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(E^(a*x)*((-a^2 + b^2 + a^3*x + a*b^2*x)*Cos[b*x] + b*(-2*a + a^2*x + b^2*x)*Sin[b*x]))/(a^2 + b^2)^2

Integrate[E^(a*x)*x*Cos[b*x], x] ==
(E^(a*x)*((-a^2 + b^2 + a^3*x + a*b^2*x)*Cos[b*x] + b*(-2*a + a^2*x + b^2*x)*Sin[b*x]))/(a^2 + b^2)^2 

           a x
Integrate[E    x Cos[b x], x] == 

  a x     2    2    3        2
(E    ((-a  + b  + a  x + a b  x) Cos[b x] + 
 
                 2      2                   2    2 2
      b (-2 a + a  x + b  x) Sin[b x])) / (a  + b )
Time to compute: 0.14 second
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