Possible values of an integral of $1/(z^2+1)$

Find all possible values of \[ \int_\gamma\frac{dz}{z^2+1}, \] where \(\gamma\) is a closed smooth curve in \(\mathbb C\setminus\{i,-i\}\).

Answer the same question when \(\gamma\) is a positively oriented circle, and describe which circles give each possible value.