Often when I solve a real integral in the complex plane I take a half circle in the upper plane an calculate the residues to get the integral. But what if theres a singular point in origo?

Here's an example of what I mean. $\displaystyle \displaystyle \int_{- \infty}^\infty \frac{\sin x}{x(x-i)(x+i)} dx$

If anyone could show a method to solve this kind of integral I go.