Integrate[Cosh[z]^2/(Sqrt[a + b Sinh[z]] (c + b Sinh[z])^2), z] == (1/(4 b (a - c))) (-((1/(Sqrt[1/(-a + I b)] b^2 (a - c))) (8 I a ((a - c) EllipticF[I ArcSinh[Sqrt[1/(-a + I b)] Sqrt[a + b Sinh[z]]], (a - I b)/(a + I b)] + c EllipticPi[(a - I b)/(a - c), I ArcSinh[Sqrt[1/(-a + I b)] Sqrt[a + b Sinh[z]]], (a - I b)/(a + I b)]) Sech[z] Sqrt[(b - I b Sinh[z])/(I a + b)] Sqrt[(b + I b Sinh[z])/ ((-I) a + b)])) + (1/(Sqrt[1/(-a + I b)] b^2 (a - c))) (I (Cosh[z] + Cosh[3 z]) (2 (a + I b) (a - c) EllipticE[I ArcSinh[Sqrt[1/(-a + I b)] Sqrt[a + b Sinh[z]]], (a - I b)/(a + I b)] + 2 (a - c) ((-I) b + c) EllipticF[I ArcSinh[Sqrt[1/(-a + I b)] Sqrt[a + b Sinh[z]]], (a - I b)/(a + I b)] + (b^2 + 2 c^2) EllipticPi[(a - I b)/(a - c), I ArcSinh[Sqrt[1/(-a + I b)] Sqrt[a + b Sinh[z]]], (a - I b)/(a + I b)]) Sech[z]^2 Sech[2 z] Sqrt[(b - I b Sinh[z])/(I a + b)] Sqrt[(b + I b Sinh[z])/((-I) a + b)]) + (6 b EllipticPi[(2 b)/(b + I c), (1/4) (Pi - 2 I z), -((2 I b)/(a - I b))] Sqrt[(a + b Sinh[z])/(a - I b)])/((b + I c) Sqrt[a + b Sinh[z]]) - (4 Cosh[z] Sqrt[a + b Sinh[z]])/(c + b Sinh[z]))

Cell[BoxData[RowBox[List[RowBox[List["\[Integral]", RowBox[List[FractionBox[SuperscriptBox[RowBox[List["Cosh", "[", "z", "]"]], "2"], RowBox[List[SqrtBox[RowBox[List["a", "+", RowBox[List["b", " ", RowBox[List["Sinh", "[", "z", "]"]]]]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["c", "+", RowBox[List["b", " ", RowBox[List["Sinh", "[", "z", "]"]]]]]], ")"]], "2"]]]], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[FractionBox["1", RowBox[List["4", " ", "b", " ", RowBox[List["(", RowBox[List["a", "-", "c"]], ")"]]]]], RowBox[List["(", RowBox[List[RowBox[List["-", RowBox[List[FractionBox["1", RowBox[List[SqrtBox[FractionBox["1", RowBox[List[RowBox[List["-", "a"]], "+", RowBox[List["\[ImaginaryI]", " ", "b"]]]]]], " ", SuperscriptBox["b", "2"], " ", RowBox[List["(", RowBox[List["a", "-", "c"]], ")"]]]]], RowBox[List["(", RowBox[List["8", " ", "\[ImaginaryI]", " ", "a", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["a", "-", "c"]], ")"]], " ", 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["Sinh", "[", "z", "]"]]]]]]]]], "]"]]]], ",", FractionBox[RowBox[List["a", "-", RowBox[List["\[ImaginaryI]", " ", "b"]]]], RowBox[List["a", "+", RowBox[List["\[ImaginaryI]", " ", "b"]]]]]]], "]"]]]], "+", RowBox[List["c", " ", RowBox[List["EllipticPi", "[", RowBox[List[FractionBox[RowBox[List["a", "-", RowBox[List["\[ImaginaryI]", " ", "b"]]]], RowBox[List["a", "-", "c"]]], ",", 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["Sinh", "[", "z", "]"]]]]]]]]], "]"]]]], ",", FractionBox[RowBox[List["a", "-", RowBox[List["\[ImaginaryI]", " ", "b"]]]], RowBox[List["a", "+", RowBox[List["\[ImaginaryI]", " ", "b"]]]]]]], "]"]]]]]], ")"]], " ", RowBox[List["Sech", "[", "z", "]"]], " ", SqrtBox[FractionBox[RowBox[List["b", "-", RowBox[List["\[ImaginaryI]", " ", "b", " ", RowBox[List["Sinh", "[", "z", "]"]]]]]], RowBox[List[RowBox[List["\[ImaginaryI]", " ", "a"]], "+", "b"]]]], " ", SqrtBox[FractionBox[RowBox[List["b", "+", RowBox[List["\[ImaginaryI]", " ", "b", " ", RowBox[List["Sinh", "[", "z", "]"]]]]]], RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "a"]], "+", "b"]]]]]], ")"]]]]]], "+", RowBox[List[FractionBox["1", RowBox[List[SqrtBox[FractionBox["1", RowBox[List[RowBox[List["-", "a"]], "+", RowBox[List["\[ImaginaryI]", " ", "b"]]]]]], " ", SuperscriptBox["b", "2"], " ", RowBox[List["(", RowBox[List["a", "-", "c"]], ")"]]]]], RowBox[List["(", RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[RowBox[List["Cosh", "[", "z", "]"]], "+", RowBox[List["Cosh", "[", RowBox[List["3", " ", "z"]], "]"]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", RowBox[List["(", RowBox[List["a", "+", RowBox[List["\[ImaginaryI]", " ", "b"]]]], ")"]], " ", RowBox[List["(", RowBox[List["a", "-", "c"]], ")"]], " ", 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["Sinh", "[", "z", "]"]]]]]]]]], "]"]]]], ",", FractionBox[RowBox[List["a", "-", RowBox[List["\[ImaginaryI]", " ", "b"]]]], RowBox[List["a", "+", RowBox[List["\[ImaginaryI]", " ", "b"]]]]]]], "]"]]]], "+", RowBox[List["2", " ", RowBox[List["(", RowBox[List["a", "-", "c"]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "b"]], "+", "c"]], ")"]], " ", 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["Sinh", "[", "z", "]"]]]]]]]]], "]"]]]], ",", FractionBox[RowBox[List["a", "-", RowBox[List["\[ImaginaryI]", " ", "b"]]]], RowBox[List["a", "+", RowBox[List["\[ImaginaryI]", " ", "b"]]]]]]], "]"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["b", "2"], "+", RowBox[List["2", " ", SuperscriptBox["c", "2"]]]]], ")"]], " ", RowBox[List["EllipticPi", "[", RowBox[List[FractionBox[RowBox[List["a", "-", RowBox[List["\[ImaginaryI]", " ", "b"]]]], RowBox[List["a", "-", "c"]]], ",", 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["Sinh", "[", "z", "]"]]]]]]]]], "]"]]]], ",", FractionBox[RowBox[List["a", "-", RowBox[List["\[ImaginaryI]", " ", "b"]]]], RowBox[List["a", "+", RowBox[List["\[ImaginaryI]", " ", "b"]]]]]]], "]"]]]]]], ")"]], " ", SuperscriptBox[RowBox[List["Sech", "[", "z", "]"]], "2"], " ", RowBox[List["Sech", "[", RowBox[List["2", " ", "z"]], "]"]], " ", SqrtBox[FractionBox[RowBox[List["b", "-", RowBox[List["\[ImaginaryI]", " ", "b", " ", RowBox[List["Sinh", "[", "z", "]"]]]]]], RowBox[List[RowBox[List["\[ImaginaryI]", " ", "a"]], "+", "b"]]]], " ", SqrtBox[FractionBox[RowBox[List["b", "+", RowBox[List["\[ImaginaryI]", " ", "b", " ", RowBox[List["Sinh", "[", "z", "]"]]]]]], RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "a"]], "+", "b"]]]]]], ")"]]]], "+", FractionBox[RowBox[List["6", " ", "b", " ", RowBox[List["EllipticPi", "[", RowBox[List[FractionBox[RowBox[List["2", " ", "b"]], RowBox[List["b", "+", RowBox[List["\[ImaginaryI]", " ", "c"]]]]], ",", RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List["\[Pi]", "-", RowBox[List["2", " ", "\[ImaginaryI]", " ", "z"]]]], ")"]]]], ",", RowBox[List["-", FractionBox[RowBox[List["2", " ", "\[ImaginaryI]", " ", "b"]], RowBox[List["a", "-", RowBox[List["\[ImaginaryI]", " ", "b"]]]]]]]]], "]"]], " ", SqrtBox[FractionBox[RowBox[List["a", "+", RowBox[List["b", " ", RowBox[List["Sinh", "[", "z", "]"]]]]]], RowBox[List["a", "-", RowBox[List["\[ImaginaryI]", " ", "b"]]]]]]]], RowBox[List[RowBox[List["(", RowBox[List["b", "+", RowBox[List["\[ImaginaryI]", " ", "c"]]]], ")"]], " ", SqrtBox[RowBox[List["a", "+", RowBox[List["b", " ", RowBox[List["Sinh", "[", "z", "]"]]]]]]]]]], "-", FractionBox[RowBox[List["4", " ", RowBox[List["Cosh", "[", "z", "]"]], " ", SqrtBox[RowBox[List["a", "+", RowBox[List["b", " ", RowBox[List["Sinh", "[", "z", "]"]]]]]]]]], RowBox[List["c", "+", RowBox[List["b", " ", RowBox[List["Sinh", "[", "z", "]"]]]]]]]]], ")"]]]]]]]]

<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> cosh </mi> <mn> 2 </mn> </msup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mrow> <msqrt> <mrow> <mi> a </mi> <mo> + </mo> <mrow> <mi> b </mi> <mo> &#8290; </mo> <mrow> <mi> sinh </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> </mrow> </msqrt> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mi> c </mi> <mo> + </mo> <mrow> <mi> b </mi> <mo> &#8290; </mo> <mrow> <mi> sinh </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </mrow> </mfrac> <mo> &#8290; </mo> <mrow> <mo> &#8518; </mo> <mi> z </mi> </mrow> </mrow> </mrow> <mo> &#10869; </mo> <mrow> <mfrac> <mn> 1 </mn> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <mi> b </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> a </mi> <mo> - </mo> <mi> c </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mfrac> <mn> 1 </mn> <mrow> <msqrt> <mfrac> <mn> 1 </mn> <mrow> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> b </mi> </mrow> <mo> - </mo> <mi> a </mi> </mrow> </mfrac> </msqrt> <mo> &#8290; </mo> <msup> <mi> b </mi> <mn> 2 </mn> </msup> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> a </mi> <mo> - </mo> <mi> c </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> cosh </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> + </mo> <mrow> <mi> cosh </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mn> 3 </mn> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> a </mi> <mo> + </mo> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> b </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> a </mi> <mo> - </mo> <mi> c </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> E </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mrow> <mi> &#8520; </mi> <mo> &#8290; </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> &#8520; </mi> <mo> &#8290; </mo> <mi> b </mi> </mrow> <mo> - </mo> <mi> a </mi> </mrow> </mfrac> </msqrt> <mo> &#8290; </mo> <msqrt> <mrow> <mi> a </mi> <mo> + </mo> <mrow> <mi> b </mi> <mo> &#8290; </mo> <mrow> <mi> sinh </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> </mrow> </msqrt> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> &#10072; </mo> <mfrac> <mrow> <mi> a </mi> <mo> - </mo> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> b </mi> </mrow> </mrow> <mrow> <mi> a </mi> <mo> + </mo> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> b </mi> </mrow> </mrow> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> a </mi> <mo> - </mo> <mi> c </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> c </mi> <mo> - </mo> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> b </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> F </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mrow> <mi> &#8520; </mi> <mo> &#8290; </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> &#8520; </mi> <mo> &#8290; </mo> <mi> b </mi> </mrow> <mo> - </mo> <mi> a </mi> </mrow> </mfrac> </msqrt> <mo> &#8290; </mo> <msqrt> <mrow> <mi> a </mi> <mo> + </mo> <mrow> <mi> b </mi> <mo> &#8290; </mo> <mrow> <mi> sinh </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> </mrow> </msqrt> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> &#10072; </mo> <mfrac> <mrow> <mi> a </mi> <mo> - </mo> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> b </mi> </mrow> </mrow> <mrow> <mi> a </mi> <mo> + </mo> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> b </mi> </mrow> </mrow> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <msup> <mi> b </mi> <mn> 2 </mn> </msup> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <msup> <mi> c </mi> <mn> 2 </mn> </msup> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> &#928; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mfrac> <mrow> <mi> a </mi> <mo> - </mo> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> b </mi> </mrow> </mrow> <mrow> <mi> a </mi> <mo> - </mo> <mi> c </mi> </mrow> </mfrac> <mo> ; </mo> <mrow> <mrow> <mi> &#8520; </mi> <mo> &#8290; </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> &#8520; </mi> <mo> &#8290; </mo> <mi> b </mi> </mrow> <mo> - </mo> <mi> a </mi> </mrow> </mfrac> </msqrt> <mo> &#8290; </mo> <msqrt> <mrow> <mi> a </mi> <mo> + </mo> <mrow> <mi> b </mi> <mo> &#8290; </mo> <mrow> <mi> sinh </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> </mrow> </msqrt> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> &#10072; </mo> <mfrac> <mrow> <mi> a </mi> <mo> - </mo> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> b </mi> </mrow> </mrow> <mrow> <mi> a </mi> <mo> + </mo> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> b </mi> </mrow> </mrow> </mfrac> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> sech </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msqrt> <mfrac> <mrow> <mi> b </mi> <mo> - </mo> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> b </mi> <mo> &#8290; </mo> <mrow> <mi> sinh </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> </mrow> <mrow> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> a </mi> </mrow> <mo> + </mo> <mi> b </mi> </mrow> </mfrac> </msqrt> <mo> &#8290; </mo> <msqrt> <mfrac> <mrow> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mrow> <mi> sinh </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mi> b </mi> </mrow> <mo> + </mo> <mi> b </mi> </mrow> <mrow> <mi> b </mi> <mo> - </mo> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> a </mi> </mrow> </mrow> </mfrac> </msqrt> <mo> &#8290; </mo> <mrow> <msup> <mi> sech </mi> <mn> 2 </mn> </msup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mfrac> <mn> 1 </mn> <mrow> <msqrt> <mfrac> <mn> 1 </mn> <mrow> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> b </mi> </mrow> <mo> - </mo> <mi> a </mi> </mrow> </mfrac> </msqrt> <mo> &#8290; </mo> <msup> <mi> b </mi> <mn> 2 </mn> </msup> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> a </mi> <mo> - </mo> <mi> c </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mn> 8 </mn> <mo> &#8290; </mo> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> a </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> a </mi> <mo> - </mo> <mi> c </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> F </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mrow> <mi> &#8520; </mi> <mo> &#8290; </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> &#8520; </mi> <mo> &#8290; </mo> <mi> b </mi> </mrow> <mo> - </mo> <mi> a </mi> </mrow> </mfrac> </msqrt> <mo> &#8290; </mo> <msqrt> <mrow> <mi> a </mi> <mo> + </mo> <mrow> <mi> b </mi> <mo> &#8290; </mo> <mrow> <mi> sinh </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> </mrow> </msqrt> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> &#10072; </mo> <mfrac> <mrow> <mi> a </mi> <mo> - </mo> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> b </mi> </mrow> </mrow> <mrow> <mi> a </mi> <mo> + </mo> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> b </mi> </mrow> </mrow> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mi> c </mi> <mo> &#8290; </mo> <mrow> <mi> &#928; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mfrac> <mrow> <mi> a </mi> <mo> - </mo> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> b </mi> </mrow> </mrow> <mrow> <mi> a </mi> <mo> - </mo> <mi> c </mi> </mrow> </mfrac> <mo> ; </mo> <mrow> <mrow> <mi> &#8520; </mi> <mo> &#8290; </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> &#8520; </mi> <mo> &#8290; </mo> <mi> b </mi> </mrow> <mo> - </mo> <mi> a </mi> </mrow> </mfrac> </msqrt> <mo> &#8290; </mo> <msqrt> <mrow> <mi> a </mi> <mo> + </mo> <mrow> <mi> b </mi> <mo> &#8290; </mo> <mrow> <mi> sinh </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> </mrow> </msqrt> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> &#10072; </mo> <mfrac> <mrow> <mi> a </mi> <mo> - </mo> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> b </mi> </mrow> </mrow> <mrow> <mi> a </mi> <mo> + </mo> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> b </mi> </mrow> </mrow> </mfrac> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msqrt> <mfrac> <mrow> <mi> b </mi> <mo> - </mo> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> b </mi> <mo> &#8290; </mo> <mrow> <mi> sinh </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> </mrow> <mrow> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> a </mi> </mrow> <mo> + </mo> <mi> b </mi> </mrow> </mfrac> </msqrt> <mo> &#8290; </mo> <msqrt> <mfrac> <mrow> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mrow> <mi> sinh </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mi> b </mi> </mrow> <mo> + </mo> <mi> b </mi> </mrow> <mrow> <mi> b </mi> <mo> - </mo> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> a </mi> </mrow> </mrow> </mfrac> </msqrt> <mo> &#8290; </mo> <mrow> <mi> sech </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mfrac> <mrow> <mn> 6 </mn> <mo> &#8290; </mo> <mi> b </mi> <mo> &#8290; </mo> <mrow> <mi> &#928; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mfrac> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> b </mi> </mrow> <mrow> <mi> b </mi> <mo> + </mo> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> c </mi> </mrow> </mrow> </mfrac> <mo> ; </mo> <mrow> <mrow> <mfrac> <mn> 1 </mn> <mn> 4 </mn> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> &#960; </mi> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> &#10072; </mo> <mrow> <mo> - </mo> <mfrac> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> b </mi> </mrow> <mrow> <mi> a </mi> <mo> - </mo> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> b </mi> </mrow> </mrow> </mfrac> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msqrt> <mfrac> <mrow> <mi> a </mi> <mo> + </mo> <mrow> <mi> b </mi> <mo> &#8290; </mo> <mrow> <mi> sinh </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> </mrow> <mrow> <mi> a </mi> <mo> - </mo> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> b </mi> </mrow> </mrow> </mfrac> </msqrt> </mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> b </mi> <mo> + </mo> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> c </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msqrt> <mrow> <mi> a </mi> <mo> + </mo> <mrow> <mi> b </mi> <mo> &#8290; </mo> <mrow> <mi> sinh </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> </mrow> </msqrt> </mrow> </mfrac> <mo> - </mo> <mfrac> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <mrow> <mi> cosh </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msqrt> <mrow> <mi> a </mi> <mo> + </mo> <mrow> <mi> b </mi> <mo> &#8290; </mo> <mrow> <mi> sinh </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> </mrow> </msqrt> </mrow> <mrow> <mi> c </mi> <mo> + </mo> <mrow> <mi> b </mi> <mo> &#8290; </mo> <mrow> <mi> sinh </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> </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> <power /> <apply> <cosh /> <ci> z </ci> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <times /> <apply> <power /> <apply> <plus /> <ci> a </ci> <apply> <times /> <ci> b </ci> <apply> <sinh /> <ci> z </ci> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <plus /> <ci> c </ci> <apply> <times /> <ci> b </ci> <apply> <sinh /> <ci> z </ci> </apply> </apply> </apply> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <times /> <cn type='integer'> 4 </cn> <ci> b </ci> <apply> <plus /> <ci> a </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> c </ci> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <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 /> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> <apply> <plus /> <ci> a </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> c </ci> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <imaginaryi /> <apply> <plus /> <apply> <cosh /> <ci> z </ci> </apply> <apply> <cosh /> <apply> <times /> <cn type='integer'> 3 </cn> <ci> z </ci> </apply> </apply> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <plus /> <ci> a </ci> <apply> <times /> <imaginaryi /> <ci> b </ci> </apply> </apply> <apply> <plus /> <ci> a </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> c </ci> </apply> </apply> <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> <sinh /> <ci> z </ci> </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> <times /> <cn type='integer'> 2 </cn> <apply> <plus /> <ci> a </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> c </ci> </apply> </apply> <apply> <plus /> <ci> c </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <imaginaryi /> <ci> b </ci> </apply> </apply> </apply> <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> <sinh /> <ci> z </ci> </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> <times /> <apply> <plus /> <apply> <power /> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <ci> c </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <ci> EllipticPi </ci> <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 /> <cn type='integer'> -1 </cn> <ci> c </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <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> <sinh /> <ci> z </ci> </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> <sech /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> z </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> <sinh /> <ci> z </ci> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <plus /> <apply> <times /> <imaginaryi /> <ci> a </ci> </apply> <ci> b </ci> </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> <sinh /> <ci> z </ci> </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> <power /> <apply> <sech /> <ci> z </ci> </apply> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <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 /> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> <apply> <plus /> <ci> a </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> c </ci> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 8 </cn> <imaginaryi /> <ci> a </ci> <apply> <plus /> <apply> <times /> <apply> <plus /> <ci> a </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> c </ci> </apply> </apply> <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> <sinh /> <ci> z </ci> </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> <times /> <ci> c </ci> <apply> <ci> EllipticPi </ci> <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 /> <cn type='integer'> -1 </cn> <ci> c </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <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> <sinh /> <ci> z </ci> </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> <power /> <apply> <times /> <apply> <plus /> <ci> b </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <imaginaryi /> <ci> b </ci> <apply> <sinh /> <ci> z </ci> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <plus /> <apply> <times /> <imaginaryi /> <ci> a </ci> </apply> <ci> b </ci> </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> <sinh /> <ci> z </ci> </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> <sech /> <ci> z </ci> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 6 </cn> <ci> b </ci> <apply> <ci> EllipticPi </ci> <apply> <times /> <cn type='integer'> 2 </cn> <ci> b </ci> <apply> <power /> <apply> <plus /> <ci> b </ci> <apply> <times /> <imaginaryi /> <ci> c </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='rational'> 1 <sep /> 4 </cn> <apply> <plus /> <pi /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <imaginaryi /> <ci> z </ci> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <imaginaryi /> <ci> b </ci> <apply> <power /> <apply> <plus /> <ci> a </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <imaginaryi /> <ci> b </ci> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <apply> <plus /> <ci> a </ci> <apply> <times /> <ci> b </ci> <apply> <sinh /> <ci> z </ci> </apply> </apply> </apply> <apply> <power /> <apply> <plus /> <ci> a </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <imaginaryi /> <ci> b </ci> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <times /> <apply> <plus /> <ci> b </ci> <apply> <times /> <imaginaryi /> <ci> c </ci> </apply> </apply> <apply> <power /> <apply> <plus /> <ci> a </ci> <apply> <times /> <ci> b </ci> <apply> <sinh /> <ci> z </ci> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 4 </cn> <apply> <cosh /> <ci> z </ci> </apply> <apply> <power /> <apply> <plus /> <ci> a </ci> <apply> <times /> <ci> b </ci> <apply> <sinh /> <ci> z </ci> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <plus /> <ci> c </ci> <apply> <times /> <ci> b </ci> <apply> <sinh /> <ci> z </ci> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["\[Integral]", RowBox[List[FractionBox[SuperscriptBox[RowBox[List["Cosh", "[", "z_", "]"]], "2"], RowBox[List[SqrtBox[RowBox[List["a_", "+", RowBox[List["b_", " ", RowBox[List["Sinh", "[", "z_", "]"]]]]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["c_", "+", RowBox[List["b_", " ", RowBox[List["Sinh", "[", "z_", "]"]]]]]], ")"]], "2"]]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List[RowBox[List["-", FractionBox[RowBox[List["8", " ", "\[ImaginaryI]", " ", "a", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["a", "-", "c"]], ")"]], " ", 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["Sinh", "[", "z", "]"]]]]]]]]], "]"]]]], ",", FractionBox[RowBox[List["a", "-", RowBox[List["\[ImaginaryI]", " ", "b"]]]], RowBox[List["a", "+", RowBox[List["\[ImaginaryI]", " ", "b"]]]]]]], "]"]]]], "+", RowBox[List["c", " ", RowBox[List["EllipticPi", "[", RowBox[List[FractionBox[RowBox[List["a", "-", RowBox[List["\[ImaginaryI]", " ", "b"]]]], RowBox[List["a", "-", "c"]]], ",", 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["Sinh", "[", "z", "]"]]]]]]]]], "]"]]]], ",", FractionBox[RowBox[List["a", "-", RowBox[List["\[ImaginaryI]", " ", "b"]]]], RowBox[List["a", "+", RowBox[List["\[ImaginaryI]", " ", "b"]]]]]]], "]"]]]]]], ")"]], " ", RowBox[List["Sech", "[", "z", "]"]], " ", SqrtBox[FractionBox[RowBox[List["b", "-", RowBox[List["\[ImaginaryI]", " ", "b", " ", RowBox[List["Sinh", "[", "z", "]"]]]]]], RowBox[List[RowBox[List["\[ImaginaryI]", " ", "a"]], "+", "b"]]]], " ", SqrtBox[FractionBox[RowBox[List["b", "+", RowBox[List["\[ImaginaryI]", " ", "b", " ", RowBox[List["Sinh", "[", "z", "]"]]]]]], RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "a"]], "+", "b"]]]]]], RowBox[List[SqrtBox[FractionBox["1", RowBox[List[RowBox[List["-", "a"]], "+", RowBox[List["\[ImaginaryI]", " ", "b"]]]]]], " ", SuperscriptBox["b", "2"], " ", RowBox[List["(", RowBox[List["a", "-", "c"]], ")"]]]]]]], "+", FractionBox[RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[RowBox[List["Cosh", "[", "z", "]"]], "+", RowBox[List["Cosh", "[", RowBox[List["3", " ", "z"]], "]"]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", RowBox[List["(", RowBox[List["a", "+", RowBox[List["\[ImaginaryI]", " ", "b"]]]], ")"]], " ", RowBox[List["(", RowBox[List["a", "-", "c"]], ")"]], " ", 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["Sinh", "[", "z", "]"]]]]]]]]], "]"]]]], ",", FractionBox[RowBox[List["a", "-", RowBox[List["\[ImaginaryI]", " ", "b"]]]], RowBox[List["a", "+", RowBox[List["\[ImaginaryI]", " ", "b"]]]]]]], "]"]]]], "+", RowBox[List["2", " ", RowBox[List["(", RowBox[List["a", "-", "c"]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "b"]], "+", "c"]], ")"]], " ", 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["Sinh", "[", "z", "]"]]]]]]]]], "]"]]]], ",", FractionBox[RowBox[List["a", "-", RowBox[List["\[ImaginaryI]", " ", "b"]]]], RowBox[List["a", "+", RowBox[List["\[ImaginaryI]", " ", "b"]]]]]]], "]"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["b", "2"], "+", RowBox[List["2", " ", SuperscriptBox["c", "2"]]]]], ")"]], " ", RowBox[List["EllipticPi", "[", RowBox[List[FractionBox[RowBox[List["a", "-", RowBox[List["\[ImaginaryI]", " ", "b"]]]], RowBox[List["a", "-", "c"]]], ",", 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["Sinh", "[", "z", "]"]]]]]]]]], "]"]]]], ",", FractionBox[RowBox[List["a", "-", RowBox[List["\[ImaginaryI]", " ", "b"]]]], RowBox[List["a", "+", RowBox[List["\[ImaginaryI]", " ", "b"]]]]]]], "]"]]]]]], ")"]], " ", SuperscriptBox[RowBox[List["Sech", "[", "z", "]"]], "2"], " ", RowBox[List["Sech", "[", RowBox[List["2", " ", "z"]], "]"]], " ", SqrtBox[FractionBox[RowBox[List["b", "-", RowBox[List["\[ImaginaryI]", " ", "b", " ", RowBox[List["Sinh", "[", "z", "]"]]]]]], RowBox[List[RowBox[List["\[ImaginaryI]", " ", "a"]], "+", "b"]]]], " ", SqrtBox[FractionBox[RowBox[List["b", "+", RowBox[List["\[ImaginaryI]", " ", "b", " ", RowBox[List["Sinh", "[", "z", "]"]]]]]], RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "a"]], "+", "b"]]]]]], RowBox[List[SqrtBox[FractionBox["1", RowBox[List[RowBox[List["-", "a"]], "+", RowBox[List["\[ImaginaryI]", " ", "b"]]]]]], " ", SuperscriptBox["b", "2"], " ", RowBox[List["(", RowBox[List["a", "-", "c"]], ")"]]]]], "+", FractionBox[RowBox[List["6", " ", "b", " ", RowBox[List["EllipticPi", "[", RowBox[List[FractionBox[RowBox[List["2", " ", "b"]], RowBox[List["b", "+", RowBox[List["\[ImaginaryI]", " ", "c"]]]]], ",", RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List["\[Pi]", "-", RowBox[List["2", " ", "\[ImaginaryI]", " ", "z"]]]], ")"]]]], ",", RowBox[List["-", FractionBox[RowBox[List["2", " ", "\[ImaginaryI]", " ", "b"]], RowBox[List["a", "-", RowBox[List["\[ImaginaryI]", " ", "b"]]]]]]]]], "]"]], " ", SqrtBox[FractionBox[RowBox[List["a", "+", RowBox[List["b", " ", RowBox[List["Sinh", "[", "z", "]"]]]]]], RowBox[List["a", "-", RowBox[List["\[ImaginaryI]", " ", "b"]]]]]]]], RowBox[List[RowBox[List["(", RowBox[List["b", "+", RowBox[List["\[ImaginaryI]", " ", "c"]]]], ")"]], " ", SqrtBox[RowBox[List["a", "+", RowBox[List["b", " ", RowBox[List["Sinh", "[", "z", "]"]]]]]]]]]], "-", FractionBox[RowBox[List["4", " ", RowBox[List["Cosh", "[", "z", "]"]], " ", SqrtBox[RowBox[List["a", "+", RowBox[List["b", " ", RowBox[List["Sinh", "[", "z", "]"]]]]]]]]], RowBox[List["c", "+", RowBox[List["b", " ", RowBox[List["Sinh", "[", "z", "]"]]]]]]]]], RowBox[List["4", " ", "b", " ", RowBox[List["(", RowBox[List["a", "-", "c"]], ")"]]]]]]]]]

2002-12-18