Convergence of the integral of $\frac{1}{(1+xy)(1+x)(1+y)}$ generalized to higher dimensions

Question

How can I show that the integral \begin{equation} \int_0^\infty \cdots \int_0^\infty \frac{1}{(1+x_1 \cdots x_n) (1+x_1) \cdots (1+x_n)} dx_1 \cdots dx_n \end{equation} is finite? For example for $n=2$, the value of the integral is $\pi^2/4$.

Answer 1

When $x\le1$, substitute $x\mapsto1/x$ to get

$$\int_0^1\frac{\mathrm dx}{(a+bx)(1+x)}=\int_1^\infty\frac{\mathrm dx}{(ax+b)(1+x)}$$

Repeat this process until all $x_i\ge1$ to get an integral of the form:

$$\int_{[1,\infty)^n}\frac{\mathrm dV}{(x_1\cdots x_k+x_{k+1}\cdots x_n)(1+x_1)\cdots(1+x_n)}$$

We can then bound each term by $1/x_i$ from above to get

$$\left(\int_1^\infty\frac{\mathrm dx}{x^2}\right)^n=1$$

and hence the integral converges absolutely.