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∫
(1/4 + I/4)*x^m*(-(Gamma[1 + m, (-1 - I)*x]/((-1 - I)*x)^m) + (I*Gamma[1 + m, (-1 + I)*x])/((-1 + I)*x)^m)

Integrate[E^x*x^m*Sin[x], x] ==
(1/4 + I/4)*x^m*(-(Gamma[1 + m, (-1 - I)*x]/ ((-1 - I)*x)^m) + (I*Gamma[1 + m, (-1 + I)*x])/ ((-1 + I)*x)^m) 

           x  m
Integrate[E  x  Sin[x], x] == 

 1   I   m    Gamma[1 + m, (-1 - I) x]
(- + -) x  (-(------------------------) + 
 4   4                         m
                   ((-1 - I) x)
 
    I Gamma[1 + m, (-1 + I) x]
    --------------------------)
                      m
          ((-1 + I) x)
Time to compute: 0.10 second
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