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


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



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

           1 + n                   1  1 + n  3 + n
-((Cos[a x]      Hypergeometric2F1[-, -----, -----, 
                                   2    2      2
 
               2
       Cos[a x] ] Sin[a x]) / 
 
                            2
    (a (1 + n) Sqrt[Sin[a x] ]))

Time to compute: 0.13 second

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