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

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

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

         1 - n                      1 - n      1 - n
(Cos[a x]      Hypergeometric2F1[1, -----, 1 + -----, 
                                      2          2
 
             2
     Cos[a x] ]) / (a (-1 + n))
Time to compute: 0.09 second
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