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


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



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

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

Time to compute: 0.14 second

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