Tangent: Integration (formula 01.08.21.0069)

Tan

$Mathematica Notation$

Elementary Functions

Tan[z]

Integration

Indefinite integration

Involving one direct function and elementary functions

Involving trigonometric and exponential functions

Involving powers of sin and exp

Involving e^{p z} sin^m(b z)

http://functions.wolfram.com/01.08.21.0069.01

Input Form

Integrate[E^(p z) Sin[b z]^m Tan[c z], z] == ((-I) ((1/(p (2 I c + p))) (Binomial[m, m/2] (E^(p z) (2 I c + p) Hypergeometric2F1[-((I p)/(2 c)), 1, 1 - (I p)/(2 c), -E^(2 I c z)] - E^((2 I c + p) z) p Hypergeometric2F1[1 - (I p)/(2 c), 1, 2 - (I p)/(2 c), -E^(2 I c z)]) (-1 + Mod[m, 2])) + Sum[(-1)^k Binomial[m, k] (-((I (-1)^m E^((2 I c - I b (-2 k + m) + p) z) Hypergeometric2F1[ (2 c + 2 b k - b m - I p)/(2 c), 1, (4 c + 2 b k - b m - I p)/ (2 c), -E^(2 I c z)])/(2 c + 2 b k - b m - I p)) + (E^((I b (-2 k + m) + p) z) Hypergeometric2F1[(-2 b k + b m - I p)/ (2 c), 1, (2 c - 2 b k + b m - I p)/(2 c), -E^(2 I c z)])/ (2 I b k - I b m - p) - (I E^((2 I c + I b (-2 k + m) + p) z) Hypergeometric2F1[(2 c - 2 b k + b m - I p)/(2 c), 1, (4 c - 2 b k + b m - I p)/(2 c), -E^(2 I c z)])/ (2 c - 2 b k + b m - I p) - ((-1)^m E^(((-I) b (-2 k + m) + p) z) Hypergeometric2F1[-((-2 b k + b m + I p)/(2 c)), 1, (2 c + 2 b k - b m - I p)/(2 c), -E^(2 I c z)])/ (2 I b k - I b m + p)), {k, 0, Floor[(1/2) (-1 + m)]}]/I^m))/2^m /; Element[m, Integers] && m > 0

Standard Form

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MathML Form

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type='integer'> -1 </cn> <apply> <times /> <imaginaryi /> <ci> p </ci> </apply> </apply> </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> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> c </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <ci> b </ci> <ci> k </ci> </apply> </apply> <apply> <times /> <ci> b </ci> <ci> m </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <imaginaryi /> <ci> p </ci> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> c </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <exponentiale /> <apply> <times /> <cn type='integer'> 2 </cn> <imaginaryi /> <ci> c </ci> <ci> z </ci> </apply> </apply> </apply> 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type='integer'> -1 </cn> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> -2 </cn> <ci> b </ci> <ci> k </ci> </apply> <apply> <times /> <ci> b </ci> <ci> m </ci> </apply> <apply> <times /> <imaginaryi /> <ci> p </ci> </apply> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> c </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <cn type='integer'> 1 </cn> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> c </ci> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <ci> b </ci> <ci> k </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> b </ci> <ci> m </ci> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <imaginaryi /> <ci> p </ci> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> c </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <exponentiale /> <apply> <times /> <cn type='integer'> 2 </cn> <imaginaryi /> <ci> c </ci> <ci> z </ci> </apply> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <imaginaryi /> <ci> b </ci> <ci> k </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <imaginaryi /> <ci> b </ci> <ci> m </ci> </apply> </apply> <ci> p </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <times /> <imaginaryi /> <apply> <power /> <exponentiale /> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <imaginaryi /> <ci> c </ci> </apply> <apply> <times /> <imaginaryi /> <ci> b </ci> <apply> <plus /> <ci> m </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <ci> k </ci> </apply> </apply> </apply> </apply> <ci> p 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<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> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> 4 </cn> <ci> c </ci> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <ci> b </ci> <ci> k </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> b </ci> <ci> m </ci> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <imaginaryi /> <ci> p </ci> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> c </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <exponentiale /> <apply> <times /> <cn type='integer'> 2 </cn> <imaginaryi /> <ci> c </ci> <ci> z </ci> </apply> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> c 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Rule Form

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

2002-12-18