Cosine: Integration (formula 01.07.21.0965)

Cos

$Mathematica Notation$

Elementary Functions

Cos[z]

Integration

Indefinite integration

Involving one direct function and elementary functions

Involving trigonometric and exponential functions

Involving rational functions of sin and exp

Involving e^{p z}(a+b sin²(d z))^-ncos(c z)

http://functions.wolfram.com/01.07.21.0965.01

Input Form

Integrate[(E^(p z) Cos[c z])/(a + b Sin[d z]^2)^2, z] == (-((1/(I c + 2 I d + p)) (E^((I c + 2 I d + p) z) ((2 a + b) (2 a + b + 2 Sqrt[a] Sqrt[a + b]) Hypergeometric2F1[ (c + 2 d - I p)/(2 d), 1, (c + 4 d - I p)/(2 d), (b E^(2 I d z))/(2 a + b - 2 Sqrt[a] Sqrt[a + b])] + (2 a + b) (-2 a - b + 2 Sqrt[a] Sqrt[a + b]) Hypergeometric2F1[ (c + 2 d - I p)/(2 d), 1, (c + 4 d - I p)/(2 d), (b E^(2 I d z))/(2 a + b + 2 Sqrt[a] Sqrt[a + b])] + 2 Sqrt[a] ((-(2 a^(3/2) + 2 Sqrt[a] b + 2 a Sqrt[a + b] + b Sqrt[a + b])) Hypergeometric2F1[(c + 2 d - I p)/(2 d), 2, (c + 4 d - I p)/(2 d), (b E^(2 I d z))/(2 a + b - 2 Sqrt[a] Sqrt[a + b])] + (2 a^(3/2) + 2 Sqrt[a] b - 2 a Sqrt[a + b] - b Sqrt[a + b]) Hypergeometric2F1[ (c + 2 d - I p)/(2 d), 2, (c + 4 d - I p)/(2 d), (b E^(2 I d z))/(2 a + b + 2 Sqrt[a] Sqrt[a + b])])))) - (1/((-I) c + 2 I d + p)) (E^(((-I) c + 2 I d + p) z) ((2 a + b) (2 a + b + 2 Sqrt[a] Sqrt[a + b]) Hypergeometric2F1[ -((c - 2 d + I p)/(2 d)), 1, -((c - 4 d + I p)/(2 d)), (b E^(2 I d z))/(2 a + b - 2 Sqrt[a] Sqrt[a + b])] + (2 a + b) (-2 a - b + 2 Sqrt[a] Sqrt[a + b]) Hypergeometric2F1[ -((c - 2 d + I p)/(2 d)), 1, -((c - 4 d + I p)/(2 d)), (b E^(2 I d z))/(2 a + b + 2 Sqrt[a] Sqrt[a + b])] + 2 Sqrt[a] ((-(2 a^(3/2) + 2 Sqrt[a] b + 2 a Sqrt[a + b] + b Sqrt[a + b])) Hypergeometric2F1[-((c - 2 d + I p)/(2 d)), 2, -((c - 4 d + I p)/(2 d)), (b E^(2 I d z))/(2 a + b - 2 Sqrt[a] Sqrt[a + b])] + (2 a^(3/2) + 2 Sqrt[a] b - 2 a Sqrt[a + b] - b Sqrt[a + b]) Hypergeometric2F1[ -((c - 2 d + I p)/(2 d)), 2, -((c - 4 d + I p)/(2 d)), (b E^(2 I d z))/(2 a + b + 2 Sqrt[a] Sqrt[a + b])]))))/ (4 a^(3/2) b (a + b)^(3/2))

Standard Form

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MathML Form

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Rule Form

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

2002-12-18