I'm trying to solve the following integral, where $a>0$, $b>0$, $y\in\mathbb{R}$ and $z\in\mathbb{R}$ are given constants: $$ \int_{-\infty}^{0} \left[ \frac{1}{(x+ia)(x-y+ib)(x-z+ib)}-\frac{1}{(x-ia)(x-y-ib)(x-z-ib)} \right] dx $$
where after the integration there is a limit of $b\to0$ but $a$ remains positive and fixed.
I have tried to do something like this: $$ \frac{1}{w_1 w_2 w_3} - \frac{1}{\bar{w_1}\bar{w_2}\bar{w_3}} = \frac{-2i}{|w_1|^2 |w_2|^2 |w_3|^2}\left(\Im(w_1)\Re(w_2)\Re(w_3)+\Re(w_1)\Im(w_2)\Re(w_3)+\Re(w_1)\Re(w_2)\Im(w_3)-\Im(w_1)\Im(w_2)\Im(w_3)\right)$$
And then feed the result into Mathematica, however, the result does not seem to converge because there are three positive logs and one negative log of $-\infty$ which don't seem to cancel out.
Is there a better approach?
I also thought about using the identity $\frac{1}{x+i\varepsilon}=\mathcal{P}\left(1/x\right)-i\pi\delta(x)$ but it doesn't really simplify things here as far as I can tell.