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http://functions.wolfram.com/01.29.21.0030.01
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Integrate[ArcCsch[a z]^4, z] == (-(1/(2 a))) (Pi^4 - 2 ArcCsch[a z]^4 -
2 a z ArcCsch[a z]^4 - 8 ArcCsch[a z]^3 Log[1 + E^(-ArcCsch[a z])] +
8 ArcCsch[a z]^3 Log[1 - E^ArcCsch[a z]] +
24 ArcCsch[a z]^2 PolyLog[2, -E^(-ArcCsch[a z])] +
24 ArcCsch[a z]^2 PolyLog[2, E^ArcCsch[a z]] +
48 ArcCsch[a z] PolyLog[3, -E^(-ArcCsch[a z])] -
48 ArcCsch[a z] PolyLog[3, E^ArcCsch[a z]] +
48 PolyLog[4, -E^(-ArcCsch[a z])] + 48 PolyLog[4, E^ArcCsch[a z]])
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Cell[BoxData[RowBox[List[RowBox[List["\[Integral]", RowBox[List[SuperscriptBox[RowBox[List["ArcCsch", "[", RowBox[List["a", " ", "z"]], "]"]], "4"], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[RowBox[List["-", FractionBox["1", RowBox[List["2", " ", "a"]]]]], RowBox[List["(", RowBox[List[SuperscriptBox["\[Pi]", "4"], "-", RowBox[List["2", " ", SuperscriptBox[RowBox[List["ArcCsch", "[", RowBox[List["a", " ", "z"]], "]"]], "4"]]], "-", RowBox[List["2", " ", "a", " ", "z", " ", SuperscriptBox[RowBox[List["ArcCsch", "[", RowBox[List["a", " ", "z"]], "]"]], "4"]]], "-", RowBox[List["8", " ", SuperscriptBox[RowBox[List["ArcCsch", "[", RowBox[List["a", " ", "z"]], "]"]], "3"], " ", RowBox[List["Log", "[", RowBox[List["1", "+", SuperscriptBox["\[ExponentialE]", RowBox[List["-", RowBox[List["ArcCsch", "[", RowBox[List["a", " ", "z"]], "]"]]]]]]], "]"]]]], "+", RowBox[List["8", " ", SuperscriptBox[RowBox[List["ArcCsch", "[", RowBox[List["a", " ", "z"]], "]"]], "3"], " ", RowBox[List["Log", "[", RowBox[List["1", "-", SuperscriptBox["\[ExponentialE]", RowBox[List["ArcCsch", "[", RowBox[List["a", " ", "z"]], "]"]]]]], "]"]]]], "+", RowBox[List["24", " ", SuperscriptBox[RowBox[List["ArcCsch", "[", RowBox[List["a", " ", "z"]], "]"]], "2"], " ", RowBox[List["PolyLog", "[", RowBox[List["2", ",", RowBox[List["-", SuperscriptBox["\[ExponentialE]", RowBox[List["-", RowBox[List["ArcCsch", "[", RowBox[List["a", " ", "z"]], "]"]]]]]]]]], "]"]]]], "+", RowBox[List["24", " ", SuperscriptBox[RowBox[List["ArcCsch", "[", RowBox[List["a", " ", "z"]], "]"]], "2"], " ", RowBox[List["PolyLog", "[", RowBox[List["2", ",", SuperscriptBox["\[ExponentialE]", RowBox[List["ArcCsch", "[", RowBox[List["a", " ", "z"]], "]"]]]]], "]"]]]], "+", RowBox[List["48", " ", RowBox[List["ArcCsch", "[", RowBox[List["a", " ", "z"]], "]"]], " ", RowBox[List["PolyLog", "[", RowBox[List["3", ",", RowBox[List["-", SuperscriptBox["\[ExponentialE]", RowBox[List["-", RowBox[List["ArcCsch", "[", RowBox[List["a", " ", "z"]], "]"]]]]]]]]], "]"]]]], "-", RowBox[List["48", " ", RowBox[List["ArcCsch", "[", RowBox[List["a", " ", "z"]], "]"]], " ", RowBox[List["PolyLog", "[", RowBox[List["3", ",", SuperscriptBox["\[ExponentialE]", RowBox[List["ArcCsch", "[", RowBox[List["a", " ", "z"]], "]"]]]]], "]"]]]], "+", RowBox[List["48", " ", RowBox[List["PolyLog", "[", RowBox[List["4", ",", RowBox[List["-", SuperscriptBox["\[ExponentialE]", RowBox[List["-", RowBox[List["ArcCsch", "[", RowBox[List["a", " ", "z"]], "]"]]]]]]]]], "]"]]]], "+", RowBox[List["48", " ", RowBox[List["PolyLog", "[", RowBox[List["4", ",", SuperscriptBox["\[ExponentialE]", RowBox[List["ArcCsch", "[", RowBox[List["a", " ", "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> <msup> <mrow> <msup> <mi> csch </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mrow> <mi> a </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mn> 4 </mn> </msup> <mo> ⁢ </mo> <mrow> <mo> ⅆ </mo> <mi> z </mi> </mrow> </mrow> </mrow> <mo> ⩵ </mo> <mrow> <mrow> <mo> - </mo> <mfrac> <mn> 1 </mn> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> a </mi> </mrow> </mfrac> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mo> - </mo> <mn> 2 </mn> </mrow> <mo> ⁢ </mo> <mi> a </mi> <mo> ⁢ </mo> <mi> z </mi> <mo> ⁢ </mo> <msup> <mrow> <msup> <mi> csch </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mrow> <mi> a </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mn> 4 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <msup> <mrow> <msup> <mi> csch </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mrow> <mi> a </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mn> 4 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 8 </mn> <mo> ⁢ </mo> <mrow> <mi> log </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> + </mo> <msup> <mi> ⅇ </mi> <mrow> <mrow> <mo> - </mo> <msup> <mi> csch </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> a </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </msup> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msup> <mrow> <msup> <mi> csch </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mrow> <mi> a </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mn> 3 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 8 </mn> <mo> ⁢ </mo> <mrow> <mi> log </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <msup> <mi> ⅇ </mi> <mrow> <msup> <mi> csch </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mrow> <mi> a </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </msup> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msup> <mrow> <msup> <mi> csch </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mrow> <mi> a </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mn> 3 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 24 </mn> <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> <msup> <mi> ⅇ </mi> <mrow> <mrow> <mo> - </mo> <msup> <mi> csch </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> a </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </msup> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msup> <mrow> <msup> <mi> csch </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mrow> <mi> a </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 24 </mn> <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> <msup> <mi> ⅇ </mi> <mrow> <msup> <mi> csch </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mrow> <mi> a </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </msup> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msup> <mrow> <msup> <mi> csch </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mrow> <mi> a </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 48 </mn> <mo> ⁢ </mo> <mrow> <msub> <semantics> <mi> Li </mi> <annotation-xml encoding='MathML-Content'> <ci> PolyLog </ci> </annotation-xml> </semantics> <mn> 3 </mn> </msub> <mo> ( </mo> <mrow> <mo> - </mo> <msup> <mi> ⅇ </mi> <mrow> <mrow> <mo> - </mo> <msup> <mi> csch </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> a </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </msup> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <msup> <mi> csch </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mrow> <mi> a </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mn> 48 </mn> <mo> ⁢ </mo> <mrow> <msub> <semantics> <mi> Li </mi> <annotation-xml encoding='MathML-Content'> <ci> PolyLog </ci> </annotation-xml> </semantics> <mn> 3 </mn> </msub> <mo> ( </mo> <msup> <mi> ⅇ </mi> <mrow> <msup> <mi> csch </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mrow> <mi> a </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </msup> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <msup> <mi> csch </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mrow> <mi> a </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <msup> <mi> π </mi> <mn> 4 </mn> </msup> <mo> + </mo> <mrow> <mn> 48 </mn> <mo> ⁢ </mo> <mrow> <msub> <semantics> <mi> Li </mi> <annotation-xml encoding='MathML-Content'> <ci> PolyLog </ci> </annotation-xml> </semantics> <mn> 4 </mn> </msub> <mo> ( </mo> <mrow> <mo> - </mo> <msup> <mi> ⅇ </mi> <mrow> <mrow> <mo> - </mo> <msup> <mi> csch </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> a </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </msup> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mn> 48 </mn> <mo> ⁢ </mo> <mrow> <msub> <semantics> <mi> Li </mi> <annotation-xml encoding='MathML-Content'> <ci> PolyLog </ci> </annotation-xml> </semantics> <mn> 4 </mn> </msub> <mo> ( </mo> <msup> <mi> ⅇ </mi> <mrow> <msup> <mi> csch </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mrow> <mi> a </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </msup> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <int /> <bvar> <ci> z </ci> </bvar> <apply> <power /> <apply> <arccsch /> <apply> <times /> <ci> a </ci> <ci> z </ci> </apply> </apply> <cn type='integer'> 4 </cn> </apply> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> a </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> -2 </cn> <ci> a </ci> <ci> z </ci> <apply> <power /> <apply> <arccsch /> <apply> <times /> <ci> a </ci> <ci> z </ci> </apply> </apply> <cn type='integer'> 4 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <apply> <arccsch /> <apply> <times /> <ci> a </ci> <ci> z </ci> </apply> </apply> <cn type='integer'> 4 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 8 </cn> <apply> <ln /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <power /> <exponentiale /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> csch </ci> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <ci> a </ci> <ci> z </ci> </apply> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <arccsch /> <apply> <times /> <ci> a </ci> <ci> z </ci> </apply> </apply> <cn type='integer'> 3 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 8 </cn> <apply> <ln /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <exponentiale /> <apply> <arccsch /> <apply> <times /> <ci> a </ci> <ci> z </ci> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <arccsch /> <apply> <times /> <ci> a </ci> <ci> z </ci> </apply> </apply> <cn type='integer'> 3 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 24 </cn> <apply> <ci> PolyLog </ci> <cn type='integer'> 2 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <exponentiale /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> csch </ci> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <ci> a </ci> <ci> z </ci> </apply> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <arccsch /> <apply> <times /> <ci> a </ci> <ci> z </ci> </apply> </apply> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 24 </cn> <apply> <ci> PolyLog </ci> <cn type='integer'> 2 </cn> <apply> <power /> <exponentiale /> <apply> <arccsch /> <apply> <times /> <ci> a </ci> <ci> z </ci> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <arccsch /> <apply> <times /> <ci> a </ci> <ci> z </ci> </apply> </apply> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 48 </cn> <apply> <ci> PolyLog </ci> <cn type='integer'> 3 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <exponentiale /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> csch </ci> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <ci> a </ci> <ci> z </ci> </apply> </apply> </apply> </apply> </apply> <apply> <arccsch /> <apply> <times /> <ci> a </ci> <ci> z </ci> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 48 </cn> <apply> <ci> PolyLog </ci> <cn type='integer'> 3 </cn> <apply> <power /> <exponentiale /> <apply> <arccsch /> <apply> <times /> <ci> a </ci> <ci> z </ci> </apply> </apply> </apply> </apply> <apply> <arccsch /> <apply> <times /> <ci> a </ci> <ci> z </ci> </apply> </apply> </apply> </apply> <apply> <power /> <pi /> <cn type='integer'> 4 </cn> </apply> <apply> <times /> <cn type='integer'> 48 </cn> <apply> <ci> PolyLog </ci> <cn type='integer'> 4 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <exponentiale /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> csch </ci> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <ci> a </ci> <ci> z </ci> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 48 </cn> <apply> <ci> PolyLog </ci> <cn type='integer'> 4 </cn> <apply> <power /> <exponentiale /> <apply> <arccsch /> <apply> <times /> <ci> a </ci> <ci> z </ci> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>
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Date Added to functions.wolfram.com (modification date)
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