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Integrate[x/Cos[a*x]^n, x] ==
(Cos[a*x]^(1 - n)*(-(2^(-1 + n)*Sqrt[Pi]*Cos[a*x]* Gamma[1 - n]*HypergeometricPFQRegularized[ {1, 1 - n/2, 1 - n/2}, {(3 - n)/2, 2 - n/2}, Cos[a*x]^2]) - (2*a*x*Hypergeometric2F1[1, 1 - n/2, (3 - n)/2, Cos[a*x]^2]*Sin[a*x])/(1 - n)))/(2*a^2)



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

         1 - n     -1 + n
(Cos[a x]      (-(2       Sqrt[Pi] Cos[a x] 
 
         Gamma[1 - n] 
 
                                              n
         HypergeometricPFQRegularized[{1, 1 - -, 
                                              2
 
               n    3 - n      n           2
           1 - -}, {-----, 2 - -}, Cos[a x] ]) - 
               2      2        2
 
                                      n  3 - n
      (2 a x Hypergeometric2F1[1, 1 - -, -----, 
                                      2    2
 
                  2                              2
          Cos[a x] ] Sin[a x]) / (1 - n))) / (2 a )

Time to compute: 0.42 second

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