Does this integral converge? WolframAlpha says that
$$\int_{-\infty}^\infty \int_{-\infty}^\infty \int_{-\infty}^\infty \frac 1{(1+x^2+y^2+z^2)^2} \, dx \, dy \, dz$$
converges, but it cannot compute integrals that are more than three variables.
Does this integral $$\int_{-\infty}^\infty \int_{-\infty}^\infty \int_{-\infty}^\infty \int_{-\infty}^\infty \frac 1{(1+x^2+y^2+z^2+w^2)^2} \, dx \, dy \, dz \, dw$$
converge?
In general, does this integral
$$\int_{\mathbb R^n} \frac 1{(1+|x|^2)^2} \, dx$$
converge?

Let $S(R)$ be the measure of $\{x_1^2+\ldots+x_n^2=R^2\}$. It is obviously $S(1)\cdot R^{n-1}$.
On the other hand:
$$ \int_{\mathbb{R}^n}\frac{1}{(1+\left|x\right|^2)^2}\,dx = \int_{0}^{+\infty}\frac{S(R)}{(1+R^2)^2}\,dR = S(1)\cdot\int_{0}^{+\infty}\frac{R^{n-1}}{(1+R^2)^2}\,dR$$
is convergent only for $n\leq 3$, and:
$$ \int_{\mathbb{R}^3}\frac{dx\,dy\,dz}{(1+x^2+y^2+z^2)^2} = 4\pi\cdot\frac{\pi}{4}=\color{red}{\pi^2}.$$
