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http://functions.wolfram.com/01.07.21.1822.01
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Integrate[E^(p z) Cos[c z]^m Cos[a z + b]^\[Nu], z] ==
(1/(p - I a \[Nu])) ((E^(p z) Binomial[m, m/2] Cos[b + a z]^\[Nu]
Hypergeometric2F1[-((I p + a \[Nu])/(2 a)), -\[Nu],
(1/2) (2 - (I p)/a - \[Nu]), -E^(2 I (b + a z))] (1 - Mod[m, 2]))/
(2^m (1 + E^(2 I (b + a z)))^\[Nu])) +
(I Cos[b + a z]^\[Nu] Sum[Binomial[m, k]
((1/(2 c k - c m + I p + a \[Nu])) (E^((I c (-2 k + m) + p) z)
Hypergeometric2F1[-((2 c k - c m + I p + a \[Nu])/(2 a)), -\[Nu],
-((2 c k - c m + I p + a (-2 + \[Nu]))/(2 a)),
-E^(2 I (b + a z))]) + (E^(((-I) c (-2 k + m) + p) z)
Hypergeometric2F1[-((c (-2 k + m) + I p + a \[Nu])/(2 a)), -\[Nu],
-((-2 c k + c m + I p + a (-2 + \[Nu]))/(2 a)),
-E^(2 I (b + a z))])/(c (-2 k + m) + I p + a \[Nu])),
{k, 0, Floor[(1/2) (-1 + m)]}])/(2^m (1 + E^(2 I (b + a z)))^\[Nu]) /;
Element[m, Integers] && m > 0
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<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mrow> <mo> ∫ </mo> <mrow> <mrow> <msup> <mi> ⅇ </mi> <mrow> <mi> p </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> </msup> <mo> ⁢ </mo> <mrow> <msup> <mi> cos </mi> <mi> m </mi> </msup> <mo> ( </mo> <mrow> <mi> c </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <msup> <mi> cos </mi> <mi> ν </mi> </msup> <mo> ( </mo> <mrow> <mi> b </mi> <mo> + </mo> <mrow> <mi> a </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ⁢ </mo> <mrow> <mo> ⅆ </mo> <mi> z </mi> </mrow> </mrow> </mrow> <mo> ⩵ </mo> <mrow> <mrow> <msup> <mn> 2 </mn> <mrow> <mo> - </mo> <mi> m </mi> </mrow> </msup> <mo> ⁢ </mo> <mi> ⅈ </mi> <mo> ⁢ </mo> <mrow> <msup> <mi> cos </mi> <mi> ν </mi> </msup> <mo> ( </mo> <mrow> <mi> b </mi> <mo> + </mo> <mrow> <mi> a </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> </mrow> <mo> ) 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