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http://functions.wolfram.com/01.23.21.0317.01
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Integrate[Csch[c z]/Sqrt[(a + b Csch[c z])^3], z] ==
-(2 (a + b Csch[c z])^(3/2) (-((b^2 Coth[c z]^2)/Sqrt[a + b Csch[c z]]) +
(1/Sqrt[1/(-a + I b)]) (I (a + I b) Sqrt[(b - I b Csch[c z])/(I a + b)]
Sqrt[(b + I b Csch[c z])/((-I) a + b)]
(EllipticE[I ArcSinh[Sqrt[1/(-a + I b)] Sqrt[a + b Csch[c z]]],
(a - I b)/(a + I b)] - EllipticF[
I ArcSinh[Sqrt[1/(-a + I b)] Sqrt[a + b Csch[c z]]],
(a - I b)/(a + I b)]))) Tanh[c z])/
(b (a^2 + b^2) c Sqrt[(a + b Csch[c z])^3])
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Cell[BoxData[RowBox[List[RowBox[List["\[Integral]", RowBox[List[FractionBox[RowBox[List["Csch", "[", RowBox[List["c", " ", "z"]], "]"]], SqrtBox[SuperscriptBox[RowBox[List["(", RowBox[List["a", "+", RowBox[List["b", " ", RowBox[List["Csch", "[", RowBox[List["c", " ", "z"]], "]"]]]]]], ")"]], "3"]]], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[RowBox[List["-", RowBox[List["(", RowBox[List["2", " ", SuperscriptBox[RowBox[List["(", RowBox[List["a", "+", RowBox[List["b", " ", RowBox[List["Csch", "[", RowBox[List["c", " ", "z"]], "]"]]]]]], ")"]], RowBox[List["3", "/", "2"]]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", FractionBox[RowBox[List[SuperscriptBox["b", "2"], " ", SuperscriptBox[RowBox[List["Coth", "[", RowBox[List["c", " ", "z"]], "]"]], "2"]]], SqrtBox[RowBox[List["a", "+", RowBox[List["b", " ", RowBox[List["Csch", "[", RowBox[List["c", " ", "z"]], "]"]]]]]]]]]], "+", RowBox[List[FractionBox["1", SqrtBox[FractionBox["1", RowBox[List[RowBox[List["-", "a"]], "+", RowBox[List["\[ImaginaryI]", " ", "b"]]]]]]], RowBox[List["(", RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List["a", "+", RowBox[List["\[ImaginaryI]", " ", "b"]]]], ")"]], " ", SqrtBox[FractionBox[RowBox[List["b", "-", RowBox[List["\[ImaginaryI]", " ", "b", " ", RowBox[List["Csch", "[", RowBox[List["c", " ", "z"]], "]"]]]]]], RowBox[List[RowBox[List["\[ImaginaryI]", " ", "a"]], "+", "b"]]]], " ", SqrtBox[FractionBox[RowBox[List["b", "+", RowBox[List["\[ImaginaryI]", " ", "b", " ", RowBox[List["Csch", "[", RowBox[List["c", " ", "z"]], "]"]]]]]], RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "a"]], "+", "b"]]]], " ", RowBox[List["(", RowBox[List[RowBox[List["EllipticE", "[", RowBox[List[RowBox[List["\[ImaginaryI]", " ", RowBox[List["ArcSinh", "[", RowBox[List[SqrtBox[FractionBox["1", RowBox[List[RowBox[List["-", "a"]], "+", RowBox[List["\[ImaginaryI]", " ", "b"]]]]]], " ", SqrtBox[RowBox[List["a", "+", RowBox[List["b", " ", RowBox[List["Csch", "[", RowBox[List["c", " ", "z"]], "]"]]]]]]]]], "]"]]]], ",", FractionBox[RowBox[List["a", "-", RowBox[List["\[ImaginaryI]", " ", "b"]]]], RowBox[List["a", "+", RowBox[List["\[ImaginaryI]", " ", "b"]]]]]]], "]"]], "-", RowBox[List["EllipticF", "[", RowBox[List[RowBox[List["\[ImaginaryI]", " ", RowBox[List["ArcSinh", "[", RowBox[List[SqrtBox[FractionBox["1", RowBox[List[RowBox[List["-", "a"]], "+", RowBox[List["\[ImaginaryI]", " ", "b"]]]]]], " ", SqrtBox[RowBox[List["a", "+", RowBox[List["b", " ", RowBox[List["Csch", "[", RowBox[List["c", " ", "z"]], "]"]]]]]]]]], "]"]]]], ",", FractionBox[RowBox[List["a", "-", RowBox[List["\[ImaginaryI]", " ", "b"]]]], RowBox[List["a", "+", RowBox[List["\[ImaginaryI]", " ", "b"]]]]]]], "]"]]]], ")"]]]], ")"]]]]]], ")"]], " ", RowBox[List["Tanh", "[", RowBox[List["c", " ", "z"]], "]"]]]], ")"]]]], "/", RowBox[List["(", RowBox[List["b", " ", RowBox[List["(", RowBox[List[SuperscriptBox["a", "2"], "+", SuperscriptBox["b", "2"]]], ")"]], " ", "c", " ", SqrtBox[SuperscriptBox[RowBox[List["(", RowBox[List["a", "+", RowBox[List["b", " ", RowBox[List["Csch", "[", RowBox[List["c", " ", "z"]], "]"]]]]]], ")"]], "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> <mfrac> <mrow> <mi> csch </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> c </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <msqrt> <msup> <mrow> <mo> ( </mo> <mrow> <mi> a </mi> <mo> + </mo> <mrow> <mi> b </mi> <mo> ⁢ </mo> <mrow> <mi> csch </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> c </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <mn> 3 </mn> </msup> </msqrt> </mfrac> <mo> ⁢ </mo> <mrow> <mo> ⅆ </mo> <mi> z </mi> </mrow> </mrow> </mrow> <mo> ⩵ </mo> <mrow> <mrow> <mo> - </mo> <mrow> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mi> a </mi> <mo> + </mo> <mrow> <mi> b </mi> <mo> ⁢ </mo> <mrow> <mi> csch </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> c </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <mrow> <mn> 3 </mn> <mo> / </mo> <mn> 2 </mn> </mrow> </msup> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mfrac> <mn> 1 </mn> <msqrt> <mfrac> <mn> 1 </mn> <mrow> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> b </mi> </mrow> <mo> - </mo> <mi> a </mi> </mrow> </mfrac> </msqrt> </mfrac> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> a </mi> <mo> + </mo> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> b </mi> </mrow> </mrow> <mtext> </mtext> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msqrt> <mfrac> <mrow> <mi> b </mi> <mo> - </mo> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> b </mi> <mo> ⁢ </mo> <mrow> <mi> csch </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> c </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mrow> <mi> b </mi> <mo> + </mo> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> a </mi> <mtext> </mtext> </mrow> </mrow> </mfrac> </msqrt> <mo> ⁢ </mo> <msqrt> <mfrac> <mrow> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mrow> <mi> csch </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> c </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mi> b </mi> </mrow> <mo> + </mo> <mi> b </mi> </mrow> <mrow> <mi> b </mi> <mo> - </mo> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> a </mi> </mrow> </mrow> </mfrac> </msqrt> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> E </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mrow> <msup> <mi> sinh </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mrow> <msqrt> <mfrac> <mn> 1 </mn> <mrow> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> b </mi> </mrow> <mo> - </mo> <mi> a </mi> </mrow> </mfrac> </msqrt> <mo> ⁢ </mo> <msqrt> <mrow> <mi> a </mi> <mo> + </mo> <mrow> <mi> b </mi> <mo> ⁢ </mo> <mrow> <mi> csch </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> c </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </msqrt> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ❘ </mo> <mfrac> <mrow> <mi> a </mi> <mo> - </mo> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> b </mi> </mrow> </mrow> <mrow> <mi> a </mi> <mo> + </mo> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> b </mi> <mtext> </mtext> </mrow> </mrow> </mfrac> </mrow> <mo> ) </mo> </mrow> <mo> - </mo> <mrow> <mi> F </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mrow> <msup> <mi> sinh </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mrow> <msqrt> <mfrac> <mn> 1 </mn> <mrow> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> b </mi> </mrow> <mo> - </mo> <mi> a </mi> </mrow> </mfrac> </msqrt> <mo> ⁢ </mo> <msqrt> <mrow> <mi> a </mi> <mo> + </mo> <mrow> <mi> b </mi> <mo> ⁢ </mo> <mrow> <mi> csch </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> c </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </msqrt> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ❘ </mo> <mfrac> <mrow> <mi> a </mi> <mo> - </mo> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> b </mi> </mrow> </mrow> <mrow> <mi> a </mi> <mo> + </mo> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> b </mi> <mtext> </mtext> </mrow> </mrow> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mfrac> <mrow> <msup> <mi> b </mi> <mn> 2 </mn> </msup> <mo> ⁢ </mo> <mrow> <msup> <mi> coth </mi> <mn> 2 </mn> </msup> <mo> ( </mo> <mrow> <mi> c </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <msqrt> <mrow> <mi> a </mi> <mo> + </mo> <mrow> <mi> b </mi> <mo> ⁢ </mo> <mrow> <mi> csch </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> c </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </msqrt> </mfrac> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> tanh </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> c </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> / </mo> <mrow> <mo> ( </mo> <mrow> <mi> b </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <msup> <mi> a </mi> <mn> 2 </mn> </msup> <mo> + </mo> <msup> <mi> b </mi> <mn> 2 </mn> </msup> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mi> c </mi> <mo> ⁢ </mo> <msqrt> <msup> <mrow> <mo> ( </mo> <mrow> <mi> a </mi> <mo> + </mo> <mrow> <mi> b </mi> <mo> ⁢ </mo> <mrow> <mi> csch </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> c </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <mn> 3 </mn> </msup> </msqrt> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <int /> <bvar> <ci> z </ci> </bvar> <apply> <times /> <apply> <csch /> <apply> <times /> <ci> c </ci> <ci> z </ci> </apply> </apply> <apply> <power /> <apply> <power /> <apply> <power /> <apply> <plus /> <ci> a </ci> <apply> <times /> <ci> b </ci> <apply> <csch /> <apply> <times /> <ci> c </ci> <ci> z </ci> </apply> </apply> </apply> </apply> <cn type='integer'> 3 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <apply> <plus /> <ci> a </ci> <apply> <times /> <ci> b </ci> <apply> <csch /> <apply> <times /> <ci> c </ci> <ci> z </ci> </apply> </apply> </apply> </apply> <cn type='rational'> 3 <sep /> 2 </cn> </apply> <apply> <plus /> <apply> <times /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <power /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <plus /> <apply> <times /> <imaginaryi /> <ci> b </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> a </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <imaginaryi /> <apply> <plus /> <ci> a </ci> <apply> <times /> <imaginaryi /> <ci> b </ci> </apply> </apply> <apply> <power /> <apply> <times /> <apply> <plus /> <ci> b </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <imaginaryi /> <ci> b </ci> <apply> <csch /> <apply> <times /> <ci> c </ci> <ci> z </ci> </apply> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <plus /> <ci> b </ci> <apply> <times /> <imaginaryi /> <ci> a </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <times /> <apply> <plus /> <apply> <times /> <imaginaryi /> <apply> <csch /> <apply> <times /> <ci> c </ci> <ci> z </ci> </apply> </apply> <ci> b </ci> </apply> <ci> b </ci> </apply> <apply> <power /> <apply> <plus /> <ci> b </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <imaginaryi /> <ci> a </ci> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <plus /> <apply> <ci> EllipticE </ci> <apply> <times /> <imaginaryi /> <apply> <arcsinh /> <apply> <times /> <apply> <power /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <plus /> <apply> <times /> <imaginaryi /> <ci> b </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> a </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <plus /> <ci> a </ci> <apply> <times /> <ci> b </ci> <apply> <csch /> <apply> <times /> <ci> c </ci> <ci> z </ci> </apply> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <plus /> <ci> a </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <imaginaryi /> <ci> b </ci> </apply> </apply> </apply> <apply> <power /> <apply> <plus /> <ci> a </ci> <apply> <times /> <imaginaryi /> <ci> b </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> EllipticF </ci> <apply> <times /> <imaginaryi /> <apply> <arcsinh /> <apply> <times /> <apply> <power /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <plus /> <apply> <times /> <imaginaryi /> <ci> b </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> a </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <plus /> <ci> a </ci> <apply> <times /> <ci> b </ci> <apply> <csch /> <apply> <times /> <ci> c </ci> <ci> z </ci> </apply> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <plus /> <ci> a </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <imaginaryi /> <ci> b </ci> </apply> </apply> </apply> <apply> <power /> <apply> <plus /> <ci> a </ci> <apply> <times /> <imaginaryi /> <ci> b </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <power /> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <coth /> <apply> <times /> <ci> c </ci> <ci> z </ci> </apply> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <power /> <apply> <plus /> <ci> a </ci> <apply> <times /> <ci> b </ci> <apply> <csch /> <apply> <times /> <ci> c </ci> <ci> z </ci> </apply> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <tanh /> <apply> <times /> <ci> c </ci> <ci> z </ci> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <ci> b </ci> <apply> <plus /> <apply> <power /> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> </apply> <ci> c </ci> <apply> <power /> <apply> <power /> <apply> <plus /> <ci> a </ci> <apply> <times /> <ci> b </ci> <apply> <csch /> <apply> <times /> <ci> c </ci> <ci> z </ci> </apply> </apply> </apply> </apply> <cn type='integer'> 3 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </annotation-xml> </semantics> </math>
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