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



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

                                   x
                           2 x Sin[-] (2 + Sin[x])
            x        x             2
(2 Log[-Cos[-] - Sin[-]] + ----------------------- - 
            2        2            x        x  3
                             (Cos[-] + Sin[-])
                                  2        2
 
    1 + 2 x + x Sin[x]
    ------------------) / 3
         x        x  2
    (Cos[-] + Sin[-])
         2        2

Time to compute: 0.43 second

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