Hyperbolic cosecant: Integration (formula 01.23.21.0305)

Csch

$Mathematica Notation$

Elementary Functions

Indefinite integration

Involving functions of the direct function

Involving products of the direct functions

http://functions.wolfram.com/01.23.21.0305.01

Input Form

Integrate[Csch[a z + b] Csch[a z], z] == (Csch[b] (Log[Sinh[a z]] - Log[Sinh[b + a z]]))/a

Standard Form

Cell[BoxData[RowBox[List[RowBox[List["\[Integral]", RowBox[List[RowBox[List["Csch", "[", RowBox[List[RowBox[List["a", " ", "z"]], "+", "b"]], "]"]], " ", RowBox[List["Csch", "[", RowBox[List["a", " ", "z"]], "]"]], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", FractionBox[RowBox[List[RowBox[List["Csch", "[", "b", "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["Log", "[", RowBox[List["Sinh", "[", RowBox[List["a", " ", "z"]], "]"]], "]"]], "-", RowBox[List["Log", "[", RowBox[List["Sinh", "[", RowBox[List["b", "+", RowBox[List["a", " ", "z"]]]], "]"]], "]"]]]], ")"]]]], "a"]]]]]

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> <mrow> <mi> csch </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> b </mi> <mo> + </mo> <mrow> <mi> a </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> csch </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> a </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> <mfrac> <mrow> <mrow> <mi> csch </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> b </mi> <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> <mrow> <mi> a </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mo> ) </mo> </mrow> <mo> - </mo> <mrow> <mi> log </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> sinh </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> b </mi> <mo> + </mo> <mrow> <mi> a </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mi> a </mi> </mfrac> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <int /> <bvar> <ci> z </ci> </bvar> <apply> <times /> <apply> <csch /> <apply> <plus /> <ci> b </ci> <apply> <times /> <ci> a </ci> <ci> z </ci> </apply> </apply> </apply> <apply> <csch /> <apply> <times /> <ci> a </ci> <ci> z </ci> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <csch /> <ci> b </ci> </apply> <apply> <plus /> <apply> <ln /> <apply> <sinh /> <apply> <times /> <ci> a </ci> <ci> z </ci> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ln /> <apply> <sinh /> <apply> <plus /> <ci> b </ci> <apply> <times /> <ci> a </ci> <ci> z </ci> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <power /> <ci> a </ci> <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[RowBox[List[RowBox[List["Csch", "[", RowBox[List[RowBox[List["a_", " ", "z_"]], "+", "b_"]], "]"]], " ", RowBox[List["Csch", "[", RowBox[List["a_", " ", "z_"]], "]"]]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List[RowBox[List["Csch", "[", "b", "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["Log", "[", RowBox[List["Sinh", "[", RowBox[List["a", " ", "z"]], "]"]], "]"]], "-", RowBox[List["Log", "[", RowBox[List["Sinh", "[", RowBox[List["b", "+", RowBox[List["a", " ", "z"]]]], "]"]], "]"]]]], ")"]]]], "a"]]]]]

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

2002-12-18