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x - Tan[(a*x)/2]/a

Integrate[Cos[a*x]/(1 + Cos[a*x]), x] ==
x - Tan[(a*x)/2]/a 

            Cos[a x]
Integrate[------------, x] == 
          1 + Cos[a x]

        a x
    Tan[---]
         2
x - --------
       a
Time to compute: 0.07 second
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