<s>[INST] Evaluating $\int_0^1 \frac{x \arctan x \log \left( 1-x^2\right)}{1+x^2}dx$ I am trying to prove that
$$\int_0^1 \frac{x \arctan x \log \left( 1-x^2\right)}{1+x^2}dx = -\frac{\pi^3}{48}-\frac{\pi}{8}\log^2 2 +G\log 2$$
where $G$ is the Catalan's Constant. Numerically, it's value is $-0.199739$.
Using the substitution $x=\tan \theta$, it can be written as 
$$
\begin{align*}
I &= \int_0^{\frac{\pi}{4}}\theta \tan \theta \log(\cos 2\theta) d\theta-2\int_0^{\frac{\pi}{4}}\theta \tan \theta \log(\cos \theta)d\theta \end{align*}
$$
Can anyone suggest a good approach to evaluate it?
 [/INST] A solution proposed by Cornel Ioan Valean
As also shown in this post, an immediate connection can be made with the integral in the current OP. So, integrating by parts, rearranging, and employing the needed results, we arrive at once at the desired closed form
$$\int_0^1 \frac{x \arctan(x) \log(1-x^2)}{1+x^2}\textrm{d}x=\frac{1}{2}\int_0^1 \left(\log\left(\frac{1+x^2}{2}\right)\right)' \arctan(x) \log(1-x^2)\textrm{d}x$$
$$=\frac{1}{2}\log(2)\underbrace{\int_0^1 \frac{\log(1-x^2)}{1+x^2}\textrm{d}x}_{\displaystyle \log(2)\pi/4-G}-\underbrace{\int_0^1 \frac{x\arctan(x) \log(2/(1+x^2))}{1-x^2}\textrm{d}x}_{\displaystyle \pi^3/192}-\frac{1}{2}\underbrace{\int_0^1\frac{\log(1-x^2)\log(1+x^2)}{1+x^2}\textrm{d}x}_{\displaystyle(\pi ^3+16 \log ^2(2)\pi  -96 \log (2) G)/32}=\log(2)G-\log^2(2)\frac{\pi}{8}-\frac{\pi^3}{48}.\tag1$$
The first remaining integral in $(1)$ may be viewed as a sums of two well-known classical (and trivial) integrals, one of them being a Putnam question, that is $\displaystyle\int_0^1 \frac{\log(1+x)}{1+x^2}\textrm{d}x=\frac{\pi}{8}\log(2)$.
The second resulting integral in $(1)$ is calculated magically here.
As regards the last integral in $(1)$, we have the simple (and magical) fact that
$$\int_0^1 \frac{\log(1-x^2)\log(1+x^2)}{1+x^2}\textrm{d}x$$
$$
=\lim_{b\to 0} \, \left(\frac{1}{16} \frac{\partial ^2}{\partial b^2}B\left(\frac{1}{4},b\right)-\frac{1}{16} \frac{\partial ^2}{\partial b^2}B\left(\frac{3}{4},b\right)\right)-\frac{1}{64} \left(\lim_{a\to \frac{1}{2}} \, \frac{\partial ^2}{\partial a^2}B\left(a,\frac{1}{2}\right)\right)$$
$$=\frac{1}{32} \left(\pi ^3+16 \pi  \log ^2(2)-96 \log (2) G\right).$$
A note: Observe that for the last integral it is enough to use the algebraic identity $ab=1/4 ((a + b)^2 - (a - b)^2)$, where $a=\log(1-x^2)$ and $b=\log(1+x^2)$. Thus, we have
$$\int_0^1 \frac{\log(1-x^2)\log(1+x^2)}{1+x^2}\textrm{d}x$$
$$=\frac{1}{4} \int_0^1 \left(\frac{\log ^2\left(1-x^4\right)}{1-x^4}-\frac{x^2 \log ^2\left(1-x^4\right)}{1-x^4}\right) \textrm{d}x-\frac{1}{4} \int_0^1 \frac{\displaystyle \log^2((1-x^2)/(1+x^2))}{1+x^2} \textrm{d}x,$$
where in the first integral we may let the variable change $x^4=y$, and in the second integral we may act in various ways, like letting $(1-x^2)/(1+x^2)=t^{1/2}$, which together lead us to the Beta function form above. So, a straightforward story essentially.
End of story
More details will be given in the sequel of (Almost) Impossible Integrals, Sums, and Series.
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