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

Time to compute: 0.10 second

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