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Integrate[x/(1 + Tan[x]), x] ==
(I*Pi*x + (2 - 2*I)*x^2 + 2*Pi*Log[1 + E^((-2*I)*x)] + Pi*Log[1 - I*E^((2*I)*x)] + 4*x*Log[1 - I*E^((2*I)*x)] - 2*Pi*Log[Cos[x]] - Pi*Log[Sin[Pi/4 + x]] - (2*I)*PolyLog[2, I*E^((2*I)*x)])/8 

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

                     2                 (-2 I) x
(I Pi x + (2 - 2 I) x  + 2 Pi Log[1 + E        ] + 
 
                  (2 I) x
    Pi Log[1 - I E       ] + 
 
                   (2 I) x
    4 x Log[1 - I E       ] - 2 Pi Log[Cos[x]] - 
 
               Pi                             (2 I) x
    Pi Log[Sin[-- + x]] - (2 I) PolyLog[2, I E       ]
               4
 
    ) / 8
Time to compute: 0.14 second
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