complex analysis - Evaluate the Cauchy Principal Value of $int_{-infty}^{infty} frac{sin x}{x(x^2-2x+2)}dx$

Evaluate the Cauchy Principal Value of $\int_{-\infty}^\infty \frac{\sin x}{x(x^2-2x+2)}dx$

so far, i have deduced that there are poles at $z=0$ and $z=1+i$ if using the upper half plane. I am considering the contour integral $\int_C \frac{e^{iz}}{z(z^2-2z+2)}dz$ I dont know how to input graphs here but it would be intended at the origin with a bigger R, semi-circle surrounding that. So, I have four contour segments.

$\int_{C_R}+\int_{-R}^{-r}+\int_{-C_r}+\int_r^R=2\pi i\operatorname{Res}[f(z)e^{iz}, 1+i]+\pi iRes[f(z)e^{iz},o]$ I think. Ok, so here is where I get stuck. Im not sure how to calculate the residue here, its not a higher pole, so not using second derivatives, not Laurent series. Which method do I use here?