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



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

  a x  (-b + c) Cos[(b - c) x] + a Sin[(b - c) x]
(E    (------------------------------------------ + 
                      2          2
                     a  + (b - c)
 
      -((b + c) Cos[(b + c) x]) + a Sin[(b + c) x]
      --------------------------------------------))\
                      2          2
                     a  + (b + c)
 
   / 2

Time to compute: 0.59 second

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