$$\begin {aligned} \int _{0}^{1}\int _{0}^{1}\int _{0}^{1}\frac {dxdydz}{\sqrt {1-x^{2}y^{2}z^{2}}} &=\sum _{n=0}^\infty \frac {\binom {2n}n}{2^{2n}}\int _{0}^{1}\int _{0}^{1}\int _{0}^{1}x^{2n}y^{2n}z^{2n}dxdydz\\ &=\sum _{n=0}^\infty \frac {\binom {2n}n}{2^{2n}(2n+1)^{3}}\\ &=\sum _{n=0}^\infty \frac {\binom {2n}n}{2^{2n}}\frac {1}2\int _{0}^{\infty }x^{2}e^{-(2n+1)x}dx\\ &=\frac {1}2\sum _{n=0}^\infty \frac {\binom {2n}n}{2^{2n}}\int _{0}^{1}\ln ^{2}\left (\frac {1}x\right )x^{2n}dx\\ &=\frac {1}2\int _{0}^{1}\frac {\ln ^{2}x}{\sqrt{1-x^{2}}}dx\\ &=\frac {1}2\int _{0}^{\frac {\pi }2}\ln ^{2}\sin xdx\\ &=\frac {1}2\left .\frac {d}{dx}B\left (x,\frac {1}2\right )\right|_{x=0}\\ &=\frac {1}2\left (\frac {\pi ^{3}}{24}+\frac {\pi \ln ^{2}2}2\right )\\ &=\textcolor {blue}{\frac {\pi ^{3}}{48}+\frac {\pi \ln ^{2}2}{4}}. \end {aligned}$$