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http://functions.wolfram.com/01.20.21.2420.01
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Integrate[Cosh[c z]^4/(a + b Cosh[c z]^2)^(3/2), z] ==
-(I (2 (2 a^2 + 3 a b + b^2) Sqrt[(2 a + b + b Cosh[2 c z])/(a + b)]
EllipticE[I c z, b/(a + b)] -
a (4 (a + b) Sqrt[(2 a + b + b Cosh[2 c z])/(a + b)]
EllipticF[I c z, b/(a + b)] + I Sqrt[2] b Sinh[2 c z])))/
(2 b^2 (a + b) c Sqrt[2 a + b + b Cosh[2 c z]])
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Cell[BoxData[RowBox[List[RowBox[List["\[Integral]", RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["Cosh", "[", RowBox[List["c", " ", "z"]], "]"]], "4"], " "]], SuperscriptBox[RowBox[List["(", RowBox[List["a", "+", RowBox[List["b", " ", SuperscriptBox[RowBox[List["Cosh", "[", RowBox[List["c", " ", "z"]], "]"]], "2"]]]]], ")"]], FractionBox["3", "2"]]], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[RowBox[List["-", RowBox[List["(", RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", SuperscriptBox["a", "2"]]], "+", RowBox[List["3", " ", "a", " ", "b"]], "+", SuperscriptBox["b", "2"]]], ")"]], " ", SqrtBox[FractionBox[RowBox[List[RowBox[List["2", " ", "a"]], "+", "b", "+", RowBox[List["b", " ", RowBox[List["Cosh", "[", RowBox[List["2", " ", "c", " ", "z"]], "]"]]]]]], RowBox[List["a", "+", "b"]]]], " ", RowBox[List["EllipticE", "[", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "c", " ", "z"]], ",", FractionBox["b", RowBox[List["a", "+", "b"]]]]], "]"]]]], "-", RowBox[List["a", " ", RowBox[List["(", RowBox[List[RowBox[List["4", " ", RowBox[List["(", RowBox[List["a", "+", "b"]], ")"]], " ", SqrtBox[FractionBox[RowBox[List[RowBox[List["2", " ", "a"]], "+", "b", "+", RowBox[List["b", " ", RowBox[List["Cosh", "[", RowBox[List["2", " ", "c", " ", "z"]], "]"]]]]]], RowBox[List["a", "+", "b"]]]], " ", RowBox[List["EllipticF", "[", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "c", " ", "z"]], ",", FractionBox["b", RowBox[List["a", "+", "b"]]]]], "]"]]]], "+", RowBox[List["\[ImaginaryI]", " ", SqrtBox["2"], " ", "b", " ", RowBox[List["Sinh", "[", RowBox[List["2", " ", "c", " ", "z"]], "]"]]]]]], ")"]]]]]], ")"]]]], ")"]]]], "/", RowBox[List["(", RowBox[List["2", " ", SuperscriptBox["b", "2"], " ", RowBox[List["(", RowBox[List["a", "+", "b"]], ")"]], " ", "c", " ", SqrtBox[RowBox[List[RowBox[List["2", " ", "a"]], "+", "b", "+", RowBox[List["b", " ", RowBox[List["Cosh", "[", RowBox[List["2", " ", "c", " ", "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> <mfrac> <mrow> <msup> <mi> cosh </mi> <mn> 4 </mn> </msup> <mo> ( </mo> <mrow> <mi> c </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mi> a </mi> <mo> + </mo> <mrow> <mi> b </mi> <mo> ⁢ </mo> <mrow> <msup> <mi> cosh </mi> <mn> 2 </mn> </msup> <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> </mfrac> <mo> ⁢ </mo> <mrow> <mo> ⅆ </mo> <mi> z </mi> </mrow> </mrow> </mrow> <mo> ⩵ </mo> <mrow> <mrow> <mo> - </mo> <mrow> <mo> ( </mo> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <msup> <mi> a </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 3 </mn> <mo> ⁢ </mo> <mi> b </mi> <mo> ⁢ </mo> <mi> a </mi> </mrow> <mo> + </mo> <msup> <mi> b </mi> <mn> 2 </mn> </msup> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msqrt> <mfrac> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> a </mi> </mrow> <mo> + </mo> <mi> b </mi> <mo> + </mo> <mrow> <mi> b </mi> <mo> ⁢ </mo> <mrow> <mi> cosh </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> <mi> b </mi> </mrow> </mfrac> </msqrt> <mo> ⁢ </mo> <mrow> <mi> E </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> c </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ❘ </mo> <mfrac> <mi> b </mi> <mrow> <mi> a </mi> <mo> + </mo> <mi> b </mi> </mrow> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mi> a </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 4 </mn> <mo> ⁢ </mo> <msqrt> <mfrac> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> a </mi> </mrow> <mo> + </mo> <mi> b </mi> <mo> + </mo> <mrow> <mi> b </mi> <mo> ⁢ </mo> <mrow> <mi> cosh </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> <mi> b </mi> </mrow> </mfrac> </msqrt> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> a </mi> <mo> + </mo> <mi> b </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> F </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> c </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ❘ </mo> <mfrac> <mi> b </mi> <mrow> <mi> a </mi> <mo> + </mo> <mi> b </mi> </mrow> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <msqrt> <mn> 2 </mn> </msqrt> <mo> ⁢ </mo> <mi> b </mi> <mo> ⁢ </mo> <mi> ⅈ </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> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> / </mo> <mrow> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <msup> <mi> b </mi> <mn> 2 </mn> </msup> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> a </mi> <mo> + </mo> <mi> b </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mi> c </mi> <mo> ⁢ </mo> <msqrt> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> a </mi> </mrow> <mo> + </mo> <mi> b </mi> <mo> + </mo> <mrow> <mi> b </mi> <mo> ⁢ </mo> <mrow> <mi> cosh </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> </msqrt> </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 /> <apply> <times /> <ci> c </ci> <ci> z </ci> </apply> </apply> <cn type='integer'> 4 </cn> </apply> <apply> <power /> <apply> <power /> <apply> <plus /> <ci> a </ci> <apply> <times /> <ci> b </ci> <apply> <power /> <apply> <cosh /> <apply> <times /> <ci> c </ci> <ci> z </ci> </apply> </apply> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <cn type='rational'> 3 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <imaginaryi /> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 3 </cn> <ci> b </ci> <ci> a </ci> </apply> <apply> <power /> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <power /> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> a </ci> </apply> <ci> b </ci> <apply> <times /> <ci> b </ci> <apply> <cosh /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> c </ci> <ci> z </ci> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <plus /> <ci> a </ci> <ci> b </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <ci> EllipticE </ci> <apply> <times /> <imaginaryi /> <ci> c </ci> <ci> z </ci> </apply> <apply> <times /> <ci> b </ci> <apply> <power /> <apply> <plus /> <ci> a </ci> <ci> b </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> a </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> 4 </cn> <apply> <power /> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> a </ci> </apply> <ci> b </ci> <apply> <times /> <ci> b </ci> <apply> <cosh /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> c </ci> <ci> z </ci> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <plus /> <ci> a </ci> <ci> b </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <plus /> <ci> a </ci> <ci> b </ci> </apply> <apply> <ci> EllipticF </ci> <apply> <times /> <imaginaryi /> <ci> c </ci> <ci> z </ci> </apply> <apply> <times /> <ci> b </ci> <apply> <power /> <apply> <plus /> <ci> a </ci> <ci> b </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <ci> b </ci> <imaginaryi /> <apply> <sinh /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> c </ci> <ci> z </ci> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> <apply> <plus /> <ci> a </ci> <ci> b </ci> </apply> <ci> c </ci> <apply> <power /> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> a </ci> </apply> <ci> b </ci> <apply> <times /> <ci> b </ci> <apply> <cosh /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> c </ci> <ci> z </ci> </apply> </apply> </apply> </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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| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["\[Integral]", RowBox[List[FractionBox[SuperscriptBox[RowBox[List["Cosh", "[", RowBox[List["c_", " ", "z_"]], "]"]], "4"], SuperscriptBox[RowBox[List["(", RowBox[List["a_", "+", RowBox[List["b_", " ", SuperscriptBox[RowBox[List["Cosh", "[", RowBox[List["c_", " ", "z_"]], "]"]], "2"]]]]], ")"]], RowBox[List["3", "/", "2"]]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List["-", FractionBox[RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", SuperscriptBox["a", "2"]]], "+", RowBox[List["3", " ", "a", " ", "b"]], "+", SuperscriptBox["b", "2"]]], ")"]], " ", SqrtBox[FractionBox[RowBox[List[RowBox[List["2", " ", "a"]], "+", "b", "+", RowBox[List["b", " ", RowBox[List["Cosh", "[", RowBox[List["2", " ", "c", " ", "z"]], "]"]]]]]], RowBox[List["a", "+", "b"]]]], " ", RowBox[List["EllipticE", "[", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "c", " ", "z"]], ",", FractionBox["b", RowBox[List["a", "+", "b"]]]]], "]"]]]], "-", RowBox[List["a", " ", RowBox[List["(", RowBox[List[RowBox[List["4", " ", RowBox[List["(", RowBox[List["a", "+", "b"]], ")"]], " ", SqrtBox[FractionBox[RowBox[List[RowBox[List["2", " ", "a"]], "+", "b", "+", RowBox[List["b", " ", RowBox[List["Cosh", "[", RowBox[List["2", " ", "c", " ", "z"]], "]"]]]]]], RowBox[List["a", "+", "b"]]]], " ", RowBox[List["EllipticF", "[", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "c", " ", "z"]], ",", FractionBox["b", RowBox[List["a", "+", "b"]]]]], "]"]]]], "+", RowBox[List["\[ImaginaryI]", " ", SqrtBox["2"], " ", "b", " ", RowBox[List["Sinh", "[", RowBox[List["2", " ", "c", " ", "z"]], "]"]]]]]], ")"]]]]]], ")"]]]], RowBox[List["2", " ", SuperscriptBox["b", "2"], " ", RowBox[List["(", RowBox[List["a", "+", "b"]], ")"]], " ", "c", " ", SqrtBox[RowBox[List[RowBox[List["2", " ", "a"]], "+", "b", "+", RowBox[List["b", " ", RowBox[List["Cosh", "[", RowBox[List["2", " ", "c", " ", "z"]], "]"]]]]]]]]]]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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