Hyperbolic tangent: Integration (formula 01.21.21.0219)

Tanh

$Mathematica Notation$

Elementary Functions

Indefinite integration

Involving functions of the direct function

Involving products of the direct functions

http://functions.wolfram.com/01.21.21.0219.01

Input Form

Integrate[Tanh[a z + b] Tanh[a z], z] == z + (Coth[b] (Log[Cosh[a z]] - Log[Cosh[b + a z]]))/a

Standard Form

Cell[BoxData[RowBox[List[RowBox[List["\[Integral]", RowBox[List[RowBox[List["Tanh", "[", RowBox[List[RowBox[List["a", " ", "z"]], "+", "b"]], "]"]], " ", RowBox[List["Tanh", "[", RowBox[List["a", " ", "z"]], "]"]], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List["z", "+", FractionBox[RowBox[List[RowBox[List["Coth", "[", "b", "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["Log", "[", RowBox[List["Cosh", "[", RowBox[List["a", " ", "z"]], "]"]], "]"]], "-", RowBox[List["Log", "[", RowBox[List["Cosh", "[", 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> tanh </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> tanh </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> <mrow> <mi> z </mi> <mo> + </mo> <mfrac> <mrow> <mrow> <mi> coth </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> cosh </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> cosh </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> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <int /> <bvar> <ci> z </ci> </bvar> <apply> <times /> <apply> <tanh /> <apply> <plus /> <ci> b </ci> <apply> <times /> <ci> a </ci> <ci> z </ci> </apply> </apply> </apply> <apply> <tanh /> <apply> <times /> <ci> a </ci> <ci> z </ci> </apply> </apply> </apply> </apply> <apply> <plus /> <ci> z </ci> <apply> <times /> <apply> <coth /> <ci> b </ci> </apply> <apply> <plus /> <apply> <ln /> <apply> <cosh /> <apply> <times /> <ci> a </ci> <ci> z </ci> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ln /> <apply> <cosh /> <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> </apply> </annotation-xml> </semantics> </math>

Rule Form

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["\[Integral]", RowBox[List[RowBox[List[RowBox[List["Tanh", "[", RowBox[List[RowBox[List["a_", " ", "z_"]], "+", "b_"]], "]"]], " ", RowBox[List["Tanh", "[", RowBox[List["a_", " ", "z_"]], "]"]]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List["z", "+", FractionBox[RowBox[List[RowBox[List["Coth", "[", "b", "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["Log", "[", RowBox[List["Cosh", "[", RowBox[List["a", " ", "z"]], "]"]], "]"]], "-", RowBox[List["Log", "[", RowBox[List["Cosh", "[", RowBox[List["b", "+", RowBox[List["a", " ", "z"]]]], "]"]], "]"]]]], ")"]]]], "a"]]]]]]]

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

2002-12-18