Numerical quadrature for an improper multiple integral In my numerical analysis course, my professor asked us to evaluate the integral
$$2 \int_{0}^{1} \cdots \int_{0}^{1} \prod_{i<j}\left(\frac{u_{i}-u_{j}}{u_{i}+u_{j}}\right)^{2} \frac{d u_{1}}{u_{1}} \cdots, \frac{d u_{5}}{u_{5}}$$
through numerical quadrature. And, an explicit form of the integral is
$2 \int_{0}^{1} \cdots \int_{0}^{1} 
    \left(\frac{u_{1}-u_{2}}{u_{1}+u_{2}}\right)^{2} \left(\frac{u_{1}-u_{3}}{u_{1}+u_{3}}\right)^{2} \left(\frac{u_{1}-u_{4}}{u_{1}+u_{4}}\right)^{2} \left(\frac{u_{1}-u_{5}}{u_{1}+u_{5}}\right)^{2} 
    \left(\frac{u_{2}-u_{3}}{u_{2}+u_{3}}\right)^{2} \left(\frac{u_{2}-u_{4}}{u_{2}+u_{4}}\right)^{2} \left(\frac{u_{2}-u_{5}}{u_{2}+u_{5}}\right)^{2} 
    \left(\frac{u_{3}-u_{4}}{u_{3}+u_{4}}\right)^{2} \left(\frac{u_{3}-u_{5}}{u_{3}+u_{5}}\right)^{2} 
    \left(\frac{u_{4}-u_{5}}{u_{4}+u_{5}}\right)^{2}$
$$\frac{d u_{1}}{u_{1}} \cdots, \frac{d u_{5}}{u_{5}}$$
He warned us that common quadrature methods, like Gauss-Legendre, and Monte Carlo methods will fail in this question. We have to design our own method to solve this specific problem. He told us that this is an important integral in Physics and is known to converge. However, I still could not find out when this integral would occur in a real world Physics problem after doing lots of research.
Common numerical quadrature methods will fail in this integral because it is improper at the end point 0: the integrand diverges to $\infty$. Therefore, the approximation result will always be bigger and bigger as the number of nodes chosen for the quadrature increases.
If I try to narrow the integral region to $\epsilon$ to 1, where $\epsilon$ is a very small number, then the integral will become proper. But the integration result will vary a lot, ranging from 0.0001 to 1000 for different values of $\epsilon$ I chose. And I think none of them gives the correct result.
I tried to search for academic articles that deals with improper integrals, but most of the articles I found are for one-dimensional cases, and I don't know if their method will work well to my problem because I do not even know the result of this integral. Some of the ideas I got from these articles include: elimination of the singularity, separation of the singular and non-singular part, change of variable. But, I don't think they will work to this problem because it is a product of 10 polynomials on both the denominator and numerator. So, I have the following questions.

*

*What are some of the references I could turn to find useful methods that deals with multiple improper integrals? And how can I know their strengths and weaknesses so that I know if I can apply it to my problem?

*What are the possible transformations that I could try to make this problem easy to deal with?

*When will this kind of integral occur in Physics and what is the background behind it? Any references available?
I am really struggling with these problems. Any thoughts on any of the above questions are more than welcome.
 A: You integral diverges.
Consider the region $u_k \in [(2k-1)\epsilon, 2k\epsilon]$. Explicitly
$$
(u_1, u_2, u_3, u_4, u_5) \in G_\epsilon =  
[\epsilon, 2\epsilon] \times
[3\epsilon, 4\epsilon] \times
[5\epsilon, 6\epsilon] \times 
[7\epsilon, 8\epsilon] \times
[9\epsilon, 10\epsilon].
$$
Each term in the product is bounded from below
$$
\left(2 |k - m| - 1\right)\epsilon \leq 
|u_k - u_m|\\
u_k + u_m \leq \left(2 k + 2 m + 1\right)\epsilon\\
\frac{1}{19} \leq \frac{2 |k - m| - 1}{2 k + 2 m + 1}
\leq \left|\frac{u_k - u_m}{u_k + u_m}\right|
$$
The integral over $G_\epsilon$ can be now estimated as
$$
\int_{\epsilon}^{2\epsilon}
\int_{3\epsilon}^{4\epsilon}
\dots
\int_{9\epsilon}^{10\epsilon}
\prod_{i<j} \left(\frac{u_i - u_j}{u_i + u_j}\right)^2 \frac{du_1}{u_1} \dots \frac{du_5}{u_5} \geq \frac{1}{19^{10}}
\int_{\epsilon}^{2\epsilon}
\int_{3\epsilon}^{4\epsilon}
\dots
\int_{9\epsilon}^{10\epsilon}
\frac{du_1}{u_1} \dots \frac{du_5}{u_5} = \\ =
\frac{\log 2 \cdot \log \frac{4}{3} \cdot \log \frac{6}{5} \cdot \log \frac{8}{7} \cdot \log \frac{10}{9}}{19^{10}} \equiv C.
$$
The integration domain contain infinitely many non-overlapping regions $G_\epsilon$. For example, take $\epsilon = \frac{1}{10^n}$, $n = 1, 2, \dots$ Integral over the each region gives at least $C$. Thus the integral diverges.
