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



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

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

Time to compute: 0.15 second

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