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



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

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

Time to compute: 0.26 second

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