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

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

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

            a x
      2 Sin[---]
             2
-----------------------
       a x        a x
a (Cos[---] + Sin[---])
        2          2
Time to compute: 0.03 second
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