Difficult integral involving $\arctan x$

We can use differentiation under the integral sign and a trick to evaluate this. First define $$ I(a,b) = \int_0^{\infty} \frac{\arctan{ax}}{x} \frac{\log{(1+b^2 x^2)}}{1+x^2} \, dx , $$ so $I(a,0)=I(0,b)=0$ and $I(1,1)$ is what we want. Differentiating one with respect to $a$ and once wrt $b$ gives $$ \partial_a\partial_b I = \int_0^{\infty} \frac{2bx^2 \, dx}{(1+x^2)(1+a^2x^2)(1+b^2x^2)}, $$ which can be done by using partial fractions and the arctangent integral a few times. When the dust settles, $$ \partial_a\partial_b I = \frac{b\pi}{(1+a)(1+b)(a+b)}, $$ and thus $$ I(1,1) = \int_0^1 \int_0^1 \frac{b\pi}{(1+a)(1+b)(a+b)} \, da \, db $$ But we can swap $a$ and $b$ and will get the same result for this integral by the symmetry of the region of integration, so we also have $$ I(1,1) = \int_0^1 \int_0^1 \frac{a\pi}{(1+a)(1+b)(a+b)} \, da \, db. $$ Adding gives $$ I(1,1) = \frac{\pi}{2}\int_0^1 \int_0^1 \frac{1}{(1+a)(1+b)} \, da \, db, $$ but this splits into a product of two copies of $\int_0^1 dy/(1+y) = \log{2}$, so $$ I(1,1) = \frac{\pi}{2}(\log{2})^2 $$ as desired.

Define $$ f(a,b) = \int \limits_0^\infty \frac{\arctan(a x) \ln (1+ b^2 x^2)}{x (1+x^2)} \, \mathrm{d} x $$ for $0 \leq a , b \leq 1$ . Then $f(1,1) = \mathfrak{I}$ and $f(0,b) = f(a,0) = 0 $ . For $0< a,b<1$ we can differentiate under the integral sign to find $$ \partial_a \partial_b f(a,b) = 2 b \int \limits_0^\infty \frac{x^2}{(1+a^2 x^2)(1+b^2 x^2)(1+x^2)} \, \mathrm{d} t = \frac{\pi b}{(1+a)(1+b)(a+b)} \, . $$ The integral can be evaluated using the residue theorem, for example. Now integrate again and exploit the symmetry of the derivative to obtain \begin{align} \mathfrak{I} &= f(1,1) = \pi \int \limits_0^1 \int \limits_0^1 \frac{b}{(1+a)(1+b)(a+b)} \, \mathrm{d}a \, \mathrm{d}b = \pi \int \limits_0^1 \int \limits_0^1 \frac{a}{(1+a)(1+b)(a+b)} \, \mathrm{d}a \, \mathrm{d}b \\ &= \frac{\pi}{2} \int \limits_0^1 \int \limits_0^1 \frac{a+b}{(1+a)(1+b)(a+b)} \, \mathrm{d}a \, \mathrm{d}b = \frac{\pi}{2} \int \limits_0^1 \int \limits_0^1 \frac{1}{(1+a)(1+b)} \, \mathrm{d}a \, \mathrm{d}b \\ &= \frac{\pi}{2} \ln^2 (2) \, . \end{align}

Notice:

$$ \Im(\log^2(1+ix))=\Im\left(\left(\frac{1}{2}\log(1+x^2)+i\arctan(x)\right)^2\right)=\frac{1}{2}\log(1+x^2)\arctan(x) $$

The integral in question is therefore (use parity) $$ I=\Im\int_{\mathbb{R}}\underbrace{\frac{\log^2(1+ix)}{x(1+x^2)}}_{f(x)} $$

Integrate around a big semicircle in the lower halfplane (to avoid the branchcut) yields

$$ I=\Im \left(2\pi i\text{Res}(f(z),z=-i)\right)=\frac{\pi}{2}\log^2(2)$$

where the residue is easy to calculate since the pole is simple.

The vansihing of the contrbutions at infinity follows from the fact that $R|f(Re^{i\phi})|\sim \log^2(R)/R^2$ in sector of $\mathbb{C}$ we are interested in.