# Does this integral converge or diverge?

I have the following integral:

$$I= \int_{-\infty}^\infty d\tau_3 \int_{-\infty}^\infty d\tau_4\ I_{13} Y_{134}, \tag{1}$$

with:

$$I_{12} := \frac{1}{(2\pi)^2 x_{12}^2} \qquad \qquad Y_{134} := \int_{\mathbb{R}^4} d^4 x_5 I_{15} I_{35} I_{45} \tag{2}$$

and with $$x_{ij}:=x_i-x_j$$, $$x_1=(1,0,0,0)$$, $$x_3=(0,0,0,\tau_3)$$, $$x_4=(0,0,0,\tau_4)$$.

The integral $$Y_{134}$$ is known and has an analytical form that can be found here for example (eq. (A.5) - it's somewhat complicated so I'd rather not type it here). I would like to know if $$(1)$$ is divergent or not. At first glance, it seems divergent, since $$Y_{134}$$ diverges logarithmically at $$\tau_3 = \tau_4$$. But we still have to integrate over $$\tau_3$$ and $$\tau_4$$, and sometimes integrals of divergent functions are finite (e.g. $$\int_0^1 dx \log x = -1$$). This is what seems to happen here when I do the integral numerically, as it was discussed in this other post of mine.

So is this integral finite or not, and how can I know in general?

$$\partial_{\tau_3} Y_{134}$$