Logarithmic moments of the Cauchy kernel
= Logarithmic moments of the Cauchy kernel
For $0<a<2$,
$$
I(a)=\int_0^\infty\frac{x^{a-1}}{1+x^2}\,dx
=\frac\pi2\csc\frac{\pi a}{2}.
$$
Differentiating at $a=1$ and expanding the secant gives
$$
\int_0^\infty\frac{(\log x)^2}{1+x^2}\,dx=\frac{\pi^3}{8},
\qquad
\int_0^\infty\frac{(\log x)^4}{1+x^2}\,dx=\frac{5\pi^5}{32}.
$$