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-(Cos[(3*a*x)/2] - 3*Sin[(a*x)/2])/(3*a*(Cos[(a*x)/2] + Sin[(a*x)/2])^3)

Integrate[(1 + Sin[a*x])^(-2), x] ==
-(Cos[(3*a*x)/2] - 3*Sin[(a*x)/2])/ (3*a*(Cos[(a*x)/2] + Sin[(a*x)/2])^3) 

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

      3 a x          a x
-(Cos[-----] - 3 Sin[---])
        2             2
--------------------------
         a x        a x  3
3 a (Cos[---] + Sin[---])
          2          2
Time to compute: 0.06 second
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