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


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



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

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

Time to compute: 0.11 second

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