Find the image of certain holomorphic function on unit circle

Find the image of certain holomorphic function on unit circle

The problem asks to show that
$$w=\log\left(\frac{1+z}{1-z}\right)+\frac{2z}{1+z^2}$$ maps the unit
circle $\mathbb{D}\subset \mathbb{C}_z$ one to one and onto the $w$-space
$\mathbb{C}_w$ with four half-line deleted by some properly chosen branch
of logarithm, and asks me to find the half-lines.
It seems unrealistic to discuss by letting $z=x+\mathrm{i}y$ since the
curve will be too complicated, and I tried to find the boundary
information by letting $z=e^{\mathrm{i}\theta}$, and observed
$\mathrm{i},-\mathrm{i},-1$ may be three of the endpoints (while the image
of $z=1$ cannot even be approximated) but I do not know how to grasp the
graph of $z\mapsto w$, is there any better approach?