<s>[INST] Is $\int_{-\pi}^{\pi}\int_{-\pi}^{\pi}\int_{-\pi}^{\pi}\frac{1}{1 - \cos x\cos y\cos z}\, dx\, dy\, dz$ finite? Is $$\int_{-\pi}^{\pi}\int_{-\pi}^{\pi}\int_{-\pi}^{\pi}\frac{1}{1 - \cos x\cos y\cos z}\, dx\, dy\, dz$$ finite? Mathematica is throwing me an error, but I suspect the integral converges, since the Taylor expansion of $\cos x\cos y\cos z$ should have a leading term of $1$ followed by a $x^{2} + y^{2} + z^{2}$ and $\frac{1}{x^{2} + y^{2} + z^{2}}$ is integrable in 3 dimensions if I switch to polar.
 [/INST] While it is a bit surprising, the integral should converge.  Around each of the corners, $(0, 0, 0)$, $(\pi, \pi, 0)$, $\dots$, etc, we have (very loose bounds)
$$
\begin{aligned}
\cos x\cos y\cos z
&\le |\cos x||\cos y||\cos z| \\
&\le \left( \frac{|\cos x| + |\cos y| + |\cos z|}{3} \right)^3
\\
&\le \left( 1 - \frac{ x^2 + y^2 + z^2 } {9} \right)^3
\quad \mathrm{for}\; |x|, |y|, |z| \in (0, \pi/2)
\\
&\le 1 - \frac{ x^2 + y^2 + z^2 } { 12 }.
\end{aligned}
$$
So
$$
\begin{aligned}
\frac{1}{1-\cos x\cos y\cos z}
&\le \frac{12}{ x^2 + y^2 + z^2 } = \frac{12}{r^2}.
\end{aligned}
$$
$$
\begin{aligned}
\int_0^{\pi/2}\int_0^{\pi/2}\int_0^{\pi/2}\frac{1}{1-\cos x\cos y\cos z}
\,dx \, dy \, dz
&\le \frac{\pi}{2} \int_0^{\pi/2} \frac{12}{r^2} r^2 \, dr = 3\pi^2.
\end{aligned}
$$
Together there are 64 such integrals, and the total integral is finite (actually 32 of them are not singular, but this is just an estimate).
For Mathematica, try this
NIntegrate[1/(1 - Cos[x] Cos[y] Cos[z]), {x, -Pi, Pi}, {y, -Pi, Pi}, {z, -Pi, Pi},
    Method -> "AdaptiveMonteCarlo", PrecisionGoal -> 3]

</s>