<s>[INST] Integral $\int_{-\infty}^{\infty} \arctan(e^x) \arctan(e^{-x})dx=\frac{7}{4}\zeta(3)$ 
How to prove that $$\int_{-\infty}^{\infty} \arctan(e^x) \arctan(e^{-x})\rm dx=\frac{7}{4}\zeta(3)?$$

This integral appeared while attempting to solve an unanswered question, namely:
Closed form of :$ \int_{-\infty}^{\infty}\arctan\left(e^{-x^2 \text{erf}(x)}\right)\,\arctan\left(e^{x^2\text{erf(x)}}\right)\,dx $
I happened to find that
$$\int_{-\infty}^{\infty} \arctan(e^x) ~\arctan(e^{-x})~dx=\frac{7}{4}\zeta(3).$$
How can you work out the answer on the right by hand?
I guess that there could be multiple approaches to solve this one.
 [/INST] Here is another slight variation on a theme.
Let
$$I = \int_{-\infty}^\infty \arctan (e^x) \arctan (e^{-x}) \, dx.$$
Setting $u = e^x$ gives
\begin{align}
I &= \int_0^\infty \arctan u \arctan \left (\frac{1}{u} \right ) \frac{du}{u}\\
&=  \int_0^1 \arctan u \arctan \left (\frac{1}{u} \right ) \frac{du}{u} +  \int_1^\infty \arctan u \arctan \left (\frac{1}{u} \right ) \frac{du}{u}\\
&= 2  \int_0^1 \arctan x \arctan \left (\frac{1}{x} \right ) \frac{dx}{x},
\end{align}
where in the second of the integrals a substitution of $x \mapsto 1/x$ has been enforced.  
As
$$\arctan \left (\frac{1}{x} \right ) = \frac{\pi}{2} - \arctan x, \qquad x > 0,$$
we can write the above integral as
$$I = \pi \int_0^1 \frac{\arctan x}{x} \, dx - 2 \int_0^1 \frac{\arctan^2 x}{x} \, dx = \pi I_1 - 2 I_2.$$
For the first integral $I_1$, we have
\begin{align}
I_1 &= \int_0^1 \frac{\arctan x}{x} \, dx\\
&= \sum_{n = 0}^\infty \frac{(-1)^n}{2n + 1} \int_0^1 x^{2n} \, dx\\
&= \sum_{n = 0}^\infty \frac{(-1)^n}{(2n + 1)^2}\\
&= \mathbf{G},
\end{align}
where $\mathbf{G}$ denotes Catalan's constant.
For the second integral $I_2$, integrating by parts gives
$$I_2 = -2 \int_0^1 \frac{\ln x \arctan x}{1 + x^2} \, dx.$$
Enforcing a substituition of $x \mapsto \arctan x$ leads to 
$$I_2 = -2 \int_0^{\frac{\pi}{4}} x \ln (\tan x) \, dx.$$
The integral appearing above has been evalauted elsewhere on this site (see, for example, here). The result is:
$$\int_0^{\frac{\pi}{4}} x \ln (\tan x) \, dx = \frac{7}{16} \zeta (3) - \frac{\pi \mathbf{G}}{4}.$$
Thus
$$I_2 = \frac{\pi \mathbf{G}}{2} - \frac{7}{8} \zeta (3),$$
giving
$$I = \pi \mathbf{G} - 2 \left (\frac{\pi \mathbf{G}}{2} - \frac{7}{8} \zeta (3) \right ),$$
or
$$\int_{-\infty}^\infty \arctan (e^x) \arctan (e^{-x}) \, dx = \frac{7}{4} \zeta (3),$$
as required.
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