Hilbert transform/contour integral $\int_{-\infty}^{\infty} \frac{\sin nx}{1-\lambda \cos x}\frac{1}{t-x} dx$

Question

Is it possible to find closed form solutions for the following integral (which is a Hilbert transform) $$P.V.\frac{1}{\pi}\int_{-\infty}^{\infty} \frac{\sin n x}{1-\lambda \cos x}\frac{1}{t-x} dx,$$ for $\lambda\in \mathbb{R}$ such that $|\lambda|<1$, $ t\in \mathbb{R}$ and $n$ a natural number. Here, P.V. means principal value.