\[\int _0 ^1 \frac{\ln (- \ln x)}{x^2 + x + 1}\,\mathrm{d}x=\frac{\pi}{\sqrt{3}}\ln\left(\frac{\Gamma(2/3)}{\Gamma(1/3)}\sqrt[3]{2\pi}\right),\]
\[\int _0 ^1 \frac{\ln (- \ln x)}{x^2 - x + 1}\,\mathrm{d}x=\frac{2\pi}{\sqrt{3}}\ln\left(\frac{\sqrt[6]{32\pi^5}}{\Gamma(1/6)}\right).\]