<s>[INST] Hard triple Integral $\int_{0}^{1}\int_{0}^{1}\int_{0}^{1}\frac{1}{2-zx^{2}-zy^{2}}dxdydz=\ln(2^{G})$ How do prove this triple integral?
$$\int_{0}^{1}\int_{0}^{1}\int_{0}^{1}\frac{1}{2-zx^{2}-zy^{2}}dxdydz=\ln(2^{G})$$
where G is Catalan's constant.
As my try I only reach to this hard single integral:
$$\int _0^1\frac{\operatorname{Li}_2\left(\frac{1+x^2}{2}\right)}{1+x^2}dx=G\ln \left(2\right)$$
 [/INST] Integrate by parts
$$I=\int _0^1\frac{\operatorname{Li}_2\left(\frac{1+x^2}{2}\right)}{1+x^2}dx
= \frac{\pi^3}{24}+\int_0^1\frac{2x\tan^{-1}x\ln\frac{1-x^2}2}{1+x^2}dx$$
Then, utilize $\int_0^1 \frac{x \tan^{-1} x \ln \left( 1-x^2\right)}{1+x^2}dx=  -\frac{\pi^3}{48}-\frac{\pi}{8}\ln^2 2 +G\ln 2$ and $\int_0^1 \frac{x\tan^{-1}x}{1+x^2}dx = \frac12G-\frac\pi2\ln2$ to obtain $I= G\ln2$.
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