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http://functions.wolfram.com/01.20.21.2434.01
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Integrate[Sqrt[(a + b Cosh[e z])/(c + d Cosh[e z])], z] ==
(2 Sqrt[(a + b Cosh[e z])/(c + d Cosh[e z])]
Sqrt[((c + d) Coth[(e z)/2]^2)/(c - d)]
Sqrt[((a + b) (c + d Cosh[e z]) Csch[(e z)/2]^2)/((-b) c + a d)]
((a + b) d EllipticF[ArcSin[Sqrt[((a + b) (c + d Cosh[e z])
Csch[(e z)/2]^2)/(-2 b c + 2 a d)]], (2 ((-b) c + a d))/
((a + b) (-c + d))] - b (c + d) EllipticPi[((-b) c + a d)/((a + b) d),
ArcSin[Sqrt[((a + b) (c + d Cosh[e z]) Csch[(e z)/2]^2)/
(-2 b c + 2 a d)]], (2 ((-b) c + a d))/((a + b) (-c + d))])
Tanh[(e z)/2])/((a + b) d e
Sqrt[((c + d) (a + b Cosh[e z]) Csch[(e z)/2]^2)/(b c - a d)])
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Cell[BoxData[RowBox[List[RowBox[List["\[Integral]", RowBox[List[SqrtBox[FractionBox[RowBox[List["a", "+", RowBox[List["b", " ", RowBox[List["Cosh", "[", RowBox[List["e", " ", "z"]], "]"]]]]]], RowBox[List["c", "+", RowBox[List["d", " ", RowBox[List["Cosh", "[", RowBox[List["e", " ", "z"]], "]"]]]]]]]], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[RowBox[List["(", RowBox[List["2", " ", SqrtBox[FractionBox[RowBox[List["a", "+", RowBox[List["b", " ", RowBox[List["Cosh", "[", RowBox[List["e", " ", "z"]], "]"]]]]]], RowBox[List["c", "+", RowBox[List["d", " ", RowBox[List["Cosh", "[", RowBox[List["e", " ", "z"]], "]"]]]]]]]], " ", SqrtBox[FractionBox[RowBox[List[RowBox[List["(", RowBox[List["c", "+", "d"]], ")"]], " ", SuperscriptBox[RowBox[List["Coth", "[", FractionBox[RowBox[List["e", " ", "z"]], "2"], "]"]], "2"]]], RowBox[List["c", "-", "d"]]]], " ", SqrtBox[FractionBox[RowBox[List[RowBox[List["(", RowBox[List["a", "+", "b"]], ")"]], " ", RowBox[List["(", RowBox[List["c", "+", RowBox[List["d", " ", RowBox[List["Cosh", "[", RowBox[List["e", " ", "z"]], "]"]]]]]], ")"]], " ", SuperscriptBox[RowBox[List["Csch", "[", FractionBox[RowBox[List["e", " ", "z"]], "2"], "]"]], "2"]]], RowBox[List[RowBox[List[RowBox[List["-", "b"]], " ", "c"]], "+", RowBox[List["a", " ", "d"]]]]]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["a", "+", "b"]], ")"]], " ", "d", " ", RowBox[List["EllipticF", "[", RowBox[List[RowBox[List["ArcSin", "[", SqrtBox[FractionBox[RowBox[List[RowBox[List["(", RowBox[List["a", "+", "b"]], ")"]], " ", RowBox[List["(", RowBox[List["c", "+", RowBox[List["d", " ", RowBox[List["Cosh", "[", RowBox[List["e", " ", "z"]], "]"]]]]]], ")"]], " ", SuperscriptBox[RowBox[List["Csch", "[", FractionBox[RowBox[List["e", " ", "z"]], "2"], "]"]], "2"]]], RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "b", " ", "c"]], "+", RowBox[List["2", " ", "a", " ", "d"]]]]]], "]"]], ",", FractionBox[RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "b"]], " ", "c"]], "+", RowBox[List["a", " ", "d"]]]], ")"]]]], RowBox[List[RowBox[List["(", RowBox[List["a", "+", "b"]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "c"]], "+", "d"]], ")"]]]]]]], "]"]]]], "-", RowBox[List["b", " ", RowBox[List["(", RowBox[List["c", "+", "d"]], ")"]], " ", RowBox[List["EllipticPi", "[", RowBox[List[FractionBox[RowBox[List[RowBox[List[RowBox[List["-", "b"]], " ", "c"]], "+", RowBox[List["a", " ", "d"]]]], RowBox[List[RowBox[List["(", RowBox[List["a", "+", "b"]], ")"]], " ", "d"]]], ",", RowBox[List["ArcSin", "[", SqrtBox[FractionBox[RowBox[List[RowBox[List["(", RowBox[List["a", "+", "b"]], ")"]], " ", RowBox[List["(", RowBox[List["c", "+", RowBox[List["d", " ", RowBox[List["Cosh", "[", RowBox[List["e", " ", "z"]], "]"]]]]]], ")"]], " ", SuperscriptBox[RowBox[List["Csch", "[", FractionBox[RowBox[List["e", " ", "z"]], "2"], "]"]], "2"]]], RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "b", " ", "c"]], "+", RowBox[List["2", " ", "a", " ", "d"]]]]]], "]"]], ",", FractionBox[RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "b"]], " ", "c"]], "+", RowBox[List["a", " ", "d"]]]], ")"]]]], RowBox[List[RowBox[List["(", RowBox[List["a", "+", "b"]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "c"]], "+", "d"]], ")"]]]]]]], "]"]]]]]], ")"]], " ", RowBox[List["Tanh", "[", FractionBox[RowBox[List["e", " ", "z"]], "2"], "]"]]]], ")"]], "/", RowBox[List["(", RowBox[List[RowBox[List["(", RowBox[List["a", "+", "b"]], ")"]], " ", "d", " ", "e", " ", SqrtBox[FractionBox[RowBox[List[RowBox[List["(", RowBox[List["c", "+", "d"]], ")"]], " ", RowBox[List["(", RowBox[List["a", "+", RowBox[List["b", " ", RowBox[List["Cosh", "[", RowBox[List["e", " ", "z"]], "]"]]]]]], ")"]], " ", SuperscriptBox[RowBox[List["Csch", "[", FractionBox[RowBox[List["e", " ", "z"]], "2"], "]"]], "2"]]], RowBox[List[RowBox[List["b", " ", "c"]], "-", RowBox[List["a", " ", "d"]]]]]]]], ")"]]]]]]]]
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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> <msqrt> <mfrac> <mrow> <mi> a </mi> <mo> + </mo> <mrow> <mi> b </mi> <mo> ⁢ </mo> <mrow> <mi> cosh </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> e </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mrow> <mi> c </mi> <mo> + </mo> <mrow> <mi> d </mi> <mo> ⁢ </mo> <mrow> <mi> cosh </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> e </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mfrac> </msqrt> <mo> ⁢ </mo> <mrow> <mo> ⅆ </mo> <mi> z </mi> </mrow> </mrow> </mrow> <mo> ⩵ </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <msqrt> <mfrac> <mrow> <mi> a </mi> <mo> + </mo> <mrow> <mi> b </mi> <mo> ⁢ </mo> <mrow> <mi> cosh </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> e </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mrow> <mi> c </mi> <mo> + </mo> <mrow> <mi> d </mi> <mo> ⁢ </mo> <mrow> <mi> cosh </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> e </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mfrac> </msqrt> <mo> ⁢ </mo> <msqrt> <mfrac> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> c </mi> <mo> + </mo> <mi> d </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <msup> <mi> coth </mi> <mn> 2 </mn> </msup> <mo> ( </mo> <mfrac> <mrow> <mi> e </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mn> 2 </mn> </mfrac> <mo> ) </mo> </mrow> </mrow> <mrow> <mi> c </mi> <mo> - </mo> <mi> d </mi> </mrow> </mfrac> </msqrt> <mo> ⁢ </mo> <msqrt> <mfrac> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> a </mi> <mo> + </mo> <mi> b </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> c </mi> <mo> + </mo> <mrow> <mi> d </mi> <mo> ⁢ </mo> <mrow> <mi> cosh </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> e </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <msup> <mi> csch </mi> <mn> 2 </mn> </msup> <mo> ( </mo> <mfrac> <mrow> <mi> e </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mn> 2 </mn> </mfrac> <mo> ) </mo> </mrow> </mrow> <mrow> <mrow> <mi> a </mi> <mo> ⁢ </mo> <mi> d </mi> </mrow> <mo> - </mo> <mrow> <mi> b </mi> <mo> ⁢ </mo> <mi> c </mi> </mrow> </mrow> </mfrac> </msqrt> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> a </mi> <mo> + </mo> <mi> b </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mi> d </mi> <mo> ⁢ </mo> <mrow> <mi> F </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mrow> <msup> <mi> sin </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <msqrt> <mfrac> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> a </mi> <mo> + </mo> <mi> b </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> c </mi> <mo> + </mo> <mrow> <mi> d </mi> <mo> ⁢ </mo> <mrow> <mi> cosh </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> e </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <msup> <mi> csch </mi> <mn> 2 </mn> </msup> <mo> ( </mo> <mfrac> <mrow> <mi> e </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mn> 2 </mn> </mfrac> <mo> ) </mo> </mrow> </mrow> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> a </mi> <mo> ⁢ </mo> <mi> d </mi> </mrow> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> b </mi> <mo> ⁢ </mo> <mi> c </mi> </mrow> </mrow> </mfrac> </msqrt> <mo> ) </mo> </mrow> <mo> ❘ </mo> <mfrac> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> a </mi> <mo> ⁢ </mo> <mi> d </mi> </mrow> <mo> - </mo> <mrow> <mi> b </mi> <mo> ⁢ </mo> <mi> c </mi> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> a </mi> <mo> + </mo> <mi> b </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> d </mi> <mo> - </mo> <mi> c </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mi> b </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> c </mi> <mo> + </mo> <mi> d </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> Π </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mfrac> <mrow> <mrow> <mi> a </mi> <mo> ⁢ </mo> <mi> d </mi> </mrow> <mo> - </mo> <mrow> <mi> b </mi> <mo> ⁢ </mo> <mi> c </mi> </mrow> </mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> a </mi> <mo> + </mo> <mi> b </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mi> d </mi> </mrow> </mfrac> <mo> ; </mo> <mrow> <mrow> <msup> <mi> sin </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <msqrt> <mfrac> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> a </mi> <mo> + </mo> <mi> b </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> c </mi> <mo> + </mo> <mrow> <mi> d </mi> <mo> ⁢ </mo> <mrow> <mi> cosh </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> e </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <msup> <mi> csch </mi> <mn> 2 </mn> </msup> <mo> ( </mo> <mfrac> <mrow> <mi> e </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mn> 2 </mn> </mfrac> <mo> ) </mo> </mrow> </mrow> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> a </mi> <mo> ⁢ </mo> <mi> d </mi> </mrow> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> b </mi> <mo> ⁢ </mo> <mi> c </mi> </mrow> </mrow> </mfrac> </msqrt> <mo> ) </mo> </mrow> <mo> ❘ </mo> <mfrac> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> a </mi> <mo> ⁢ </mo> <mi> d </mi> </mrow> <mo> - </mo> <mrow> <mi> b </mi> <mo> ⁢ </mo> <mi> c </mi> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> a </mi> <mo> + </mo> <mi> b </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> d </mi> <mo> - </mo> <mi> c </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mfrac> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> tanh </mi> <mo> ⁡ </mo> <mo> ( </mo> <mfrac> <mrow> <mi> e </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mn> 2 </mn> </mfrac> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> / </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> a </mi> <mo> + </mo> <mi> b </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mi> d </mi> <mo> ⁢ </mo> <mi> e </mi> <mo> ⁢ </mo> <msqrt> <mfrac> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> c </mi> <mo> + </mo> <mi> d </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> a </mi> <mo> + </mo> <mrow> <mi> b </mi> <mo> ⁢ </mo> <mrow> <mi> cosh </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> e </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <msup> <mi> csch </mi> <mn> 2 </mn> </msup> <mo> ( </mo> <mfrac> <mrow> <mi> e </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mn> 2 </mn> </mfrac> <mo> ) </mo> </mrow> </mrow> <mrow> <mrow> <mi> b </mi> <mo> ⁢ </mo> <mi> c </mi> </mrow> <mo> - </mo> <mrow> <mi> a </mi> <mo> ⁢ </mo> <mi> d </mi> </mrow> </mrow> </mfrac> </msqrt> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <int /> <bvar> <ci> z </ci> </bvar> <apply> <power /> <apply> <times /> <apply> <plus /> <ci> a </ci> <apply> <times /> <ci> b </ci> <apply> <cosh /> <apply> <times /> <ci> e </ci> <ci> z </ci> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <plus /> <ci> c </ci> <apply> <times /> <ci> d </ci> <apply> <cosh /> <apply> <times /> <ci> e </ci> <ci> z </ci> </apply> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <apply> <times /> <apply> <plus /> <ci> a </ci> <apply> <times /> <ci> b </ci> <apply> <cosh /> <apply> <times /> <ci> e </ci> <ci> z </ci> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <plus /> <ci> c </ci> <apply> <times /> <ci> d </ci> <apply> <cosh /> <apply> <times /> <ci> e </ci> <ci> z </ci> </apply> </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> c </ci> <ci> d </ci> </apply> <apply> <power /> <apply> <coth /> <apply> <times /> <ci> e </ci> <ci> z </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <plus /> <ci> c </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> d </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <times /> <apply> <plus /> <ci> a </ci> <ci> b </ci> </apply> <apply> <plus /> <ci> c </ci> <apply> <times /> <ci> d </ci> <apply> <cosh /> <apply> <times /> <ci> e </ci> <ci> z </ci> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <csch /> <apply> 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<apply> <times /> <cn type='integer'> 2 </cn> <ci> a </ci> <ci> d </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <ci> b </ci> <ci> c </ci> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <plus /> <apply> <times /> <ci> a </ci> <ci> d </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> b </ci> <ci> c </ci> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <apply> <plus /> <ci> a </ci> <ci> b </ci> </apply> <apply> <plus /> <ci> d </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> c </ci> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <tanh /> <apply> <times /> <ci> e </ci> <ci> z </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> 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</semantics> </math>
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Date Added to functions.wolfram.com (modification date)
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){kind=small}
