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I was able to calculate: $$\int_0^\infty\arctan\left(e^{-x}\right)\,dx=G$$ $$\int_0^\infty\arctan^2\left(e^{-x}\right)\,dx=\frac\pi2\,G-\frac78\zeta(3)$$ $G$ is the Catalan constant. In both cases Maple is able to find corresponding indefinite integral, and the above results can be obtained by taking limits. They also can be done using identities $$\arctan\left(e^{-x}\right)=\Im\left(\log\left(1+ie^{-x}\right)\right)\color{gray}{,\quad x\in\mathbb{R}}$$ $$\arctan^2\left(e^{-x}\right)=\frac14\log^2\left(1+e^{-2x}\right)-\Re\left(\log^2\left(1+ie^{-x}\right)\right)\color{gray}{,\quad x\in\mathbb{R}}$$ I'm also interested in this value: $$\int_0^\infty\arctan\left(e^{-x}\right)\,\arctan\left(e^{-2x}\right)\,dx$$ Could you help me with it? Maple cannot do it.

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    $\begingroup$ Mathematica will grudgingly handle the last one, although the output looks like a huge pile of polylog functions. (It reduces to only a couple of lines of polylog functions at the root of a certain quartic equation, plus some temrs involving the Catalan constant and arcsinh, for low values of 'reduces'.) Maybe it simplifies, but trying to do so directly doesn't yield anything useful. Where does this sort of ugly integral come from? $\endgroup$ – anomaly Jan 8 '16 at 23:56
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    $\begingroup$ Inverse symbolic calculations do not find anything looking like the numerical value $0.25617307025323739218841203901935947736527563268609$ $\endgroup$ – Claude Leibovici Jan 9 '16 at 3:17
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    $\begingroup$ @ClaudeLeibovici Unfortunately, inverse symbolic calculators are not perfect. See the closed form in my answer below. $\endgroup$ – Vladimir Reshetnikov Jan 10 '16 at 2:00
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This is a fascinating integral! Following generally the same approach as shown in David H's answer, with help from Mathematica and manual simplification using known di- and trilogarithm identities, I was able to establish the following: $$\int_0^x\arctan\left(e^{-z}\right)\,\arctan\left(e^{-2z}\right)\,dz=x\operatorname{arccot}\left(e^x\right)\operatorname{arccot}\left(e^{2x}\right)\\ +\frac{\pi^2}{32}\left(\vphantom{\Large|}\!\ln\left(1+e^{4x}\right)-10\ln\left(e^{2x}+e^x\sqrt2+1\right)\right)\\ +\frac\pi2\left(\vphantom{\Large|}\!\operatorname{arccot}\left(1+e^x\sqrt2\right)+\arctan\left(e^x\right)\right)\cdot\ln\left(e^{2x}+e^x\sqrt2+1\right)\\ +\frac i4\left\{\left(\vphantom{\Large|}2\operatorname{arccot}\left(e^x\right)+\operatorname{arccot}\left(e^{2x}\right)\right)\cdot\\ \left[\operatorname{Li}_2\left(\tfrac{(-1)^{3/4}+e^x}{-i+e^x}\right)-\operatorname{Li}_2\left(\tfrac{-(-1)^{1/4}+e^x}{i+e^x}\right) +\operatorname{Li}_2\left(\tfrac{i-(-1)^{1/4}e^x}{i+e^x}\right)-\operatorname{Li}_2\left(\tfrac{1-(-1)^{1/4}e^x}{1+ie^x}\right)\right]\\ +\left(\vphantom{\Large|}\!\operatorname{arccot}\left(e^{2x}\right)+2\operatorname{arccot}\left(1+e^x\sqrt2\right)-2\operatorname{arccot}\left(e^x\right)\right)\cdot\\ \left[\operatorname{Li}_2\left(\tfrac{-(-1)^{1/4}+e^x}{-i+e^x}\right)-\operatorname{Li}_2\left(\tfrac{(-1)^{3/4}+e^x}{i+e^x}\right)+\operatorname{Li}_2\left(\tfrac{-1+(-1)^{1/4}e^x}{-1+ie^x}\right)-\operatorname{Li}_2\left(\tfrac{-i+(-1)^{1/4}e^x}{-i+e^x}\right)\right]\right\}\\ +\frac i2\left\{\operatorname{arccot}\left(1+e^x\sqrt2\right)\cdot\\ \left[\operatorname{Li}_2\left(\tfrac{-(-1)^{3/4}+e^x}{i+e^x}\right)-\operatorname{Li}_2\left(\tfrac{(-1)^{1/4}+e^x}{-i+e^x}\right)+\operatorname{Li}_2\left(\tfrac{i+(-1)^{1/4}e^x}{i-e^x}\right)-\operatorname{Li}_2\left(\tfrac{i+(-1)^{3/4}e^x}{i+e^x}\right)\right]+\left(\vphantom{\Large|}\!\operatorname{arccot}\left(e^x\right)-\operatorname{arccot}\left(1+e^x\sqrt2\right)\right)\cdot\left[\vphantom{\Large|}\operatorname{Li}_2\left(1-(-1)^{1/4}e^x\right)-\operatorname{Li}_2\left(1+(-1)^{1/4}e^x\right)\\ +\operatorname{Li}_2\left(1-(-1)^{3/4}e^x\right)-\operatorname{Li}_2\left(1+(-1)^{3/4}e^x\right)\right]\right\}\\ +\frac14\left[\operatorname{Li}_3\left(\tfrac{-(-1)^{1/4}+e^x}{-i+e^x}\right)-\operatorname{Li}_3\left(\tfrac{-(-1)^{1/4}+e^x}{i+e^x}\right)+\operatorname{Li}_3\left(\tfrac{(-1)^{1/4}+e^x}{-i+e^x}\right)-\operatorname{Li}_3\left(\tfrac{(-1)^{1/4}+e^x}{i+e^x}\right)\\ -\operatorname{Li}_3\left(\tfrac{-(-1)^{3/4}+e^x}{-i+e^x}\right)+\operatorname{Li}_3\left(\tfrac{-(-1)^{3/4}+e^x}{i+e^x}\right) -\operatorname{Li}_3\left(\tfrac{(-1)^{3/4}+e^x}{-i+e^x}\right)+\operatorname{Li}_3\left(\tfrac{(-1)^{3/4}+e^x}{i+e^x}\right)\\ +\operatorname{Li}_3\left(\tfrac{i-(-1)^{1/4}e^x}{i+e^x}\right)-\operatorname{Li}_3\left(\tfrac{-1+(-1)^{1/4}e^x}{-1+ie^x}\right)-\operatorname{Li}_3\left(\tfrac{-i+(-1)^{1/4}e^x}{-i+e^x}\right)-\operatorname{Li}_3\left(-\tfrac{i+(-1)^{1/4}e^x}{-i+e^x}\right)\\ +\operatorname{Li}_3\left(\tfrac{i+(-1)^{1/4}e^x}{i+e^x}\right)+\operatorname{Li}_3\left(\tfrac{-i+(-1)^{3/4}e^x}{-i+e^x}\right)+\operatorname{Li}_3\left(-\tfrac{i+(-1)^{3/4}e^x}{-i+e^x}\right) -\operatorname{Li}_3\left(\tfrac{i+(-1)^{3/4}e^x}{i+e^x}\right)\right]\\ +\frac\pi{64}\left[\left(2+\sqrt2\right)\cdot\left(16G+\pi^2\sqrt2\right)-\sqrt2\,\psi^{(1)}\!\left(\tfrac18\right)\right]$$ (here is the corresponding Mathematica expression)

If I'm not mistaken, it must hold for all real $x$, and might hold for some complex $x$, but in general there are some branch cuts that need to be dealt with (I was not yet able to analyze them completely). A possible proof consists of taking a derivative, a (tedious) simplification and computing a limit to establish the constant of integration. I relied on Mathematica for some steps, followed by high-precision numerical validation.

So, the final answer is

$$\int_0^\infty\arctan\left(e^{-x}\right)\,\arctan\left(e^{-2x}\right)\,dx=\frac\pi{64}\left[\left(2+\sqrt2\right)\cdot\left(16G+\pi^2\sqrt2\right)-\sqrt2\,\psi^{(1)}\!\left(\tfrac18\right)\right]$$

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    $\begingroup$ inverse symbolic calculators are not perfect, for sure .... but ... you are ! $\endgroup$ – Claude Leibovici Jan 10 '16 at 2:19
  • $\begingroup$ I wonder if the antiderivative could be simplified. I feel that possibly the number of polylogarithm terms and complex numbers can be reduced if we put to work $\operatorname{Cl}_n(z)$ and $\operatorname{Ti}_n(z)$. Unfortunately, Mathematica does not have native support for these functions. $\endgroup$ – Vladimir Reshetnikov Jan 10 '16 at 2:26
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    $\begingroup$ @VladimirReshetnikov +1) The final value you found is much tidier than I was anticipating! In general, the reduction of logarithmic integrals of the form $\int\frac{\ln{(1+ax)}\ln{(1+bx)}}{x}dx$ will produce five trilogarithmic terms prior to simplification. Regarding the integral in the OP, I managed to reduce it to a sum of four logarithmic integrals, meaning $4\times5=20$ trilogs! This is part of the reason I gave up on finishing my answer. :) $\endgroup$ – David H Jan 10 '16 at 12:44
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We'll have use for the following trigonometric identity, which may be verified by the well-known arctangent addition law:

$$\small{\arctan{\left(x^{2}\right)}=\arctan{\left(\frac{x}{\sqrt{2}-x}\right)}-\arctan{\left(\frac{x}{\sqrt{2}+x}\right)}};~~~\small{\left|x\right|<\sqrt{2}}.$$

We'll also need the definition of the arctangent function in terms of complex logarithms:

$$\arctan{\left(z\right)}=\frac{\ln{\left(\frac{1+iz}{1-iz}\right)}}{2i}.$$

Then,

$$\begin{align} I &=\int_{0}^{\infty}\arctan{\left(e^{-y}\right)}\arctan{\left(e^{-2y}\right)}\,\mathrm{d}y\\ &=\int_{0}^{1}\frac{\arctan{\left(x\right)}\arctan{\left(x^{2}\right)}}{x}\,\mathrm{d}x;~~~\small{\left[e^{-y}=x\right]}\\ &=\int_{0}^{1}\frac{\arctan{\left(x\right)}\left[\arctan{\left(\frac{x}{\sqrt{2}-x}\right)}-\arctan{\left(\frac{x}{\sqrt{2}+x}\right)}\right]}{x}\,\mathrm{d}x\\ &=\int_{0}^{1}\frac{\arctan{\left(x\right)}\left[\ln{\left(1-e^{-\frac{i\pi}{4}}x\right)}-\ln{\left(1-e^{\frac{i\pi}{4}}x\right)}\right]}{2ix}\,\mathrm{d}x\\ &~~~~~-\int_{0}^{1}\frac{\arctan{\left(x\right)}\left[\ln{\left(1+e^{\frac{i\pi}{4}}x\right)}-\ln{\left(1+e^{-\frac{i\pi}{4}}x\right)}\right]}{2ix}\,\mathrm{d}x\\ &=\small{\int_{0}^{1}\frac{\left[\ln{\left(1+ix\right)}-\ln{\left(1-ix\right)}\right]\left[\ln{\left(1+e^{\frac{i\pi}{4}}x\right)}-\ln{\left(1+e^{-\frac{i\pi}{4}}x\right)}\right]}{4x}\,\mathrm{d}x}\\ &~~~~~\small{-\int_{0}^{1}\frac{\left[\ln{\left(1+ix\right)}-\ln{\left(1-ix\right)}\right]\left[\ln{\left(1+e^{\frac{3i\pi}{4}}x\right)}-\ln{\left(1+e^{-\frac{3i\pi}{4}}x\right)}\right]}{4x}\,\mathrm{d}x}\\ &=\small{\int_{0}^{1}\frac{\Re{\left[\ln{\left(1+ix\right)}\ln{\left(1+e^{\frac{i\pi}{4}}x\right)}-\ln{\left(1-ix\right)}\ln{\left(1+e^{\frac{i\pi}{4}}x\right)}\right]}}{2x}\,\mathrm{d}x}\\ &~~~~~\small{-\int_{0}^{1}\frac{\Re{\left[\ln{\left(1+ix\right)}\ln{\left(1+e^{\frac{3i\pi}{4}}x\right)}-\ln{\left(1-ix\right)}\ln{\left(1+e^{\frac{3i\pi}{4}}x\right)}\right]}}{2x}\,\mathrm{d}x}.\\ \end{align}$$

Any integral of the form $\int\frac{\ln{\left(ax+b\right)}\ln{\left(cx+d\right)}}{px+q}\,\mathrm{d}x$ can be systematically reduced to trilogarithms, dilogarithms, and elementary functions. Thus, we recognize that $I$ can be decomposed into a sum of such integrals, and so in principle we are done except for the tedium.

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Using the series for arctan, it is $$\sum_{i=0}^{\infty}\sum_{j=0}^{\infty}\frac{(-1)^{i+j}}{(2i+1)(4j+2)(2i+4j+3)}$$

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Too long for a comment :

As far as definite integrals are concerned, there are no currently known methods of actually proving that they do not possess a closed form. $($ In other words, there are no equivalents of Liouville's theorem or the Risch algorithm for definite integrals $).$ So what I'm about to write
does not constitute “proof” of anything, but serves merely as an intuitive explanation for why
it is unlikely that the integral you posted to have a “meaningful” closed form. Notice that the
other two integrands possess an inverse, i.e., $$y~=~\arctan^n(e^{-x})~\iff~x~=~-\ln\tan\sqrt[n]y~=~\ln\cos\sqrt[n]y~-~\ln\sin\sqrt[n]y~.$$ At the same time, remember the well-known relation between the definite integral of a function, and that of its inverse:

$\qquad\qquad\qquad\quad$

Applying it to the former two cases, it yields something manageable, following a substitution of the form $y=t^n.$ The latter integrand, however, does not exhibit such auspicious traits, since its inverse cannot be expressed in terms of any known functions, be they either special or elementary.

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