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

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

                  -n
Integrate[Cos[a x]  , x] == 

         1 - n
(Cos[a x]      Csc[a x] 
 
                      1  1 - n  3 - n          2
    Hypergeometric2F1[-, -----, -----, Cos[a x] ] 
                      2    2      2
 
                 2
    Sqrt[Sin[a x] ]) / (a (-1 + n))
Time to compute: 0.17 second
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