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

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

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

           a x
-2 Log[Cos[---]]
            2
----------------
       a
Time to compute: < 0.01 second
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