Previous try
It should be $\pi^2/2$ but showing this requires invoking the Cauchy residue theorem, as is often the case when $\pi$ creeps in integrals.
The demonstration proceeds in two steps.
I - Contour integration
- First refer to C. Leibovici's answer:
$$ \int \frac{1}{\sqrt{x^2+2 x}} \, dx = 2 \tanh ^{-1}\left(\sqrt{\frac{x}{x+2}}\right)$$
then integrate by parts using this result, which yields:
$$I = \int_{0}^{1/\sqrt{3}} \frac{8 \tanh ^{-1}(v)}{v \left(1-v^4\right)} \, dv$$
- Now notice that the (simple) poles of the integrand $f(v)$ are:
$$ \{-1,0,-i,i,1\} $$
Choose a rectangular integration contour such as:
$$ \{\frac{-1}{\sqrt{3}}, 0^{+}, \frac{1}{\sqrt{3}}, c\, i + \frac{1}{\sqrt{3}}, c \, i -\frac{1}{\sqrt{3}}\},$$
with: $\, \, c \rightarrow \, +\infty $
$0^{+}$ intends to symbolize a little dent around the 0 pole, this is to make it clean but anyway the residue there is null.
Compute the residue at $i$, this is easy and I'll leave it to you. Hint: notice that $ \tanh^{-1}(i) = -i \frac{\pi}{4} $, which is the true reason why you get in $\pi$ along the way.
Check that at infinity along the upper side of the rectangle contour, $f(v)$ is negligible, which is fairly easy too.
The hard bit is to show that the real part of the contour integration along one or the other of the vertical sides of the rectangle is null just at the points chosen. With other points, it is not null, and it does not cancel (it adds up) when the contour integration along the two vertical sides is performed:
$$ \Re{ \int_{\pm\frac{1}{\sqrt{3}}}^{\pm\frac{1}{\sqrt{3}} + i \infty} f(v) \, dv} = 0 \,\,\, (\mathscr{C})$$
The easy bit is that, in any case, the imaginary part does obviously cancel, owing to the fact that $f(v)$ is even.
The hard bit is not demonstrated yet, but does not look intractable.
It is anyway an improvement to replace a conjecture on a multiple of $\pi^2$ by a conjecture about a null value, with a hint at optimization (for which there are lots of methods around).
- Applying Cauchy's residue theorem you now get the result:
$$ \int_R f(z) \, dz = 2i\pi \, Res(f, i) = 2i\pi (-(i \frac{\pi}{2})) = \pi^2$$
- Considering the fact that the integral vanishes along the three sides above the real axis when you let $c$ grow to infinity, you now are left with the integration along the real segment $[-\frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}}]$, whose value is 2I, owing to the parity of $f$. Hence the result.
II - Towards the proof of conjecture ($\mathscr{C}$)
Brute-force integration of $f(v)$ along a vertical rectangular side (let us take the right one) $[\frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}} + i y]$:
$$f(\frac{1}{\sqrt{3}} + i \,y) = \frac{8 \tanh ^{-1}\left(\frac{1}{\sqrt{3}}+i y\right)}{\left(1-\left(\frac{1}{\sqrt{3}}+i y\right)^4\right) \left(\frac{1}{\sqrt{3}}+i y\right)}$$
This gives the imaginary part (owing to contour integration, $f$ must be multiplied by $i$, the derivative of the parametrized contour side):
$$g(y) = \frac{18 \left(90 \sqrt{3} y^4 \tan ^{-1}(\frac{y}{\frac{1}{\sqrt{3}}-1})-2 \sqrt{3} \left(-45 y^4+30 y^2+8\right) \tan ^{-1}(\frac{y}{1+\frac{1}{\sqrt{3}}})+4 \sqrt{3} \left(15 y^2+4\right) \tan ^{-1}\left(\frac{1}{2} \left(\sqrt{3}+3\right) y\right)+3 \left(-9 y^4+30 y^2+4\right) y \log \left(1-\frac{4 \sqrt{3}}{3 y^2+2 \sqrt{3}+4}\right)\right)}{\left(3 y^2+1\right) \left(336 y^2+27 \left(y^2+2\right) \left(3 y^2-2\right) y^4+64\right)}$$
Integration between $y = 0$ and $y= N$ (let us denote this integral $J(N)$) then using an asymptotic series when N tends to infinity reduces $\mathscr{(C)}$ to $\mathscr{(C')}$:
$$J(N) \rightarrow 0 \,\,\, \mathscr{(C')}$$
(numerically this much is proved).
Plot of $J(N)$ as $N \rightarrow +\infty$
![Plot of J(N) as N \rightarrow +\infty](https://cdn.statically.io/img/i.sstatic.net/rZHSc.png)
When proved this will prove ($\mathscr{C}$) and complete the demonstration. I'll leave this for further edits.
This attack procedure may well work if one finds a good absolutely dominating function for $\Im{f(\frac{1}{\sqrt{3}} + i \,y)}$ for which integration and asymptotic approximation may work without too much toil.
III - Edit: going further and final comments
Integrating $g(y)$ in II is a bit cumbersome, to say the least, but can be done.
Doing this goes through the following steps:
1. Decompose the denominator into simple poles to find the values for $\alpha$ above.
2. Integrate:
$$I_k(N) = \int_0^N \frac{u^k \tan^{-1}(u)}{u - \alpha} \, du, \,\, k=0, 2, 4, \, \alpha \in \mathbb{R} $$
3. Integrate:
$$ L_k(N) = \int_0^N y^k \frac{\log(1-\frac{4 \sqrt{3}}{3 y^2+2 \sqrt{3}+4})}{y - \alpha} \, dy, k=1, 3, 5, \, \alpha \in \mathbb{R} $$
4. Expand each of the above results by N and sum the different polynomial contributions
5. Simplify the resulting expression and let N tend to infinity.
On these steps:
Point 1
The poles of the denominator are :
$$\left\{-\frac{i}{\sqrt{3}}, \frac{i}{\sqrt{3}}, -i \sqrt{\frac{4}{3}-\frac{2}{\sqrt{3}}}, i \sqrt{\frac{4}{3}-\frac{2}{\sqrt{3}}}, \frac{1}{3} \left(-3-i \sqrt{3}\right),\frac{1}{3} \left(3-i \sqrt{3}\right), \frac{1}{3} \left(-3+i \sqrt{3}\right), \frac{1}{3} \left(3+i \sqrt{3}\right), -i \sqrt{\frac{2}{3} \left(\sqrt{3}+2\right)}, i \sqrt{\frac{2}{3} \left(\sqrt{3}+2\right)}\right\}$$
So the denominator easily simplifies into a sum of 10 simple poles. Each must be decomposed into 3 arctangent-type and 1 logarithm-type numerator. Then the polynomial factors are expanded to end up with $I_k(N)$ and $L_k(N)$ expressions. In all there must be 60 $I_k$-ype integrals and 30 $L_k$-type integrals to consider.
Point 2
Actually the case $k = 0$ is well-known: it is a generalized inverse tangent integral of second order (see 1), for which an asymptotic expansion is known:
$$Ti_2(N, -\alpha) = \frac{1}{2} \pi \ln{N} - \frac{1}{4} \pi \ln{(1 + \alpha^2)} - Ti_2(\alpha) + \tan^{-1}(\alpha) \ln{|\alpha|} + O(\frac{1}{N})$$
The other cases can be obtained from this starting point by classic integration techniques (integration by parts and auxiliary parameter derivation), so there is reason to consider that an asymptotic expansion of the arctangent part of $J(N)$ is algorithmically doable.
Also, maybe the good strategy would be to find an asymptotic expansion for $I_k(N)$ from the above formula before tackling the 60 $I_k$-type integrals. This would simplify the task by an order of magnitude.
Further, $L_k(N)$ is a sum of classic integrals that can be found in Gradshtein or Prudnikov.
There is reason to believe that proving or refuting conjecture $\mathscr{(C')}$ is an algorithmically doable task. It is just quite cumbersome and will take a few dozen classic operations; yet not anything that exceeds the power of CAS systems, so there must be a conclusion in the end of the daunting calculus: true
or false
.
We expect the asymptotic logarithms and arctangents to cancel out easily; however we will be stranded with the second-order inverse tangent expressions taken at the 10 poles. These are closely associated values, so we expect them to cancel out; but to reach this conclusion we will need formulas relating $Ti_2$ values of comparable complexity as the above solution using the five-term Kummerian $Li_2$ relation (see 1, appendix). This may be doable again but, at the end of the day, looks like a much-complexified version of the same solution.
It looks fascinating that for such integrals, there seems to be no way out other than by manipulating polylogs and associated functions.
1 Leonard Lewin, Polylogarithms and associated functions, 1981, North Holland, chapter 3 and expansion p. 288 (11).