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


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



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

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

Time to compute: 0.09 second

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