Evaluating the series $\sum\limits_{n=1}^{\infty} \frac{1}{n^{3} \binom{2n}{n}} $

Using the Taylor expansion for $\arcsin^2(x)$ and integrating, it's easy to show (as you made) that $$ \begin{align} \sum_{n=1}^{\infty} \frac{1}{n^{3} \binom{2n}{n}} &= 4 \int_{0}^{\frac{1}{2}} \frac{\arcsin^{2}(x)}{x} \operatorname{d}x = - 8 \int_{0}^{\frac{\pi}{6}} u \ln ( 2 \sin u ) \operatorname{d}u \\ &= \frac{2 \pi}{3} \sum_{n=1}^{\infty} \frac{\sin (\frac{n \pi}{3})}{n^{2}} + 2 \sum_{n=1}^{\infty} \frac{\cos (\frac{n \pi}{3})}{n^{3}} - 2 \zeta(3)\\ &=\frac{2 \pi}{3}\operatorname{Cl}_2\left(\frac{\pi}{3}\right)+2\operatorname{Cl}_3\left(\frac{\pi}{3}\right)-2 \zeta(3) \end{align} $$ recalling that the Clausen function are defined as $$ \operatorname{Cl}_{m}(\theta)= \begin{cases}\displaystyle \sum_{n=1}^{\infty} \frac{\sin (n\theta)}{n^{m}}& \text{for } m\text{ even}\\ \displaystyle\sum_{n=1}^{\infty} \frac{\cos(n\theta)}{n^{m}}& \text{for } m\text{ odd} \end{cases} $$ The value of the Clausen function $\operatorname{Cl}_3$ at $\frac{\pi}{3}$ is $$ \operatorname{Cl}_3\left(\frac{\pi}{3}\right)=\frac{1}{2}\left(1-2^{-2}\right)\left(1-3^{-2}\right)\zeta(3)=\frac{1}{3}\zeta(3) $$ From the duplication formula $$ \operatorname{Cl}_{2m}(2\theta)=2^{2m-1}\left[\operatorname{Cl}_{2m}(\theta)-\operatorname{Cl}_{2m}(\pi-\theta)\right] $$ we find $$ \operatorname{Cl}_{2}\left(\frac{2\pi}{3}\right)=\frac{2}{3}\operatorname{Cl}_{2}\left(\frac{\pi}{3}\right). $$ From the identities for the trigamma function at $1/3$ and $2/3$ $$ \begin{align} \psi_1\left(\frac{1}{3}\right) &=\frac{2\pi^2}{3}+3\sqrt 3\operatorname{Cl}_{2}\left(\frac{2\pi}{3}\right)\\ \psi_1\left(\frac{2}{3}\right) &=\frac{2\pi^2}{3}-3\sqrt 3\operatorname{Cl}_{2}\left(\frac{2\pi}{3}\right) \end{align} $$ one has $$ \psi_1\left(\frac{1}{3}\right)-\psi_1\left(\frac{2}{3}\right)=6\sqrt 3\operatorname{Cl}_{2}\left(\frac{2\pi}{3}\right) $$ and then $$ \operatorname{Cl}_{2}\left(\frac{\pi}{3}\right)=\frac{\sqrt 3}{12}\left[\psi_1\left(\frac{1}{3}\right)-\psi_1\left(\frac{2}{3}\right)\right] $$

Finally, putting all together, we have $$ \begin{align} \sum_{n=1}^{\infty} \frac{1}{n^{3} \binom{2n}{n}} &=\frac{2 \pi}{3}\operatorname{Cl}_2\left(\frac{\pi}{3}\right)+2\operatorname{Cl}_3\left(\frac{\pi}{3}\right)-2 \zeta(3)\\ &=\frac{2 \pi}{3}\frac{\sqrt 3}{12}\left[\psi_1\left(\frac{1}{3}\right)-\psi_1\left(\frac{2}{3}\right)\right]+2\frac{1}{3}\zeta(3)-2\zeta(3)\\ &=\frac{\pi\sqrt 3}{18}\left[\psi_1\left(\frac{1}{3}\right)-\psi_1\left(\frac{2}{3}\right)\right]-\frac{4}{3}\zeta(3). \end{align} $$

$\newcommand{\angles}[1]{\left\langle\, #1 \,\right\rangle} \newcommand{\braces}[1]{\left\lbrace\, #1 \,\right\rbrace} \newcommand{\bracks}[1]{\left\lbrack\, #1 \,\right\rbrack} \newcommand{\ceil}[1]{\,\left\lceil\, #1 \,\right\rceil\,} \newcommand{\dd}{{\rm d}} \newcommand{\ds}[1]{\displaystyle{#1}} \newcommand{\expo}[1]{\,{\rm e}^{#1}\,} \newcommand{\fermi}{\,{\rm f}} \newcommand{\floor}[1]{\,\left\lfloor #1 \right\rfloor\,} \newcommand{\half}{{1 \over 2}} \newcommand{\ic}{{\rm i}} \newcommand{\iff}{\Longleftrightarrow} \newcommand{\imp}{\Longrightarrow} \newcommand{\pars}[1]{\left(\, #1 \,\right)} \newcommand{\partiald}[3][]{\frac{\partial^{#1} #2}{\partial #3^{#1}}} \newcommand{\pp}{{\cal P}} \newcommand{\root}[2][]{\,\sqrt[#1]{\vphantom{\large A}\,#2\,}\,} \newcommand{\sech}{\,{\rm sech}} \newcommand{\sgn}{\,{\rm sgn}} \newcommand{\totald}[3][]{\frac{{\rm d}^{#1} #2}{{\rm d} #3^{#1}}} \newcommand{\verts}[1]{\left\vert\, #1 \,\right\vert}$ $\ds{\sum_{n=1}^{\infty} \frac{1}{n^{3} \binom{2n}{n}} = \frac{ \pi \sqrt{3}}{18} \Big[ \psi_{1} \left(\frac{1}{3} \right) - \psi_{1} \left(\frac{2}{3} \right) \Big]- \frac{4}{3} \zeta(3)}$

\begin{align} &\sum_{n = 1}^{\infty}{x^{n} \over n{2n \choose n}} =\sum_{n = 1}^{\infty}x^{n}\, {\Gamma\pars{n}\Gamma\pars{n + 1} \over \Gamma\pars{2n + 1}} =\sum_{n = 1}^{\infty}x^{n}\int_{0}^{1}t^{n - 1}\pars{1 - t}^{n}\,\dd t \\[3mm] & = \int_{0}^{1}\sum_{n = 1}^{\infty}\bracks{tx\pars{1 - t}}^{n} \,{\dd t \over t} =\int_{0}^{1}\bracks{{1 \over 1 - tx\pars{1 - t}} - 1}\,{\dd t \over t} \\[3mm] & = \int_{0}^{1}{x\pars{1 - t} \over 1 - tx\pars{1 - t}}\,\dd t \end{align}

\begin{align} &\sum_{n = 1}^{\infty}{x^{n} \over n^{2}{2n \choose n}} =\int_{0}^{x}\dd y\int_{0}^{1}{1 - t \over 1 - ty\pars{1 - t}}\,\dd t =-\int_{0}^{1}{\ln\pars{1 - \bracks{1 - t}tx} \over t}\,\dd t \end{align}

\begin{align} &\color{#66f}{\large\sum_{n = 1}^{\infty}{1 \over n^{3}{2n \choose n}}} =-\int_{0}^{1}{\dd x \over x} \int_{0}^{1}{\ln\pars{1 - \bracks{1 - t}tx} \over t}\,\dd t \\[3mm] = &\ \int_{0}^{1} {{\rm Li}_{2}\pars{t\bracks{1 - t}} \over t}\,\dd t \\[3mm] = &\ \color{#66f}{{\pi \over 36\root{3}}\, \left[\psi ^{(1)}\left(\frac{1}{3}\right)-\psi ^{(1)}\left(\frac{2}{3}\right)+\psi ^{(1)}\left(\frac{1}{6}\right)-\psi ^{(1)}\left(\frac{5}{6}\right)\right]-\frac{4 \zeta (3)}{3}} \\[3mm] \approx &\ {\tt 0.5229} \end{align}