Symbolic Integration along contour: branch cut problem?
The numeric result ($2π \cdot i$) is correct by the residue theorem, since
In[25]:= Residue[1/x, {x, 0}]
Out[25]= 1
and your curve's winding number is 1, so
$$∮_{C} \frac{1}{z} \mathrm{d} z = 2π \cdot i \cdot \mathrm{Res}(1/z; z=0) = 2π \cdot i$$
So it looks like your symbolic ContourIntegrate
is buggy.