Hyperbolic tangent: Integration (formula 01.21.21.0077)

Tanh

$Mathematica Notation$

Elementary Functions

Indefinite integration

Involving one direct function and elementary functions

Involving hyperbolic functions

Involving power of sinh

Involving sinh^mu(b z)

http://functions.wolfram.com/01.21.21.0077.01

Input Form

Integrate[Sinh[z]^10 Tanh[z], z] == (1/15360) (8430 Cosh[2 z] - 1740 Cosh[4 z] + 335 Cosh[6 z] - 45 Cosh[8 z] + 3 Cosh[10 z] - 15360 Log[Cosh[z]])

Standard Form

Cell[BoxData[RowBox[List[RowBox[List["\[Integral]", RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["Sinh", "[", "z", "]"]], "10"], " ", RowBox[List["Tanh", "[", "z", "]"]]]], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[FractionBox["1", "15360"], RowBox[List["(", RowBox[List[RowBox[List["8430", " ", RowBox[List["Cosh", "[", RowBox[List["2", " ", "z"]], "]"]]]], "-", RowBox[List["1740", " ", RowBox[List["Cosh", "[", RowBox[List["4", " ", "z"]], "]"]]]], "+", RowBox[List["335", " ", RowBox[List["Cosh", "[", RowBox[List["6", " ", "z"]], "]"]]]], "-", RowBox[List["45", " ", RowBox[List["Cosh", "[", RowBox[List["8", " ", "z"]], "]"]]]], "+", RowBox[List["3", " ", RowBox[List["Cosh", "[", RowBox[List["10", " ", "z"]], "]"]]]], "-", RowBox[List["15360", " ", RowBox[List["Log", "[", RowBox[List["Cosh", "[", "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> <mrow> <msup> <mi> sinh </mi> <mn> 10 </mn> </msup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> tanh </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mo> ⁢ </mo> <mrow> <mo> ⅆ </mo> <mi> z </mi> </mrow> </mrow> </mrow> <mo> ⩵ </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 15360 </mn> </mfrac> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 8430 </mn> <mo> ⁢ </mo> <mrow> <mi> cosh </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mn> 1740 </mn> <mo> ⁢ </mo> <mrow> <mi> cosh </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mn> 4 </mn> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mn> 335 </mn> <mo> ⁢ </mo> <mrow> <mi> cosh </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mn> 6 </mn> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mn> 45 </mn> <mo> ⁢ </mo> <mrow> <mi> cosh </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mn> 8 </mn> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mn> 3 </mn> <mo> ⁢ </mo> <mrow> <mi> cosh </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mn> 10 </mn> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mn> 15360 </mn> <mo> ⁢ </mo> <mrow> <mi> log </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> cosh </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </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> <sinh /> <ci> z </ci> </apply> <cn type='integer'> 10 </cn> </apply> <apply> <tanh /> <ci> z </ci> </apply> </apply> </apply> <apply> <times /> <cn type='rational'> 1 <sep /> 15360 </cn> <apply> <plus /> <apply> <times /> <cn type='integer'> 8430 </cn> <apply> <cosh /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> z </ci> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 1740 </cn> <apply> <cosh /> <apply> <times /> <cn type='integer'> 4 </cn> <ci> z </ci> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 335 </cn> <apply> <cosh /> <apply> <times /> <cn type='integer'> 6 </cn> <ci> z </ci> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 45 </cn> <apply> <cosh /> <apply> <times /> <cn type='integer'> 8 </cn> <ci> z </ci> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 3 </cn> <apply> <cosh /> <apply> <times /> <cn type='integer'> 10 </cn> <ci> z </ci> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 15360 </cn> <apply> <ln /> <apply> <cosh /> <ci> z </ci> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

Rule Form

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["\[Integral]", RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["Sinh", "[", "z_", "]"]], "10"], " ", RowBox[List["Tanh", "[", "z_", "]"]]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List[RowBox[List["8430", " ", RowBox[List["Cosh", "[", RowBox[List["2", " ", "z"]], "]"]]]], "-", RowBox[List["1740", " ", RowBox[List["Cosh", "[", RowBox[List["4", " ", "z"]], "]"]]]], "+", RowBox[List["335", " ", RowBox[List["Cosh", "[", RowBox[List["6", " ", "z"]], "]"]]]], "-", RowBox[List["45", " ", RowBox[List["Cosh", "[", RowBox[List["8", " ", "z"]], "]"]]]], "+", RowBox[List["3", " ", RowBox[List["Cosh", "[", RowBox[List["10", " ", "z"]], "]"]]]], "-", RowBox[List["15360", " ", RowBox[List["Log", "[", RowBox[List["Cosh", "[", "z", "]"]], "]"]]]]]], "15360"]]]]]

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

2002-12-18