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


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



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

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

Time to compute: 0.22 second

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