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http://functions.wolfram.com/01.20.21.4602.01
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Integrate[(z Sinh[2 c z])/(a Sinh[c z] + b Cosh[c z]), z] ==
(1/((a + b) c^2)) (2 ((2 a b ArcTan[(a + b Tanh[(c z)/2])/Sqrt[-a^2 + b^2]]
ArcTanh[b/a])/((a - b) Sqrt[-a^2 + b^2]) + ((a + b c z) Cosh[c z])/
(-a + b) -
(b ((c z + ArcTanh[b/a]) (Log[1 - E^((-c) z - ArcTanh[b/a])] -
Log[1 + E^((-c) z - ArcTanh[b/a])]) +
PolyLog[2, -E^((-c) z - ArcTanh[b/a])] -
PolyLog[2, E^((-c) z - ArcTanh[b/a])]))/((a - b) Sqrt[1 - b^2/a^2]) +
((b + a c z) Sinh[c z])/(a - b)))
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Cell[BoxData[RowBox[List[RowBox[List["\[Integral]", RowBox[List[FractionBox[RowBox[List["z", " ", RowBox[List["Sinh", "[", RowBox[List["2", " ", "c", " ", "z"]], "]"]]]], RowBox[List[RowBox[List["a", " ", RowBox[List["Sinh", "[", RowBox[List["c", " ", "z"]], "]"]]]], "+", RowBox[List["b", " ", RowBox[List["Cosh", "[", RowBox[List["c", " ", "z"]], "]"]]]]]]], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[FractionBox["1", RowBox[List[RowBox[List["(", RowBox[List["a", "+", "b"]], ")"]], " ", SuperscriptBox["c", "2"]]]], RowBox[List["(", RowBox[List["2", " ", RowBox[List["(", RowBox[List[FractionBox[RowBox[List["2", " ", "a", " ", "b", " ", RowBox[List["ArcTan", "[", FractionBox[RowBox[List["a", "+", RowBox[List["b", " ", RowBox[List["Tanh", "[", FractionBox[RowBox[List["c", " ", "z"]], "2"], "]"]]]]]], SqrtBox[RowBox[List[RowBox[List["-", SuperscriptBox["a", "2"]]], "+", SuperscriptBox["b", "2"]]]]], "]"]], " ", RowBox[List["ArcTanh", "[", FractionBox["b", "a"], "]"]]]], RowBox[List[RowBox[List["(", RowBox[List["a", "-", "b"]], ")"]], " ", SqrtBox[RowBox[List[RowBox[List["-", SuperscriptBox["a", "2"]]], "+", SuperscriptBox["b", "2"]]]]]]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["a", "+", RowBox[List["b", " ", "c", " ", "z"]]]], ")"]], " ", RowBox[List["Cosh", "[", RowBox[List["c", " ", "z"]], "]"]]]], RowBox[List[RowBox[List["-", "a"]], "+", "b"]]], "-", RowBox[List[RowBox[List["(", RowBox[List["b", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["c", " ", "z"]], "+", RowBox[List["ArcTanh", "[", FractionBox["b", "a"], "]"]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["Log", "[", RowBox[List["1", "-", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List[RowBox[List["-", "c"]], " ", "z"]], "-", RowBox[List["ArcTanh", "[", FractionBox["b", "a"], "]"]]]]]]], "]"]], "-", RowBox[List["Log", "[", RowBox[List["1", "+", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List[RowBox[List["-", "c"]], " ", "z"]], "-", RowBox[List["ArcTanh", "[", FractionBox["b", "a"], "]"]]]]]]], "]"]]]], ")"]]]], "+", RowBox[List["PolyLog", "[", RowBox[List["2", ",", RowBox[List["-", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List[RowBox[List["-", "c"]], " ", "z"]], "-", RowBox[List["ArcTanh", "[", FractionBox["b", "a"], "]"]]]]]]]]], "]"]], "-", RowBox[List["PolyLog", "[", RowBox[List["2", ",", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List[RowBox[List["-", "c"]], " ", "z"]], "-", RowBox[List["ArcTanh", "[", FractionBox["b", "a"], "]"]]]]]]], "]"]]]], ")"]]]], ")"]], "/", RowBox[List["(", RowBox[List[RowBox[List["(", RowBox[List["a", "-", "b"]], ")"]], " ", SqrtBox[RowBox[List["1", "-", FractionBox[SuperscriptBox["b", "2"], SuperscriptBox["a", "2"]]]]]]], ")"]]]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["b", "+", RowBox[List["a", " ", "c", " ", "z"]]]], ")"]], " ", RowBox[List["Sinh", "[", RowBox[List["c", " ", "z"]], "]"]]]], RowBox[List["a", "-", "b"]]]]], ")"]]]], ")"]]]]]]]]
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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> z </mi> <mo> ⁢ </mo> <mrow> <mi> sinh </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> c </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mrow> <mrow> <mi> a </mi> <mo> ⁢ </mo> <mrow> <mi> sinh </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> c </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mi> b </mi> <mo> ⁢ </mo> <mrow> <mi> cosh </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> c </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mfrac> <mo> ⁢ </mo> <mrow> <mo> ⅆ </mo> <mi> z </mi> </mrow> </mrow> </mrow> <mo> ⩵ </mo> <mrow> <mfrac> <mn> 1 </mn> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> a </mi> <mo> + </mo> <mi> b </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msup> <mi> c </mi> <mn> 2 </mn> </msup> </mrow> </mfrac> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mfrac> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> a </mi> <mo> ⁢ </mo> <mi> b </mi> <mo> ⁢ </mo> <mrow> <msup> <mi> tan </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mfrac> <mrow> <mi> a </mi> <mo> + </mo> <mrow> <mi> b </mi> <mo> ⁢ </mo> <mrow> <mi> tanh </mi> <mo> ⁡ </mo> <mo> ( </mo> <mfrac> <mrow> <mi> c </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mn> 2 </mn> </mfrac> <mo> ) </mo> </mrow> </mrow> </mrow> <msqrt> <mrow> <msup> <mi> b </mi> <mn> 2 </mn> </msup> <mo> - </mo> <msup> <mi> a </mi> <mn> 2 </mn> </msup> </mrow> </msqrt> </mfrac> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <msup> <mi> tanh </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mfrac> <mi> b </mi> <mi> a </mi> </mfrac> <mo> ) </mo> </mrow> </mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> a </mi> <mo> - </mo> <mi> b </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msqrt> <mrow> <msup> <mi> b </mi> <mn> 2 </mn> </msup> <mo> - </mo> <msup> <mi> a </mi> <mn> 2 </mn> </msup> </mrow> </msqrt> </mrow> </mfrac> <mo> + </mo> <mfrac> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> a </mi> <mo> + </mo> <mrow> <mi> b </mi> <mo> ⁢ </mo> <mi> c </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> cosh </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> c </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mrow> <mi> b </mi> <mo> - </mo> <mi> a </mi> </mrow> </mfrac> <mo> + </mo> <mfrac> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> b </mi> <mo> + </mo> <mrow> <mi> a </mi> <mo> ⁢ </mo> <mi> c </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> sinh </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> c </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mrow> <mi> a </mi> <mo> - </mo> <mi> b </mi> </mrow> </mfrac> <mo> - </mo> <mrow> <mfrac> <mn> 1 </mn> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> a </mi> <mo> - </mo> <mi> b </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msqrt> <mrow> <mn> 1 </mn> <mo> - </mo> <mfrac> <msup> <mi> b </mi> <mn> 2 </mn> </msup> <msup> <mi> a </mi> <mn> 2 </mn> </msup> </mfrac> </mrow> </msqrt> </mrow> </mfrac> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> b </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> c </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> + </mo> <mrow> <msup> <mi> tanh </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mfrac> <mi> b </mi> <mi> a </mi> </mfrac> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> log </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <msup> <mi> ⅇ </mi> <mrow> <mrow> <mrow> <mo> - </mo> <mi> c </mi> </mrow> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> - </mo> <mrow> <msup> <mi> tanh </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mfrac> <mi> b </mi> <mi> a </mi> </mfrac> <mo> ) </mo> </mrow> </mrow> </msup> </mrow> <mo> ) </mo> </mrow> <mo> - </mo> <mrow> <mi> log </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> + </mo> <msup> <mi> ⅇ </mi> <mrow> <mrow> <mrow> <mo> - </mo> <mi> c </mi> </mrow> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> - </mo> <mrow> <msup> <mi> tanh </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mfrac> <mi> b </mi> <mi> a </mi> </mfrac> <mo> ) </mo> </mrow> </mrow> </msup> </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> <mo> - </mo> <msup> <mi> ⅇ </mi> <mrow> <mrow> <mrow> <mo> - </mo> <mi> c </mi> </mrow> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> - </mo> <mrow> <msup> <mi> tanh </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mfrac> <mi> b </mi> <mi> a </mi> </mfrac> <mo> ) </mo> </mrow> </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> <msup> <mi> ⅇ </mi> <mrow> <mrow> <mrow> <mo> - </mo> <mi> c </mi> </mrow> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> - </mo> <mrow> <msup> <mi> tanh </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mfrac> <mi> b </mi> <mi> a </mi> </mfrac> <mo> ) </mo> </mrow> </mrow> </msup> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <int /> <bvar> <ci> z </ci> </bvar> <apply> <times /> <ci> z </ci> <apply> <sinh /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> c </ci> <ci> z </ci> </apply> </apply> <apply> <power /> <apply> <plus /> <apply> <times /> <ci> a </ci> <apply> <sinh /> <apply> <times /> <ci> c </ci> <ci> z </ci> </apply> </apply> </apply> <apply> <times /> <ci> b </ci> <apply> <cosh /> <apply> <times /> <ci> c </ci> <ci> z </ci> </apply> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <times /> <apply> <plus /> <ci> a </ci> <ci> b </ci> </apply> <apply> <power /> <ci> c </ci> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> a </ci> <ci> b </ci> <apply> <arctan /> <apply> <times /> <apply> <plus /> <ci> a </ci> <apply> <times /> <ci> b </ci> <apply> <tanh /> <apply> <times /> <ci> c </ci> <ci> z </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <power /> <apply> <plus /> <apply> <power /> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> a </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> <apply> <arctanh /> <apply> <times /> <ci> b </ci> <apply> <power /> <ci> a </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <apply> <plus /> <ci> a </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> b </ci> </apply> </apply> <apply> <power /> <apply> <plus /> <apply> <power /> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <apply> <plus /> <ci> a </ci> <apply> <times /> <ci> b </ci> <ci> c </ci> <ci> z </ci> </apply> </apply> <apply> <cosh /> <apply> <times /> <ci> c </ci> <ci> z </ci> </apply> </apply> <apply> <power /> <apply> <plus /> <ci> b </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> a </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <apply> <plus /> <ci> b </ci> <apply> <times /> <ci> a </ci> <ci> c </ci> <ci> z </ci> </apply> </apply> <apply> <sinh /> <apply> <times /> <ci> c </ci> <ci> z </ci> </apply> </apply> <apply> <power /> <apply> <plus /> <ci> a </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> b </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <times /> <apply> <plus /> <ci> a </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> b </ci> </apply> </apply> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <power /> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <power /> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <ci> b </ci> <apply> <plus /> <apply> <times /> <apply> <plus /> <apply> <times /> <ci> c </ci> <ci> z </ci> </apply> <apply> <arctanh /> <apply> <times /> <ci> b </ci> <apply> <power /> <ci> a </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <plus /> <apply> <ln /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <exponentiale /> <apply> <plus /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> c </ci> </apply> <ci> z </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <arctanh /> <apply> <times /> <ci> b </ci> <apply> <power /> <ci> a </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ln /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <power /> <exponentiale /> <apply> <plus /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> c </ci> </apply> <ci> z </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <arctanh /> <apply> <times /> <ci> b </ci> <apply> <power /> <ci> a </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <ci> PolyLog </ci> <cn type='integer'> 2 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <exponentiale /> <apply> <plus /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> c </ci> </apply> <ci> z </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <arctanh /> <apply> <times /> <ci> b </ci> <apply> <power /> <ci> a </ci> <cn type='integer'> -1 </cn> </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> <power /> <exponentiale /> <apply> <plus /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> c </ci> </apply> <ci> z </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <arctanh /> <apply> <times /> <ci> b </ci> <apply> <power /> <ci> a </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </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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){kind=small}
