After a calculation with perturbation theory in a many-body problem, I end up with the following integral which I cannot compute:
$$I=\int_{-D}^{E}\int_{E}^{D}\int_{-D}^{E} dx_1 dx_2 dx_3 \dfrac{1}{x-x_1+x_2-x_3}$$
It seems simple, but the moment you make a first integral (e.g, with respect to $x_3$), you've got the logarithms to integrate and it gets very messy very quickly, and the integral is undoable.
Another thing I tried is to rewrite the integrand as some other integral and then use Fubini. For instance, it seemed a good idea to use the fact: $$\dfrac{1}{a}=\text{sign}(a) \int_0^\infty\text{e}^{-\vert a \vert y} dy$$ and write: $$\int_0^\infty \int_{-D}^{E}\int_{E}^{D}\int_{-D}^{E} dx_1 dx_2 dx_3 dy \ \text{sign}(x-x_1+x_2-x_3) \ \text{e}^{-\vert x-x_1+x_2-x_3 \vert y},$$ but, again, the sign-dependence of the limits in the integral (after doing the first one) makes the problem untractable.
I think that something in this line (rewrite the integrand as a different integral and use Fubini) could work, but I would need some representation which is not sign-dependent.
I would like to at least reduce the 3-dimensional integral to a simple 1-dimensional integral (that I could then compute easily numerically)
Thanks!
PS: I add the tag 'mathematical physics' because this is a very typical sort of integrals one finds in feynman diagrams calculations or related things.
It's straightforward to integrate logarithms indefinitely, and more generally integration by parts shows $$\int x^n \log x \ {\rm d}x = \frac{x^{n+1}}{n+1} \times \log x - \int \frac{x^{n+1}}{n+1} \times \frac{1}{x} \ {\rm d}x = \frac{x^{n+1} \log x}{n+1} - \frac{x^{n+1}}{(n+1)^2}$$ As far as I can see, this should be all you need to work out the answer. It might be ugly, but it's straightforward!