I think that will not be useful to compute the Apéry's constant as $$\zeta(3)=\frac{4}{\pi}\sum_{n=1}^{\infty}\frac{1}{n^2}\int_0^{\frac{\pi}{2n}}\sin^2(3x)dx+\frac{1}{3\pi}\left(\sum_{n=1}^{\infty}\frac{\sin(\frac{3\pi}{n})}{n^2}\right),$$ which is easily deduced from $$\int_0^{\frac{\pi}{2n}}\sin^2(3x)dx=\frac{1}{12}\left(\frac{3\pi}{n}-\sin(\frac{3\pi}{n})\right).$$
For deduce this last identity I've used an online tool of symbolic calculus. I say that couldn't be useful since neither I don't know how evaluate an alternative of the series of integrals $\sum_{n\geq 1} \frac{1}{n^2}\int_0^{\frac{\pi}{2n}}\sin^2(3x)dx$ (if it is feasible, to compute in a distinct form).
In any case I would like to know
Question. It is know, or it is possible to get a closed-form for $$\sum_{n=1}^{\infty}\frac{\sin(\frac{3\pi}{n})}{n^2}$$ in terms of known constants? Thanks in advance.
When I've do more computations, using some trigonometric tricks I've found also with $\sum_{n=1}^{\infty}\frac{\sin(\frac{\pi}{n})}{n^2}$ and $\sum_{n=1}^{\infty}\frac{\sin(\frac{2\pi}{n})}{n^2}$. It is easy to check the absolute convergence of such series, thus by comparision test these series are convergents. Using another time Wolfram Alpha (its Online Series Calculator) I don't to get a closed-form (only an approximation is given for such series) in terms of known constant, this is as exact value. Are know these series? When I've used this computation in its (General) Calculator, yes then I've obtained some closed-form, but the Series Calculator doesn't sure any closed-form.
My attemps were using partial summation and Euler-MacLaurin (first approximation), but I believe that I can not find this real value with these methods.