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Integrate[E^(a*x)*Cos[b*x]^n, x] ==
(E^(a*x)*(E^((-I)*b*x) + E^(I*b*x))^n* Hypergeometric2F1[-n, -(I*a + b*n)/(2*b), 1 - ((I/2)*a)/b - n/2, -E^((2*I)*b*x)])/ (2^n*(1 + E^((2*I)*b*x))^n*(a - I*b*n))



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

  a x   -I b x    I b x n
(E    (E       + E     )  
 
                                            I
                                            - a
                          -(I a + b n)      2     n
    Hypergeometric2F1[-n, ------------, 1 - --- - -, 
                              2 b            b    2
 
       (2 I) b x
     -E         ]) / 
 
    n       (2 I) b x n
  (2  (1 + E         )  (a - I b n))

Time to compute: 0.15 second

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