Hyperbolic secant: Integration (formula 01.24.21.0389)

Sech

$Mathematica Notation$

Elementary Functions

Indefinite integration

Involving functions of the direct function and hyperbolic functions

Involving powers of the direct function and hyperbolic functions

Involving cosh(b z)

http://functions.wolfram.com/01.24.21.0389.01

Input Form

Integrate[Cosh[4 z] Sech[z]^5, z] == (1/8) (70 ArcTan[Tanh[z/2]] - 29 Sech[z] Tanh[z] + 2 Sech[z]^3 Tanh[z])

Standard Form

Cell[BoxData[RowBox[List[RowBox[List["\[Integral]", RowBox[List[RowBox[List[RowBox[List["Cosh", "[", RowBox[List["4", " ", "z"]], "]"]], " ", SuperscriptBox[RowBox[List["Sech", "[", "z", "]"]], "5"]]], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[FractionBox["1", "8"], " ", RowBox[List["(", RowBox[List[RowBox[List["70", " ", RowBox[List["ArcTan", "[", RowBox[List["Tanh", "[", FractionBox["z", "2"], "]"]], "]"]]]], "-", RowBox[List["29", " ", RowBox[List["Sech", "[", "z", "]"]], " ", RowBox[List["Tanh", "[", "z", "]"]]]], "+", RowBox[List["2", " ", SuperscriptBox[RowBox[List["Sech", "[", "z", "]"]], "3"], " ", RowBox[List["Tanh", "[", "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> <mi> cosh </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mn> 4 </mn> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <msup> <mi> sech </mi> <mn> 5 </mn> </msup> <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> 8 </mn> </mfrac> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mrow> <mi> tanh </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <msup> <mi> sech </mi> <mn> 3 </mn> </msup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mn> 29 </mn> <mo> ⁢ </mo> <mrow> <mi> tanh </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> sech </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mn> 70 </mn> <mo> ⁢ </mo> <mrow> <msup> <mi> tan </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mrow> <mi> tanh </mi> <mo> ⁡ </mo> <mo> ( </mo> <mfrac> <mi> z </mi> <mn> 2 </mn> </mfrac> <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> <cosh /> <apply> <times /> <cn type='integer'> 4 </cn> <ci> z </ci> </apply> </apply> <apply> <power /> <apply> <sech /> <ci> z </ci> </apply> <cn type='integer'> 5 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='rational'> 1 <sep /> 8 </cn> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <tanh /> <ci> z </ci> </apply> <apply> <power /> <apply> <sech /> <ci> z </ci> </apply> <cn type='integer'> 3 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 29 </cn> <apply> <tanh /> <ci> z </ci> </apply> <apply> <sech /> <ci> z </ci> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 70 </cn> <apply> <arctan /> <apply> <tanh /> <apply> <times /> <ci> z </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </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[RowBox[List["Cosh", "[", RowBox[List["4", " ", "z_"]], "]"]], " ", SuperscriptBox[RowBox[List["Sech", "[", "z_", "]"]], "5"]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox["1", "8"], " ", RowBox[List["(", RowBox[List[RowBox[List["70", " ", RowBox[List["ArcTan", "[", RowBox[List["Tanh", "[", FractionBox["z", "2"], "]"]], "]"]]]], "-", RowBox[List["29", " ", RowBox[List["Sech", "[", "z", "]"]], " ", RowBox[List["Tanh", "[", "z", "]"]]]], "+", RowBox[List["2", " ", SuperscriptBox[RowBox[List["Sech", "[", "z", "]"]], "3"], " ", RowBox[List["Tanh", "[", "z", "]"]]]]]], ")"]]]]]]]]

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

2002-12-18