Wolfram Researchfunctions.wolfram.comOther Wolfram Sites
Search Site
Function CategoriesGraphics GalleryNotationsGeneral IdentitiesAbout This Site Email Comments

View Related Information In
The Documentation Center
MathWorld

Download All Formulas For This Function
Mathematica Notebook
PDF File

 

Developed with Mathematica -- Download a Free Trial Version
 











ArcSech






Mathematica Notation

Traditional Notation









Elementary Functions > ArcSech[z] > Integration > Indefinite integration > Involving one direct function and elementary functions > Involving power function > Involving power > Linear argument





http://functions.wolfram.com/01.30.21.0009.01









  


  










Input Form





Integrate[ArcSech[a z + b]/z, z] == -4 I ArcSin[Sqrt[1 - 1/b]/Sqrt[2]] ArcTanh[((1 + b) Tanh[(1/2) ArcSech[b + a z]])/Sqrt[1 - b^2]] - ArcSech[b + a z] Log[1 + E^(-2 ArcSech[b + a z])] + ArcSech[b + a z] Log[1 + (-1 + Sqrt[1 - b^2])/(b E^ArcSech[b + a z])] + 2 I ArcSin[Sqrt[1 - 1/b]/Sqrt[2]] Log[1 + (-1 + Sqrt[1 - b^2])/(b E^ArcSech[b + a z])] + ArcSech[b + a z] Log[1 - (1 + Sqrt[1 - b^2])/(b E^ArcSech[b + a z])] - 2 I ArcSin[Sqrt[1 - 1/b]/Sqrt[2]] Log[1 - (1 + Sqrt[1 - b^2])/(b E^ArcSech[b + a z])] + (1/2) PolyLog[2, -E^(-2 ArcSech[b + a z])] - PolyLog[2, -((-1 + Sqrt[1 - b^2])/(b E^ArcSech[b + a z]))] - PolyLog[2, (1 + Sqrt[1 - b^2])/(b E^ArcSech[b + a z])]










Standard Form





Cell[BoxData[RowBox[List[RowBox[List["\[Integral]", RowBox[List[FractionBox[RowBox[List["ArcSech", "[", RowBox[List[RowBox[List["a", " ", "z"]], "+", "b"]], "]"]], "z"], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[RowBox[List[RowBox[List["-", "4"]], " ", "\[ImaginaryI]", " ", RowBox[List["ArcSin", "[", FractionBox[SqrtBox[RowBox[List["1", "-", FractionBox["1", "b"]]]], SqrtBox["2"]], "]"]], " ", RowBox[List["ArcTanh", "[", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["1", "+", "b"]], ")"]], " ", RowBox[List["Tanh", "[", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["ArcSech", "[", RowBox[List["b", "+", RowBox[List["a", " ", "z"]]]], "]"]]]], "]"]]]], SqrtBox[RowBox[List["1", "-", SuperscriptBox["b", "2"]]]]], "]"]]]], "-", RowBox[List[RowBox[List["ArcSech", "[", RowBox[List["b", "+", RowBox[List["a", " ", "z"]]]], "]"]], " ", RowBox[List["Log", "[", RowBox[List["1", "+", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "2"]], " ", RowBox[List["ArcSech", "[", RowBox[List["b", "+", RowBox[List["a", " ", "z"]]]], "]"]]]]]]], "]"]]]], "+", RowBox[List[RowBox[List["ArcSech", "[", RowBox[List["b", "+", RowBox[List["a", " ", "z"]]]], "]"]], " ", RowBox[List["Log", "[", RowBox[List["1", "+", FractionBox[RowBox[List[RowBox[List["-", "1"]], "+", SqrtBox[RowBox[List["1", "-", SuperscriptBox["b", "2"]]]]]], RowBox[List["b", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["ArcSech", "[", RowBox[List["b", "+", RowBox[List["a", " ", "z"]]]], "]"]]]]]]]], "]"]]]], "+", RowBox[List["2", " ", "\[ImaginaryI]", " ", RowBox[List["ArcSin", "[", FractionBox[SqrtBox[RowBox[List["1", "-", FractionBox["1", "b"]]]], SqrtBox["2"]], "]"]], " ", RowBox[List["Log", "[", RowBox[List["1", "+", FractionBox[RowBox[List[RowBox[List["-", "1"]], "+", SqrtBox[RowBox[List["1", "-", SuperscriptBox["b", "2"]]]]]], RowBox[List["b", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["ArcSech", "[", RowBox[List["b", "+", RowBox[List["a", " ", "z"]]]], "]"]]]]]]]], "]"]]]], "+", RowBox[List[RowBox[List["ArcSech", "[", RowBox[List["b", "+", RowBox[List["a", " ", "z"]]]], "]"]], " ", RowBox[List["Log", "[", RowBox[List["1", "-", FractionBox[RowBox[List["1", "+", SqrtBox[RowBox[List["1", "-", SuperscriptBox["b", "2"]]]]]], RowBox[List["b", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["ArcSech", "[", RowBox[List["b", "+", RowBox[List["a", " ", "z"]]]], "]"]]]]]]]], "]"]]]], "-", RowBox[List["2", " ", "\[ImaginaryI]", " ", RowBox[List["ArcSin", "[", FractionBox[SqrtBox[RowBox[List["1", "-", FractionBox["1", "b"]]]], SqrtBox["2"]], "]"]], " ", RowBox[List["Log", "[", RowBox[List["1", "-", FractionBox[RowBox[List["1", "+", SqrtBox[RowBox[List["1", "-", SuperscriptBox["b", "2"]]]]]], RowBox[List["b", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["ArcSech", "[", RowBox[List["b", "+", RowBox[List["a", " ", "z"]]]], "]"]]]]]]]], "]"]]]], "+", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["PolyLog", "[", RowBox[List["2", ",", RowBox[List["-", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "2"]], " ", RowBox[List["ArcSech", "[", RowBox[List["b", "+", RowBox[List["a", " ", "z"]]]], "]"]]]]]]]]], "]"]]]], "-", RowBox[List["PolyLog", "[", RowBox[List["2", ",", RowBox[List["-", FractionBox[RowBox[List[RowBox[List["-", "1"]], "+", SqrtBox[RowBox[List["1", "-", SuperscriptBox["b", "2"]]]]]], RowBox[List["b", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["ArcSech", "[", RowBox[List["b", "+", RowBox[List["a", " ", "z"]]]], "]"]]]]]]]]]], "]"]], "-", RowBox[List["PolyLog", "[", RowBox[List["2", ",", FractionBox[RowBox[List["1", "+", SqrtBox[RowBox[List["1", "-", SuperscriptBox["b", "2"]]]]]], RowBox[List["b", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["ArcSech", "[", RowBox[List["b", "+", RowBox[List["a", " ", "z"]]]], "]"]]]]]]]], "]"]]]]]]]]










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> <msup> <mi> sech </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mrow> <mrow> <mi> a </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> + </mo> <mi> b </mi> </mrow> <mo> ) </mo> </mrow> <mi> z </mi> </mfrac> <mo> &#8290; </mo> <mrow> <mo> &#8518; </mo> <mi> z </mi> </mrow> </mrow> </mrow> <mo> &#10869; </mo> <mrow> <mrow> <mrow> <mo> - </mo> <mn> 4 </mn> </mrow> <mo> &#8290; </mo> <mi> &#8520; </mi> <mo> &#8290; </mo> <mrow> <msup> <mi> sin </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mrow> <mfrac> <mn> 1 </mn> <msqrt> <mn> 2 </mn> </msqrt> </mfrac> <mo> &#8290; </mo> <msqrt> <mrow> <mn> 1 </mn> <mo> - </mo> <mfrac> <mn> 1 </mn> <mi> b </mi> </mfrac> </mrow> </msqrt> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <msup> <mi> tanh </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mrow> <mfrac> <mrow> <mi> b </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <msqrt> <mrow> <mn> 1 </mn> <mo> - </mo> <msup> <mi> b </mi> <mn> 2 </mn> </msup> </mrow> </msqrt> </mfrac> <mo> &#8290; </mo> <mrow> <mi> tanh </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> &#8290; </mo> <mrow> <msup> <mi> sech </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mrow> <mi> b </mi> <mo> + </mo> <mrow> <mi> a </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mrow> <msup> <mi> sech </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mrow> <mi> b </mi> <mo> + </mo> <mrow> <mi> a </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> + </mo> <msup> <mi> &#8519; </mi> <mrow> <mrow> <mo> - </mo> <mn> 2 </mn> </mrow> <mo> &#8290; </mo> <mrow> <msup> <mi> sech </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mrow> <mi> b </mi> <mo> + </mo> <mrow> <mi> a </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </msup> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mrow> <msup> <mi> sech </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mrow> <mi> b </mi> <mo> + </mo> <mrow> <mi> a </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mfrac> <mrow> <msqrt> <mrow> <mn> 1 </mn> <mo> - </mo> <msup> <mi> b </mi> <mn> 2 </mn> </msup> </mrow> </msqrt> <mo> - </mo> <mn> 1 </mn> </mrow> <mrow> <mi> b </mi> <mo> &#8290; </mo> <msup> <mi> &#8519; </mi> <mrow> <msup> <mi> sech </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mrow> <mi> b </mi> <mo> + </mo> <mrow> <mi> a </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> </mrow> <mo> ) </mo> </mrow> </msup> </mrow> </mfrac> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> &#8520; </mi> <mo> &#8290; </mo> <mrow> <msup> <mi> sin </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mrow> <mfrac> <mn> 1 </mn> <msqrt> <mn> 2 </mn> </msqrt> </mfrac> <mo> &#8290; </mo> <msqrt> <mrow> <mn> 1 </mn> <mo> - </mo> <mfrac> <mn> 1 </mn> <mi> b </mi> </mfrac> </mrow> </msqrt> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mfrac> <mrow> <msqrt> <mrow> <mn> 1 </mn> <mo> - </mo> <msup> <mi> b </mi> <mn> 2 </mn> </msup> </mrow> </msqrt> <mo> - </mo> <mn> 1 </mn> </mrow> <mrow> <mi> b </mi> <mo> &#8290; </mo> <msup> <mi> &#8519; </mi> <mrow> <msup> <mi> sech </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mrow> <mi> b </mi> <mo> + </mo> <mrow> <mi> a </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> </mrow> <mo> ) </mo> </mrow> </msup> </mrow> </mfrac> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mrow> <msup> <mi> sech </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mrow> <mi> b </mi> <mo> + </mo> <mrow> <mi> a </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mfrac> <mrow> <msqrt> <mrow> <mn> 1 </mn> <mo> - </mo> <msup> <mi> b </mi> <mn> 2 </mn> </msup> </mrow> </msqrt> <mo> + </mo> <mn> 1 </mn> </mrow> <mrow> <mi> b </mi> <mo> &#8290; </mo> <msup> <mi> &#8519; </mi> <mrow> <msup> <mi> sech </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mrow> <mi> b </mi> <mo> + </mo> <mrow> <mi> a </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> </mrow> <mo> ) </mo> </mrow> </msup> </mrow> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> &#8520; </mi> <mo> &#8290; </mo> <mrow> <msup> <mi> sin </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mrow> <mfrac> <mn> 1 </mn> <msqrt> <mn> 2 </mn> </msqrt> </mfrac> <mo> &#8290; </mo> <msqrt> <mrow> <mn> 1 </mn> <mo> - </mo> <mfrac> <mn> 1 </mn> <mi> b </mi> </mfrac> </mrow> </msqrt> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mfrac> <mrow> <msqrt> <mrow> <mn> 1 </mn> <mo> - </mo> <msup> <mi> b </mi> <mn> 2 </mn> </msup> </mrow> </msqrt> <mo> + </mo> <mn> 1 </mn> </mrow> <mrow> <mi> b </mi> <mo> &#8290; </mo> <msup> <mi> &#8519; </mi> <mrow> <msup> <mi> sech </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mrow> <mi> b </mi> <mo> + </mo> <mrow> <mi> a </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> </mrow> <mo> ) </mo> </mrow> </msup> </mrow> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> &#8290; </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> <mo> - </mo> <msup> <mi> &#8519; </mi> <mrow> <mrow> <mo> - </mo> <mn> 2 </mn> </mrow> <mo> &#8290; </mo> <mrow> <msup> <mi> sech </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mrow> <mi> b </mi> <mo> + </mo> <mrow> <mi> a </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </msup> </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> <mfrac> <mrow> <msqrt> <mrow> <mn> 1 </mn> <mo> - </mo> <msup> <mi> b </mi> <mn> 2 </mn> </msup> </mrow> </msqrt> <mo> + </mo> <mn> 1 </mn> </mrow> <mrow> <mi> b </mi> <mo> &#8290; </mo> <msup> <mi> &#8519; </mi> <mrow> <msup> <mi> sech </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mrow> <mi> b </mi> <mo> + </mo> <mrow> <mi> a </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> </mrow> <mo> ) </mo> </mrow> </msup> </mrow> </mfrac> <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> <mo> - </mo> <mfrac> <mrow> <msqrt> <mrow> <mn> 1 </mn> <mo> - </mo> <msup> <mi> b </mi> <mn> 2 </mn> </msup> </mrow> </msqrt> <mo> - </mo> <mn> 1 </mn> </mrow> <mrow> <mi> b </mi> <mo> &#8290; </mo> <msup> <mi> &#8519; </mi> <mrow> <msup> <mi> sech </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mrow> <mi> b </mi> <mo> + </mo> <mrow> <mi> a </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> </mrow> <mo> ) </mo> </mrow> </msup> </mrow> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <int /> <bvar> <ci> z </ci> </bvar> <apply> <times /> <apply> <arcsech /> <apply> <plus /> <apply> <times /> <ci> a </ci> <ci> z </ci> </apply> <ci> b </ci> </apply> </apply> <apply> <power /> <ci> z </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> -4 </cn> <imaginaryi /> <apply> <arcsin /> <apply> <times /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <ci> b </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> <apply> <arctanh /> <apply> <times /> <apply> <times /> <apply> <plus /> <ci> b </ci> <cn type='integer'> 1 </cn> </apply> <apply> <power /> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <tanh /> <apply> <times /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <arcsech /> <apply> <plus /> <ci> b </ci> <apply> <times /> <ci> a </ci> <ci> z </ci> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <arcsech /> <apply> <plus /> <ci> b </ci> <apply> <times /> <ci> a </ci> <ci> z </ci> </apply> </apply> </apply> <apply> <ln /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <power /> <exponentiale /> <apply> <times /> <cn type='integer'> -2 </cn> <apply> <arcsech /> <apply> <plus /> <ci> b </ci> <apply> <times /> <ci> a </ci> <ci> z </ci> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <arcsech /> <apply> <plus /> <ci> b </ci> <apply> <times /> <ci> a </ci> <ci> z </ci> </apply> </apply> </apply> <apply> <ln /> <apply> <plus /> <apply> <times /> <apply> <plus /> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> <apply> <power /> <apply> <times /> <ci> b </ci> <apply> <power /> <exponentiale /> <apply> <arcsech /> <apply> <plus /> <ci> b </ci> <apply> <times /> <ci> a </ci> <ci> z </ci> </apply> </apply> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='integer'> 1 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <imaginaryi /> <apply> <arcsin /> <apply> <times /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <ci> b </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> <apply> <ln /> <apply> <plus /> <apply> <times /> <apply> <plus /> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> <apply> <power /> <apply> <times /> <ci> b </ci> <apply> <power /> <exponentiale /> <apply> <arcsech /> <apply> <plus /> <ci> b </ci> <apply> <times /> <ci> a </ci> <ci> z </ci> </apply> </apply> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='integer'> 1 </cn> </apply> </apply> </apply> <apply> <times /> <apply> <arcsech /> <apply> <plus /> <ci> b </ci> <apply> <times /> <ci> a </ci> <ci> z </ci> </apply> </apply> </apply> <apply> <ln /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <plus /> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <power /> <apply> <times /> <ci> b </ci> <apply> <power /> <exponentiale /> <apply> <arcsech /> <apply> <plus /> <ci> b </ci> <apply> <times /> <ci> a </ci> <ci> z </ci> </apply> </apply> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <imaginaryi /> <apply> <arcsin /> <apply> <times /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <ci> b </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> <apply> <ln /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <plus /> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <power /> <apply> <times /> <ci> b </ci> <apply> <power /> <exponentiale /> <apply> <arcsech /> <apply> <plus /> <ci> b </ci> <apply> <times /> <ci> a </ci> <ci> z </ci> </apply> </apply> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <ci> PolyLog </ci> <cn type='integer'> 2 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <exponentiale /> <apply> <times /> <cn type='integer'> -2 </cn> <apply> <arcsech /> <apply> <plus /> <ci> b </ci> <apply> <times /> <ci> a </ci> <ci> z </ci> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> PolyLog </ci> <cn type='integer'> 2 </cn> <apply> <times /> <apply> <plus /> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <power /> <apply> <times /> <ci> b </ci> <apply> <power /> <exponentiale /> <apply> <arcsech /> <apply> <plus /> <ci> b </ci> <apply> <times /> <ci> a </ci> <ci> z </ci> </apply> </apply> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> PolyLog </ci> <cn type='integer'> 2 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <plus /> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> <apply> <power /> <apply> <times /> <ci> b </ci> <apply> <power /> <exponentiale /> <apply> <arcsech /> <apply> <plus /> <ci> b </ci> <apply> <times /> <ci> a </ci> <ci> z </ci> </apply> </apply> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["\[Integral]", RowBox[List[FractionBox[RowBox[List["ArcSech", "[", RowBox[List[RowBox[List["a_", " ", "z_"]], "+", "b_"]], "]"]], "z_"], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List["-", "4"]], " ", "\[ImaginaryI]", " ", RowBox[List["ArcSin", "[", FractionBox[SqrtBox[RowBox[List["1", "-", FractionBox["1", "b"]]]], SqrtBox["2"]], "]"]], " ", RowBox[List["ArcTanh", "[", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["1", "+", "b"]], ")"]], " ", RowBox[List["Tanh", "[", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["ArcSech", "[", RowBox[List["b", "+", RowBox[List["a", " ", "z"]]]], "]"]]]], "]"]]]], SqrtBox[RowBox[List["1", "-", SuperscriptBox["b", "2"]]]]], "]"]]]], "-", RowBox[List[RowBox[List["ArcSech", "[", RowBox[List["b", "+", RowBox[List["a", " ", "z"]]]], "]"]], " ", RowBox[List["Log", "[", RowBox[List["1", "+", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "2"]], " ", RowBox[List["ArcSech", "[", RowBox[List["b", "+", RowBox[List["a", " ", "z"]]]], "]"]]]]]]], "]"]]]], "+", RowBox[List[RowBox[List["ArcSech", "[", RowBox[List["b", "+", RowBox[List["a", " ", "z"]]]], "]"]], " ", RowBox[List["Log", "[", RowBox[List["1", "+", FractionBox[RowBox[List[RowBox[List["-", "1"]], "+", SqrtBox[RowBox[List["1", "-", SuperscriptBox["b", "2"]]]]]], RowBox[List["b", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["ArcSech", "[", RowBox[List["b", "+", RowBox[List["a", " ", "z"]]]], "]"]]]]]]]], "]"]]]], "+", RowBox[List["2", " ", "\[ImaginaryI]", " ", RowBox[List["ArcSin", "[", FractionBox[SqrtBox[RowBox[List["1", "-", FractionBox["1", "b"]]]], SqrtBox["2"]], "]"]], " ", RowBox[List["Log", "[", RowBox[List["1", "+", FractionBox[RowBox[List[RowBox[List["-", "1"]], "+", SqrtBox[RowBox[List["1", "-", SuperscriptBox["b", "2"]]]]]], RowBox[List["b", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["ArcSech", "[", RowBox[List["b", "+", RowBox[List["a", " ", "z"]]]], "]"]]]]]]]], "]"]]]], "+", RowBox[List[RowBox[List["ArcSech", "[", RowBox[List["b", "+", RowBox[List["a", " ", "z"]]]], "]"]], " ", RowBox[List["Log", "[", RowBox[List["1", "-", FractionBox[RowBox[List["1", "+", SqrtBox[RowBox[List["1", "-", SuperscriptBox["b", "2"]]]]]], RowBox[List["b", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["ArcSech", "[", RowBox[List["b", "+", RowBox[List["a", " ", "z"]]]], "]"]]]]]]]], "]"]]]], "-", RowBox[List["2", " ", "\[ImaginaryI]", " ", RowBox[List["ArcSin", "[", FractionBox[SqrtBox[RowBox[List["1", "-", FractionBox["1", "b"]]]], SqrtBox["2"]], "]"]], " ", RowBox[List["Log", "[", RowBox[List["1", "-", FractionBox[RowBox[List["1", "+", SqrtBox[RowBox[List["1", "-", SuperscriptBox["b", "2"]]]]]], RowBox[List["b", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["ArcSech", "[", RowBox[List["b", "+", RowBox[List["a", " ", "z"]]]], "]"]]]]]]]], "]"]]]], "+", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["PolyLog", "[", RowBox[List["2", ",", RowBox[List["-", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "2"]], " ", RowBox[List["ArcSech", "[", RowBox[List["b", "+", RowBox[List["a", " ", "z"]]]], "]"]]]]]]]]], "]"]]]], "-", RowBox[List["PolyLog", "[", RowBox[List["2", ",", RowBox[List["-", FractionBox[RowBox[List[RowBox[List["-", "1"]], "+", SqrtBox[RowBox[List["1", "-", SuperscriptBox["b", "2"]]]]]], RowBox[List["b", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["ArcSech", "[", RowBox[List["b", "+", RowBox[List["a", " ", "z"]]]], "]"]]]]]]]]]], "]"]], "-", RowBox[List["PolyLog", "[", RowBox[List["2", ",", FractionBox[RowBox[List["1", "+", SqrtBox[RowBox[List["1", "-", SuperscriptBox["b", "2"]]]]]], RowBox[List["b", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["ArcSech", "[", RowBox[List["b", "+", RowBox[List["a", " ", "z"]]]], "]"]]]]]]]], "]"]]]]]]]]










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





2001-10-29