Integrate: $ \int_0^\infty \frac{\log(x)}{(1+x^2)^2} \, dx $ without using complex analysis methods Can this integral be solved without using any complex analysis methods: $$ \int_0^\infty \frac{\log(x)}{(1+x^2)^2} \, dx $$
Thanks.

A really easy method using the substitution $x=\tan t$ offered by Jack D'Aurizio but followed by elementary integration by parts without using Fourier series.
$$\begin{align}
&\int_0^\frac{\pi}{2}\cos^2x\ln\tan x dx=\frac12 \int_0^\frac\pi2 \cos2x\ln\tan xdx\\
=&\frac14\sin2x\ln\tan x\bigg|^\frac{\pi}{2}_0-\frac14\int_0^\frac{\pi}2\frac{\sin 2x}{\sin x \cos x}dx=-\frac14\int_0^\frac\pi22dx=\color{blue}{-\frac\pi 4}
\end{align}$$
