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

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

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

        a x              a x        a x
 -2 Cot[---]   2 Log[Cos[---] - Sin[---]]
         2                2          2
(----------- - -------------------------- + 
      a                    a
 
              a x        a x
    2 Log[Cos[---] + Sin[---]]
               2          2
    --------------------------) / 2
                a
Time to compute: 0.03 second
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