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

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

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

                     a x        a x
(-(a x) + 2 Log[-Cos[---] - Sin[---]] + 
                      2          2
 
                a x
      2 a x Sin[---]
                 2          2
    -------------------) / a
        a x        a x
    Cos[---] + Sin[---]
         2          2
Time to compute: 0.12 second
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