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

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

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

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