Hyperbolic secant: Integration (formula 01.24.21.0233)

Sech

$Mathematica Notation$

Elementary Functions

Sech[z]

Integration

Indefinite integration

Involving one direct function and elementary functions

Involving hyperbolic, exponential and a power functions

Involving cosh, exp and power

Involving zⁿ e^{p z}cosh(b z) sech( c z)

http://functions.wolfram.com/01.24.21.0233.01

Input Form

Integrate[z^n E^(p z) Cosh[b z] Sech[c z], z] == n! E^(c z) (E^((-b + p) z) Sum[(((-1)^j z^(-j + n) (-b + p + c)^(-1 - j))/ (-j + n)!) HypergeometricPFQ[{Subscript[b, 1], \[Ellipsis], Subscript[b, j + 1], 1}, {1 + Subscript[b, 1], \[Ellipsis], 1 + Subscript[b, j + 1]}, -E^(2 c z)], {j, 0, n}] + E^((b + p) z) Sum[(((-1)^j z^(-j + n) (b + p + c)^(-1 - j))/(-j + n)!) HypergeometricPFQ[{Subscript[c, 1], \[Ellipsis], Subscript[c, j + 1], 1}, {1 + Subscript[c, 1], \[Ellipsis], 1 + Subscript[c, j + 1]}, -E^(2 c z)], {j, 0, n}]) /; Subscript[b, 1] == Subscript[b, 2] == \[Ellipsis] == Subscript[b, n + 1] == (c + p - b)/(2 c) && Subscript[c, 1] == Subscript[c, 2] == \[Ellipsis] == Subscript[c, n + 1] == (c + p + b)/(2 c) && Element[n, Integers] && n >= 0

Standard Form

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MathML Form

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</cn> </list> <list> <apply> <plus /> <apply> <times /> <apply> <plus /> <ci> c </ci> <ci> p </ci> <ci> b </ci> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> c </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='integer'> 1 </cn> </apply> <ci> … </ci> <apply> <plus /> <apply> <times /> <apply> <plus /> <ci> c </ci> <ci> p </ci> <ci> b </ci> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> c </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='integer'> 1 </cn> </apply> </list> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <exponentiale /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> c </ci> <ci> z </ci> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <in /> <ci> n </ci> <ci> ℕ </ci> </apply> </apply> </annotation-xml> </semantics> </math>

Rule Form

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

2002-12-18