$$
\begin{align}
\int_0^{\pi/2}\{\log(\tan(x))\}\,\mathrm{d}x
&=\int_{-\infty}^\infty\{u\}\frac{e^u\,\mathrm{d}u}{1+e^{2u}}\\
&=\frac12\int_{-\infty}^\infty\frac{e^u\,\mathrm{d}u}{1+e^{2u}}\\
&=\int_0^\infty\frac{e^u\,\mathrm{d}u}{1+e^{2u}}\\
&=\int_0^\infty\left(e^{-u}-e^{-3u}+e^{-5u}-\dots\right)\mathrm{d}u\\
&=1-\frac13+\frac15-\dots\\
&=\frac\pi4
\end{align}
$$
