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

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

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