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


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



                  1 + 2 m
Integrate[Sin[a x]       , x] == 

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

Time to compute: 0.27 second

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