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(Hypergeometric2F1[1, (1 + n)/2, (3 + n)/2, -Tan[a*x]^2]*Tan[a*x]^(1 + n))/(a + a*n)

Integrate[Tan[a*x]^n, x] ==
(Hypergeometric2F1[1, (1 + n)/2, (3 + n)/2, -Tan[a*x]^2]*Tan[a*x]^(1 + n))/(a + a*n) 

                  n
Integrate[Tan[a x] , x] == 

                      1 + n  3 + n           2
(Hypergeometric2F1[1, -----, -----, -Tan[a x] ] 
                        2      2
 
            1 + n
    Tan[a x]     ) / (a + a n)
Time to compute: 0.12 second
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