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

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

                p
Integrate[Sin[x] , x] == 

                           1  1 - p  3        2
-(Cos[x] Hypergeometric2F1[-, -----, -, Cos[x] ] 
                           2    2    2
 
          1 + p        2 (-1 - p)/2
    Sin[x]      (Sin[x] )          )
Time to compute: 0.13 second
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