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


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



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

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

Time to compute: 0.16 second

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