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-x + 2*Log[-2*(1 + E^x)]

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

                x
          -1 + E
Integrate[-------, x] == 
               x
          1 + E

                    x
-x + 2 Log[-2 (1 + E )]
Time to compute: < 0.01 second
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