I am facing the following definite integral:
$$\int_{-\infty}^{\infty} d\tau_3 \int_{-\infty}^{\infty} d\tau_4 \frac{1}{x_1^2+\tau_3^2}\frac{1}{a \tau_3^2+b\tau_4^2+c\tau_3\tau_4+d} \tag{1}$$
with $x_1^2 > 0$, $a$, $b$, $c$ and $d$ are real coefficients and I know (maybe it's useful) that $d$ is positive.
I have no idea how to tackle this integral, but I have put it in Mathematica to see what happens. It gives back $0$ unambiguously if I type it in the following way:
$$\text{Integrate}\left[\frac{1}{x_1^2+\tau_3^2} \frac{1}{a \tau_3^2+b\tau_4^2+c\tau_3\tau_4+d}, \lbrace \tau_3,-\infty,\infty \rbrace,\lbrace \tau_4,-\infty,\infty \rbrace\right]$$
However if I invert the integration measures, it gives me also $0$, but only if some weird condition is fulfilled:
$$\text{Im} \left[\frac{i c x_1 \pm \sqrt{-4bd+4abx_1^2-c^2x_1^2}}{b} \right] > 0\tag{2}$$
The strictly positive sign as well as the $\pm$ make this condition quite selective. For example, for $c=0$ there is no way to fulfill this constraint I think, and this value is not excluded in my situation as far as I can see.
So my questions are:
1) Is the above integral $0$ for all values of $a$, $b$, $c$ and $d$? If not, what are the possible results and the associated constraints?
2) How can I compute this integral?