$$\iiint\limits_{x,y,z=0}^\infty \frac{xy^2z^3}{1-\cos(x^2)\cos(y^3)\cos(z^4)}\,dx\,dy\,dz$$
I think this integral is solved by making variable changes
$$ u=\tan(x/2),\:v=\tan(y/2),\:w=\tan(z/2)$$
and another change of variables to spherical coordinates and with the beta or gamma functions.
Use $X=x^2,\,Y=y^3,\,Z=z^4$ to write the integral as$$\frac{1}{24}\int_{[0,\,\infty)^3}\frac{dXdYdZ}{1-\cos X\cos Y\cos Z}\ge\frac{1}{48}\int_{[0,\,\infty)^3}dXdYdZ=\infty,$$where we've used $0\le1-\cos X\cos Y\cos Z\le2\implies\frac{1}{1-\cos X\cos Y\cos Z}\ge\frac12$. So the integral is $\infty$.