$$\begin {aligned} \int _{0}^{\infty }\ln \left (\frac {1+x^{2}+x^{4}+x^{6}}{1+x^{6}}\right )dx &=\int _{0}^{\infty }\ln \left (\frac {\left (1+x^{2}\right )\left (1+x^{4}\right )}{1+x^{6}}\right )dx\\ &=\int _{0}^{\infty }\ln \left (\frac {1+x^{4}}{1+x^{4}-x^{2}}\right )dx\\ &=\int _{0}^{\infty }\ln \left (\frac {x^{2}+x^{-2}}{x^{2}+x^{-2}-1}\right )dx\\ &=\int _{0}^{\infty }\ln \left (\frac {\left (x-\frac {1}x\right )^2+2}{\left (x-\frac {1}x\right )^2+1}\right )dx\\ &=\int _{0}^{\infty }\ln \left (\frac {x^{2}+2}{x^{2}+1}\right )dx \quad(\text{Cauchy-Schlömilch変換})\\ &=\int _{0}^{\infty }\int _{1}^{2}\frac {dt}{x^{2}+t}dx\\ &=\int _{1}^{2}\int _{0}^{\infty }\frac {dx}{x^2+t}dt\\ &=\int _{1}^{2}\frac {\pi }{2\sqrt t}dt\\ &=\textcolor {blue}{\pi \left (\sqrt 2-1\right )}. \end {aligned}$$