Hyperbolic cotangent: Integration (formula 01.22.21.0128)

Coth

$Mathematica Notation$

Elementary Functions

Indefinite integration

Involving one direct function and elementary functions

Involving hyperbolic functions

Involving algebraic functions of cosh

Involving (a+b cosh(c z))^beta

http://functions.wolfram.com/01.22.21.0128.01

Input Form

Integrate[Coth[c z]/Sqrt[a + a Cosh[c z]], z] == (-1 + Cosh[(c z)/2] (-Log[Cosh[(c z)/4]] + Log[Sinh[(c z)/4]]))/ (c Sqrt[a (1 + Cosh[c z])])

Standard Form

Cell[BoxData[RowBox[List[RowBox[List["\[Integral]", RowBox[List[FractionBox[RowBox[List["Coth", "[", RowBox[List["c", " ", "z"]], "]"]], SqrtBox[RowBox[List["a", "+", RowBox[List["a", " ", RowBox[List["Cosh", "[", RowBox[List["c", " ", "z"]], "]"]]]]]]]], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", FractionBox[RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List[RowBox[List["Cosh", "[", FractionBox[RowBox[List["c", " ", "z"]], "2"], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", RowBox[List["Log", "[", RowBox[List["Cosh", "[", FractionBox[RowBox[List["c", " ", "z"]], "4"], "]"]], "]"]]]], "+", RowBox[List["Log", "[", RowBox[List["Sinh", "[", FractionBox[RowBox[List["c", " ", "z"]], "4"], "]"]], "]"]]]], ")"]]]]]], RowBox[List["c", " ", SqrtBox[RowBox[List["a", " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["Cosh", "[", RowBox[List["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> <mfrac> <mrow> <mi> coth </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> c </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <msqrt> <mrow> <mrow> <mrow> <mi> cosh </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> c </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mi> a </mi> </mrow> <mo> + </mo> <mi> a </mi> </mrow> </msqrt> </mfrac> <mo> ⁢ </mo> <mrow> <mo> ⅆ </mo> <mi> z </mi> </mrow> </mrow> </mrow> <mo> ⩵ </mo> <mfrac> <mrow> <mrow> <mrow> <mi> cosh </mi> <mo> ⁡ </mo> <mo> ( </mo> <mfrac> <mrow> <mi> c </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mn> 2 </mn> </mfrac> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> log </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> sinh </mi> <mo> ⁡ </mo> <mo> ( </mo> <mfrac> <mrow> <mi> c </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mn> 4 </mn> </mfrac> <mo> ) </mo> </mrow> <mo> ) </mo> </mrow> <mo> - </mo> <mrow> <mi> log </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> cosh </mi> <mo> ⁡ </mo> <mo> ( </mo> <mfrac> <mrow> <mi> c </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mn> 4 </mn> </mfrac> <mo> ) </mo> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mrow> <mi> c </mi> <mo> ⁢ </mo> <msqrt> <mrow> <mi> a </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> cosh </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> c </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </msqrt> </mrow> </mfrac> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <int /> <bvar> <ci> z </ci> </bvar> <apply> <times /> <apply> <coth /> <apply> <times /> <ci> c </ci> <ci> z </ci> </apply> </apply> <apply> <power /> <apply> <power /> <apply> <plus /> <apply> <times /> <apply> <cosh /> <apply> <times /> <ci> c </ci> <ci> z </ci> </apply> </apply> <ci> a </ci> </apply> <ci> a </ci> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <apply> <plus /> <apply> <times /> <apply> <cosh /> <apply> <times /> <ci> c </ci> <ci> z </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <plus /> <apply> <ln /> <apply> <sinh /> <apply> <times /> <ci> c </ci> <ci> z </ci> <apply> <power /> <cn type='integer'> 4 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ln /> <apply> <cosh /> <apply> <times /> <ci> c </ci> <ci> z </ci> <apply> <power /> <cn type='integer'> 4 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> <apply> <power /> <apply> <times /> <ci> c </ci> <apply> <power /> <apply> <times /> <ci> a </ci> <apply> <plus /> <apply> <cosh /> <apply> <times /> <ci> c </ci> <ci> z </ci> </apply> </apply> <cn type='integer'> 1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

Rule Form

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["\[Integral]", RowBox[List[FractionBox[RowBox[List["Coth", "[", RowBox[List["c_", " ", "z_"]], "]"]], SqrtBox[RowBox[List["a_", "+", RowBox[List["a_", " ", RowBox[List["Cosh", "[", RowBox[List["c_", " ", "z_"]], "]"]]]]]]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List[RowBox[List["Cosh", "[", FractionBox[RowBox[List["c", " ", "z"]], "2"], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", RowBox[List["Log", "[", RowBox[List["Cosh", "[", FractionBox[RowBox[List["c", " ", "z"]], "4"], "]"]], "]"]]]], "+", RowBox[List["Log", "[", RowBox[List["Sinh", "[", FractionBox[RowBox[List["c", " ", "z"]], "4"], "]"]], "]"]]]], ")"]]]]]], RowBox[List["c", " ", SqrtBox[RowBox[List["a", " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["Cosh", "[", RowBox[List["c", " ", "z"]], "]"]]]], ")"]]]]]]]]]]]]

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

2002-12-18