Plot[{I (Log[x + I] - 
              Log[x - I]), 2  ArcTan[x]}, {x, -2, 
        2}]

Plot[
      Evaluate[{I (Log[u + I] - 
                  Log[u - I]), 2  ArcTan[u]} /. 
          u -> (((x^3) - 2  x)/((x^2) - 1))], {x, -2, 
        2}]

simp = 
      ComplexExpand[
        2  ArcTan[
              z] - I (Log[z + I] - 
                Log[z - I])]

simp = FullSimplify[simp, z > -1 && z < 1]

FullSimplify[Table[simp, {z, -3, 3}]]