Finding a contour to evaluate$\int_{-\infty}^{\infty}\frac{x\sin x}{x^2+a^2}\,dx$

I am looking to evaluate to evaluate the real integral $I=\int_{-\infty}^{\infty}\frac{x\sin x}{x^2+a^2}\,dx$ ($a>0$) using Cauchy's residue formula. My strategy is to use the residue theorem to calculate the value of a contour that encircles the a pole and has a side coincides with the real axis. My hope was that the other sides of the contour would be simple to evaluate, but I am struggling to find an appropriate contour.

Finding a Pole

Defining $f(z)=\frac{z\sin z}{z^2+a^2}$, we quickly see that $1/f$ vanishes at $z_{0}=ai$. Moreover one can check that $g(z)=\begin{cases} \frac{1}{f(z)} \text{ if } z\neq z_{0},\\ 0 \text{ if } z= z_{0}.\end{cases}$ is holomorphic, so $ai$ is indeed a pole of $f$.

Calculating the Residue