox[List["\[ImaginaryI]", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["-", "z"]]]]]]], "]"]]]], ")"]]]], "+", RowBox[List["PolyLog", "[", RowBox[List["2", ",", RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["-", "z"]]]]]]], "]"]], "-", RowBox[List["PolyLog", "[", RowBox[List["2", ",", RowBox[List["\[ImaginaryI]", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["-", "z"]]]]]]], "]"]]]], ")"]], " ", SqrtBox[RowBox[List["-", SuperscriptBox[RowBox[List["Tanh", "[", FractionBox["z", "2"], "]"]], "2"]]]]]]]]]]]]








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> <mrow> <msup> <mi> sech </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mrow> <mi> cosh </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> &#8518; </mo> <mi> z </mi> </mrow> </mrow> </mrow> <mo> &#10869; </mo> <mrow> <mrow> <mi> z </mi> <mo> &#8290; </mo> <mrow> <msup> <mi> sech </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mrow> <mi> cosh </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mrow> <mi> coth </mi> <mo> &#8289; </mo> <mo> ( </mo> <mfrac> <mi> z </mi> <mn> 2 </mn> </mfrac> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> z </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <msup> <mi> &#8519; </mi> <mrow> <mo> - </mo> <mi> z </mi> </mrow> </msup> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> - </mo> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> + </mo> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <msup> <mi> &#8519; </mi> <mrow> <mo> - </mo> <mi> z </mi> </mrow> </msup> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <msub> <semantics> <mi> Li </mi> <annotation-xml encoding='MathML-Content'> <ci> PolyLog </ci> </annotation-xml> </semantics> <mn> 2 </mn> </msub> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mi> &#8520; </mi> </mrow> <mo> &#8290; </mo> <msup> <mi> &#8519; </mi> <mrow> <mo> - </mo> <mi> z </mi> </mrow> </msup> </mrow> <mo> ) </mo> </mrow> <mo> - </mo> <mrow> <msub> <semantics> <mi> Li </mi> <annotation-xml encoding='MathML-Content'> <ci> PolyLog </ci> </annotation-xml> </semantics> <mn> 2 </mn> </msub> <mo> ( </mo> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <msup> <mi> &#8519; </mi> <mrow> <mo> - </mo> <mi> z </mi> </mrow> </msup> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msqrt> <mrow> <mrow> <mo> - </mo> <msup> <mi> tanh </mi> <mn> 2 </mn> </msup> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mfrac> <mi> z </mi> <mn> 2 </mn> </mfrac> <mo> ) </mo> </mrow> </mrow> </msqrt> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <int /> <bvar> <ci> z </ci> </bvar> <apply> <arcsech /> <apply> <cosh /> <ci> z </ci> </apply> </apply> </apply> <apply> <plus /> <apply> <times /> <ci> z </ci> <apply> <arcsech /> <apply> <cosh /> <ci> z </ci> </apply> </apply> </apply> <apply> <times /> <imaginaryi /> <apply> <coth /> <apply> <times /> <ci> z </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <plus /> <apply> <times /> <ci> z </ci> <apply> <plus /> <apply> <ln /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <imaginaryi /> <apply> <power /> <exponentiale /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> z </ci> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ln /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <imaginaryi /> <apply> <power /> <exponentiale /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> z </ci> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <ci> PolyLog </ci> <cn type='integer'> 2 </cn> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <imaginaryi /> </apply> <apply> <power /> <exponentiale /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> z </ci> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> PolyLog </ci> <cn type='integer'> 2 </cn> <apply> <times /> <imaginaryi /> <apply> <power /> <exponentiale /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> z </ci> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> tanh </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <ci> z </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["\[Integral]", RowBox[List[RowBox[List["ArcSech", "[", RowBox[List["Cosh", "[", "z_", "]"]], "]"]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["z", " ", RowBox[List["ArcSech", "[", RowBox[List["Cosh", "[", "z", "]"]], "]"]]]], "+", RowBox[List["\[ImaginaryI]", " ", RowBox[List["Coth", "[", FractionBox["z", "2"], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["z", " ", RowBox[List["(", RowBox[List[RowBox[List["Log", "[", RowBox[List["1", "-", RowBox[List["\[ImaginaryI]", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["-", "z"]]]]]]], "]"]], "-", RowBox[List["Log", "[", RowBox[List["1", "+", RowBox[List["\[ImaginaryI]", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["-", "z"]]]]]]], "]"]]]], ")"]]]], "+", RowBox[List["PolyLog", "[", RowBox[List["2", ",", RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["-", "z"]]]]]]], "]"]], "-", RowBox[List["PolyLog", "[", RowBox[List["2", ",", RowBox[List["\[ImaginaryI]", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["-", "z"]]]]]]], "]"]]]], ")"]], " ", SqrtBox[RowBox[List["-", SuperscriptBox[RowBox[List["Tanh", "[", FractionBox["z", "2"], "]"]], "2"]]]]]]]]]]]]


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





2001-10-29