-4\pi i \int\limits_{0}^{+\infty} \frac{x \ln x}{x^3+1}dx +4\pi^2 \int\limits_{0}^{+\infty} \frac{dx}{x^3+1}=2\pi i \left(\frac{\pi^2}{3}-\frac{\pi^2}{27}e^{-i \frac{\pi}{3}}-\frac{25\pi^2}{27}e^{i \frac{\pi}{3}}  \right)