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Integrate[E^(a*x)*x^m*Cos[b*x], x] ==
(-(x^(1 + m)*(-((a - I*b)*x))^(-1 - m)* Gamma[1 + m, -((a - I*b)*x)]) - x^(1 + m)*(-((a + I*b)*x))^(-1 - m)* Gamma[1 + m, -((a + I*b)*x)])/2



           a x  m
Integrate[E    x  Cos[b x], x] == 

    1 + m                 -1 - m
(-(x      (-((a - I b) x))       
 
       Gamma[1 + m, -((a - I b) x)]) - 
 
     1 + m                 -1 - m
    x      (-((a + I b) x))       
 
     Gamma[1 + m, -((a + I b) x)]) / 2

Time to compute: 0.21 second

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