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



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

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

Time to compute: 0.14 second

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