I have been attempting this question for some time.
My reason is as follow:
The contour C (−i, −2 − i, −2 −4i, −4i) is a rectangle on the left of the Imaginary axis (vertical axis).
For any z inside and on the contour C, $Re(z) >= -2$ but $Re(z) <=0$
Moreover, $Re(Log(z)) = ln(z) <= \ln|-2-4i| = \ln(2\sqrt(5))$.
And $Re(Log(z)) = \ln(z) >= ln|0 - i| = 0$
Can I conclude that there is a point where $z + Log(z) = 0$ ?
Because of that, $z + Log(z)$ is not analytic on $[-2, 0]$, which is inside the contour X. This means the function $\frac{\sin(z)}{z + Log(z)}$ is not analytic everywhere inside and on the contour C.
By the Cauchy-Goursat theorem, can I justify that $\int_y\frac{\sin(z)}{z + Log(z)} \neq 0$ ?