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


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



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

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

Time to compute: 0.13 second

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