integrals.wolfram.com   Use one of Mathematica's 2500+ functions FREE online

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

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

             a x
       2 Cot[---]
              2
-2 x - ----------
           a
-----------------
        2