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


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



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

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

Time to compute: 0.07 second

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