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



                   1
Integrate[-------------------, x] == 
                  n         m
          Cos[a x]  Sin[a x]

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

Time to compute: 0.34 second

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