Can you prove or disprove the following claim:



> 
> **Claim:**
> $$\frac{\sqrt{3} \pi}{24}=\displaystyle\sum\_{n=0}^{\infty}\frac{1}{(6n+1)(6n+5)}$$
> 
> 
> 


The SageMath cell that demonstrates this claim can be found [here](https://sagecell.sagemath.org/?z=eJwrTbI1MgACa65i2-LSXI08WwOd0iQdQ30NDTOtPG1DTS0wbaqpqWnNVVCUmVeiUaxlqIfgFBaVaBhragVk6huZaFoDAD15FQQ=&lang=gp&interacts=eJyLjgUAARUAuQ==).



An infinite series that converges to $\frac{\sqrt{3}\pi}{24}$
Here is an elementary proof. We rewrite the series as
$$\frac{1}{4}\int\_0^1\frac{1-x^4}{1-x^6}\,dx=\frac{1}{8}\int\_0^1\frac{dx}{1-x+x^2}+\frac{1}{8}\int\_0^1\frac{dx}{1+x+x^2}.$$
It is straightforward to show that
\begin{align\*}
\int\_0^1\frac{dx}{1-x+x^2}&=\frac{2\pi}{3\sqrt{3}},\\
\int\_0^1\frac{dx}{1+x+x^2}&=\frac{\pi}{3\sqrt{3}},
\end{align\*}
so we are done.


