$$
\begin{align}
\int_0^\infty\frac{\arctan^3(\pi x)-\arctan^3(x)}{x}\,\mathrm{d}x
&=\int_0^\infty\frac{\arctan^3(\pi x)}{x}\,\mathrm{d}x
-\int_0^\infty\frac{\arctan^3(x)}{x}\,\mathrm{d}x\\
&=\lim_{N\to\infty}\int_N^{\pi N}\frac{\frac{\pi^3}{8}}{x}\,\mathrm{d}x\\
&=\frac{\pi^3}{8}\log(\pi)
\end{align}
$$
