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((-I/2)*(((-I)*a*x)^m*Gamma[1 - m, (-I)*a*x] - (I*a*x)^m*Gamma[1 - m, I*a*x]))/(a*x^m)

Integrate[Cos[a*x]/x^m, x] ==
((-I/2)*(((-I)*a*x)^m*Gamma[1 - m, (-I)*a*x] - (I*a*x)^m*Gamma[1 - m, I*a*x]))/(a*x^m) 

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

 -I          m
(-- ((-I a x)  Gamma[1 - m, -I a x] - 
 2
 
             m                             m
      (I a x)  Gamma[1 - m, I a x])) / (a x )
Time to compute: 0.05 second
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