I would like to numerically evaluate the following integral:

$ $ I = \int_{-\infty}^\infty d\tau_3 \int_{-\infty}^\infty d\tau_4 \frac{1}{1+\tau_3^2} \left\lbrace \frac{2}{1+\tau_4^2} \log (\tau_3 – \tau_4)^2 + \left(\frac{1}{1+\tau_3^2} + \frac{1}{1+\tau_4^2} \right) \phi(\tau_3,\tau_4) \right\rbrace \tag{1}$ $

with $ \phi(r,s)$ a complicated function as defined in the code below. Note that the first term with the log is divergent, but that this divergence is canceled by another divergence present in the 2nd term with the $ \phi$ -function. When I try to evaluate the integral, `NIntegrate`

stays unevaluated. Why is that, and what is the numerical value of this integral?

Here is the code I used so far:

`S[\[Tau]3_, \[Tau]4_] := (\[Tau]3 - \[Tau]4)^2/(1 + \[Tau]3^2); a[\[Tau]3_, \[Tau]4_] := 1/4 Sqrt[4*R[\[Tau]3, \[Tau]4]*S[\[Tau]3, \[Tau]4] - (1 - R[\[Tau]3, \[Tau]4] - S[\[Tau]3, \[Tau]4])^2] ; F[\[Tau]3_, \[Tau]4_] := I Sqrt[-((1 - R[\[Tau]3, \[Tau]4] - S[\[Tau]3, \[Tau]4] - 4 I*a[\[Tau]3, \[Tau]4])/(1 - R[\[Tau]3, \[Tau]4] - S[\[Tau]3, \[Tau]4] + 4 I*a[\[Tau]3, \[Tau]4]))]; phi[\[Tau]3_, \[Tau]4_] := 1/a[\[Tau]3, \[Tau]4] Im[PolyLog[2, F[\[Tau]3, \[Tau]4]*Sqrt[R[\[Tau]3, \[Tau]4]/S[\[Tau]3, \[Tau]4]]] + Log[Sqrt[R[\[Tau]3, \[Tau]4]/S[\[Tau]3, \[Tau]4]]] Log[1 - F[\[Tau]3, \[Tau]4]*Sqrt[R[\[Tau]3, \[Tau]4]/S[\[Tau]3, \[Tau]4]]]]; NIntegrate[1/(1^2 + \[Tau]3^2) (2/(1^2 + \[Tau]4^2)Log[(\[Tau]3 - \[Tau]4)^2] + (1/(1^2 + \[Tau]3^2) + 1/(1^2 + \[Tau]4^2)) phi[\[Tau]3, \[Tau]4]), {\[Tau]3, -\[Infinity], \[Infinity]}, {\[Tau]4, -\[Infinity], \[Infinity]}] `