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

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

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

                a x
          2 Sin[---]
                 2
x - -----------------------
           a x        a x
    a (Cos[---] + Sin[---])
            2          2
Time to compute: 0.07 second
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