Mapping unit circle with the map $\omega= \frac{z\ln z}{z^{2}+1}$

Question

In an exercise involving conformal mappings, the task is to determine the line in the $\omega$ plane traced by the point $\omega= \frac{z\ln z}{z^{2}+1}$, as $z$ traces the unit circle. I am not sure whether this is a trick question because I feel like I need to deform the given contour to a dogbone contour as $arg\left ( z \right ) \in\left [-\pi,\pi \right )$