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


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



                  -m
Integrate[Sin[a x]  , x] == 

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

Time to compute: 0.27 second

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