Contour integral of log x over x squared plus 2x plus 2

Use contour integration to evaluate $\int_0^{\infty}\frac{\log x}{x^2+2x+2}\,dx.$

$\tfrac{\pi\ln2}{8}\approx0.2722$. Integrate $\frac{(\log z)^2}{z^2+2z+2}$ around a keyhole contour (branch cut on the positive axis); the poles are $z=-1\pm i=\sqrt2e^{\pm3\pi i/4}$. Matching imaginary parts with the residue sum isolates the log integral as $\tfrac{\pi\ln 2}{8}$, which numerical quadrature confirms.