Hyperbolic secant: Integration (formula 01.24.21.0382)

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 power of sinh

Involving sinh^u(b z) sech^nu(c z)

http://functions.wolfram.com/01.24.21.0382.01

Input Form

Integrate[Sqrt[Sinh[2 c z]] Sech[c z]^3, z] == (2 Sech[c z] Sqrt[Sinh[2 c z]] Tanh[c z])/(3 c)

Standard Form

Cell[BoxData[RowBox[List[RowBox[List["\[Integral]", RowBox[List[SqrtBox[RowBox[List["Sinh", "[", RowBox[List["2", " ", "c", " ", "z"]], "]"]]], SuperscriptBox[RowBox[List["Sech", "[", RowBox[List["c", " ", "z"]], "]"]], "3"], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", FractionBox[RowBox[List["2", " ", RowBox[List["Sech", "[", RowBox[List["c", " ", "z"]], "]"]], " ", SqrtBox[RowBox[List["Sinh", "[", RowBox[List["2", " ", "c", " ", "z"]], "]"]]], " ", RowBox[List["Tanh", "[", RowBox[List["c", " ", "z"]], "]"]]]], RowBox[List["3", " ", "c"]]]]]]]

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> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </msup> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> c </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <msup> <mi> sech </mi> <mn> 3 </mn> </msup> <mo> ( </mo> <mrow> <mi> c </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> <mn> 2 </mn> <mo> ⁢ </mo> <mrow> <mi> sech </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> c </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <msup> <mi> sinh </mi> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </msup> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> c </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> tanh </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> c </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mrow> <mn> 3 </mn> <mo> ⁢ </mo> <mi> c </mi> </mrow> </mfrac> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <int /> <bvar> <ci> z </ci> </bvar> <apply> <times /> <apply> <power /> <apply> <sinh /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> c </ci> <ci> z </ci> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <sech /> <apply> <times /> <ci> c </ci> <ci> z </ci> </apply> </apply> <cn type='integer'> 3 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <sech /> <apply> <times /> <ci> c </ci> <ci> z </ci> </apply> </apply> <apply> <power /> <apply> <sinh /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> c </ci> <ci> z </ci> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <tanh /> <apply> <times /> <ci> c </ci> <ci> z </ci> </apply> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 3 </cn> <ci> c </ci> </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[RowBox[List[SqrtBox[RowBox[List["Sinh", "[", RowBox[List["2", " ", "c_", " ", "z_"]], "]"]]], " ", SuperscriptBox[RowBox[List["Sech", "[", RowBox[List["c_", " ", "z_"]], "]"]], "3"]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List["2", " ", RowBox[List["Sech", "[", RowBox[List["c", " ", "z"]], "]"]], " ", SqrtBox[RowBox[List["Sinh", "[", RowBox[List["2", " ", "c", " ", "z"]], "]"]]], " ", RowBox[List["Tanh", "[", RowBox[List["c", " ", "z"]], "]"]]]], RowBox[List["3", " ", "c"]]]]]]]

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

2002-12-18