Hyperbolic cosecant: Integration (formula 01.23.21.0400)

Csch

$Mathematica Notation$

Elementary Functions

Csch[z]

Integration

Indefinite integration

Involving functions of the direct function and hyperbolic functions

Involving powers of the direct function and hyperbolic functions

Involving algebraic functions of cosh

http://functions.wolfram.com/01.23.21.0400.01

Input Form

Integrate[Sqrt[a + b Cosh[2 c z]] Csch[c z]^4, z] == ((2 (a^2 - 2 a b - b^2) Cosh[2 c z] - a (4 a + b - b Cosh[4 c z])) Coth[c z] Csch[c z]^2 + 4 I a (a + b) Sqrt[(a + b Cosh[2 c z])/(a + b)] EllipticE[I c z, (2 b)/(a + b)] - 4 I (a^2 - b^2) Sqrt[(a + b Cosh[2 c z])/(a + b)] EllipticF[I c z, (2 b)/(a + b)])/ (6 (a + b) c Sqrt[a + b Cosh[2 c z]])

Standard Form

Cell[BoxData[RowBox[List[RowBox[List["\[Integral]", RowBox[List[SqrtBox[RowBox[List["a", "+", RowBox[List["b", " ", RowBox[List["Cosh", "[", RowBox[List["2", " ", "c", " ", "z"]], "]"]]]]]]], " ", SuperscriptBox[RowBox[List["Csch", "[", RowBox[List["c", " ", "z"]], "]"]], "4"], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", " ", RowBox[List["(", RowBox[List[SuperscriptBox["a", "2"], "-", RowBox[List["2", " ", "a", " ", "b"]], "-", SuperscriptBox["b", "2"]]], ")"]], " ", RowBox[List["Cosh", "[", RowBox[List["2", " ", "c", " ", "z"]], "]"]]]], "-", RowBox[List["a", " ", RowBox[List["(", RowBox[List[RowBox[List["4", " ", "a"]], "+", "b", "-", RowBox[List["b", " ", RowBox[List["Cosh", "[", RowBox[List["4", " ", "c", " ", "z"]], "]"]]]]]], ")"]]]]]], ")"]], " ", RowBox[List["Coth", "[", RowBox[List["c", " ", "z"]], "]"]], " ", SuperscriptBox[RowBox[List["Csch", "[", RowBox[List["c", " ", "z"]], "]"]], "2"]]], "+", RowBox[List["4", " ", "\[ImaginaryI]", " ", "a", " ", RowBox[List["(", RowBox[List["a", "+", "b"]], ")"]], " ", SqrtBox[FractionBox[RowBox[List["a", "+", 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[RowBox[List["2", " ", "b"]], RowBox[List["a", "+", "b"]]]]], "]"]]]], "-", RowBox[List["4", " ", "\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[SuperscriptBox["a", "2"], "-", SuperscriptBox["b", "2"]]], ")"]], " ", SqrtBox[FractionBox[RowBox[List["a", "+", 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[RowBox[List["2", " ", "b"]], RowBox[List["a", "+", "b"]]]]], "]"]]]]]], ")"]], "/", RowBox[List["(", RowBox[List["6", " ", RowBox[List["(", RowBox[List["a", "+", "b"]], ")"]], " ", "c", " ", SqrtBox[RowBox[List["a", "+", RowBox[List["b", " ", RowBox[List["Cosh", "[", RowBox[List["2", " ", "c", " ", "z"]], "]"]]]]]]]]], ")"]]]]]]]]

MathML Form

<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> <mrow> <msqrt> <mrow> <mi> a </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> <mo> ⁢ </mo> <mrow> <msup> <mi> csch </mi> <mn> 4 </mn> </msup> <mo> ( </mo> <mrow> <mi> c </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ⁢ </mo> <mrow> <mo> ⅆ </mo> <mi> z </mi> </mrow> </mrow> </mrow> <mo> ⩵ </mo> <mrow> <mfrac> <mn> 1 </mn> <mrow> <mn> 6 </mn> <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> <mi> a </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> </mfrac> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <msup> <mi> a </mi> <mn> 2 </mn> </msup> <mo> - </mo> <mrow> <mn> 2 </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> <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> <mo> - </mo> <mrow> <mi> a </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 4 </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> 4 </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> <mo> ⁢ </mo> <mrow> <mi> coth </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> c </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <msup> <mi> csch </mi> <mn> 2 </mn> </msup> <mo> ( </mo> <mrow> <mi> c </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mn> 4 </mn> <mo> ⁢ </mo> <mi> a </mi> <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> <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> <mi> ⅈ </mi> <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> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> b </mi> </mrow> <mrow> <mi> a </mi> <mo> + </mo> <mi> b </mi> </mrow> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mn> 4 </mn> <mo> ⁢ </mo> <mi> ⅈ </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> <msqrt> <mfrac> <mrow> <mi> a </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> 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> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> b </mi> </mrow> <mrow> <mi> a </mi> <mo> + </mo> <mi> b </mi> </mrow> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> </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> <plus /> <ci> a </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> <power /> <apply> <csch /> <apply> <times /> <ci> c </ci> <ci> z </ci> </apply> </apply> <cn type='integer'> 4 </cn> </apply> </apply> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <times /> <cn type='integer'> 6 </cn> <apply> <plus /> <ci> a </ci> <ci> b </ci> </apply> <ci> c </ci> <apply> <power /> <apply> <plus /> <ci> a </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> <plus /> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <plus /> <apply> <power /> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <ci> b </ci> <ci> a </ci> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <cosh /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> c </ci> <ci> z </ci> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> a </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> 4 </cn> <ci> a </ci> </apply> <ci> b </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> b </ci> <apply> <cosh /> <apply> <times /> <cn type='integer'> 4 </cn> <ci> c </ci> <ci> z </ci> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <coth /> <apply> <times /> <ci> c </ci> <ci> z </ci> </apply> </apply> <apply> <power /> <apply> <csch /> <apply> <times /> <ci> c </ci> <ci> z </ci> </apply> </apply> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 4 </cn> <ci> a </ci> <apply> <power /> <apply> <times /> <apply> <plus /> <ci> a </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> <imaginaryi /> <apply> <ci> EllipticE </ci> <apply> <times /> <imaginaryi /> <ci> c </ci> <ci> z </ci> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <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 /> <cn type='integer'> 4 </cn> <imaginaryi /> <apply> <plus /> <apply> <power /> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <apply> <plus /> <ci> a </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> EllipticF </ci> <apply> <times /> <imaginaryi /> <ci> c </ci> <ci> z </ci> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <ci> b </ci> <apply> <power /> <apply> <plus /> <ci> a </ci> <ci> b </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

Rule Form

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Date Added to functions.wolfram.com (modification date)

2002-12-18