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


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



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

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

Time to compute: 0.02 second

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