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



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

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

Time to compute: 0.16 second

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