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



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

         1 - n                   1 - m  1 - n  3 - n
(Cos[a x]      Hypergeometric2F1[-----, -----, -----, 
                                   2      2      2
 
             2          1 + m          2 -(1/2) - m/2
     Cos[a x] ] Sin[a x]      (Sin[a x] )            )
 
    / (a (-1 + n))

Time to compute: 0.27 second

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