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-(1 + 2*Cos[(a*x)/2]^2*(Log[Cos[(a*x)/2]] - Log[Sin[(a*x)/2]]))/(2*a*(1 + Cos[a*x]))


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



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

            a x 2          a x             a x
-(1 + 2 Cos[---]  (Log[Cos[---]] - Log[Sin[---]]))
             2              2               2
--------------------------------------------------
                2 a (1 + Cos[a x])

Time to compute: 0.09 second

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