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



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

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

Time to compute: 0.15 second

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