complex analysis - Finding contour integral $int_gamma frac{mathrm{Im} (z)}{z - alpha} dz$

I'm trying to find the contour integral

$$\int_\gamma \frac{\mathrm{Im} (z)}{z - \alpha} dz$$
where $\alpha$ is a complex number such that $0 < |\alpha| < 2$ and $\gamma$ is the circle oriented in the positive sense, centred at the circle with radius 3.

I can find that

$$\int_\gamma \frac{\mathrm{Im} (z)}{z - \alpha} dz = \int_0^{2\pi} \frac{e^{it}-e^{-it}}{2i} \frac{1}{e^{it}-\alpha} i e^{it} dz$$

but the denominator is making it difficult to find the value of the contour integral. How can I proceed in this?