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


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



                  -2 m
Integrate[Cos[a x]    , x] == 

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

Time to compute: 0.22 second

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