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(2*Log[2*Cos[(a*x)/2]] - Log[Cos[a*x]])/a

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

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

            a x
2 Log[2 Cos[---]] - Log[Cos[a x]]
             2
---------------------------------
                a
Time to compute: 0.03 second
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