Wolfram MathWorld states that $$ \sum_{n=1}^{\infty} \frac{1}{n^{3} \binom{2n}{n}} = \frac{ \pi \sqrt{3}}{18} \Bigg[ \psi_{1} \left(\frac{1}{3} \right) – \psi_{1} \left(\frac{2}{3} \right) \Bigg]- \frac{4}{3} \zeta(3) \, , $$
where $\psi_{1}(x)$ is the trigamma function.
I'm having difficulty getting the result in that form.
Using the Taylor expansion $$ \arcsin^{2}(x) = \frac{1}{2} \sum_{n=1}^{\infty} \frac{1}{n^{2} \binom{2n}{n}} (2x)^{2n},$$ I get
$$ \sum_{n=1}^{\infty} \frac{1}{n^{3} \binom{2n}{n}} = 4 \int_{0}^{\frac{1}{2}} \frac{\arcsin^{2}(x)}{x} \, \mathrm dx. $$
Then I integrating by parts, I get
$$ \begin{align} &4 \int_{0}^{\frac{1}{2}} \frac{\arcsin^{2}(x)}{x} \, \mathrm dx \\ &= \frac{\pi^{2}}{9} \ln \left(\frac{1}{2} \right) – 8 \int_{0}^{\frac{1}{2}} \frac{\arcsin (x) \ln (x)}{\sqrt{1-x^{2}}} \, \mathrm dx \\ &= – \frac{\pi^{2}}{9} \ln 2 – 8 \int_{0}^{\frac{\pi}{6}} u \ln (\sin u ) \, \mathrm du \\ &= – 8 \ln 2 \int_{0}^{\frac{\pi}{6}} u \, \mathrm du – 8\int_{0}^{\frac{\pi}{6}} u \ln (\sin u ) \, \mathrm du \\ &= – 8 \int_{0}^{\frac{\pi}{6}} u \ln ( 2 \sin u ) \, \mathrm du \\ &= -8 \ \text{Re} \int_{0}^{\frac{\pi}{6}} u \ln (1-e^{2iu}) \, \mathrm du \\ &= 8 \ \text{Re} \int_{0}^{\frac{\pi}{6}} u \sum_{n=1}^{\infty} \frac{e^{2in u}}{n} \, \mathrm du \\ &= 8 \ \sum_{n=1}^{\infty} \frac{1}{n} \int_{0}^{\frac{\pi}{6}} u \cos (2nu) \, du \\ &= \frac{2 \pi}{3} \sum_{n=1}^{\infty} \frac{\sin (\frac{n \pi}{3})}{n^{2}} + 2 \sum_{n=1}^{\infty} \frac{\cos (\frac{n \pi}{3})}{n^{3}} – 2 \zeta(3) \\ &= \frac{2 \pi}{3} \Bigg( \frac{\sqrt{3}}{2} \sum_{n=0}^{\infty} \frac{1}{(6n+1)^{2}} + \frac{\sqrt{3}}{2} \sum_{n=0}^{\infty} \frac{1}{(6n+2)^{2}} – \frac{\sqrt{3}}{2} \sum_{n=0}^{\infty} \frac{1}{(6n+4)^{2}} – \frac{\sqrt{3}}{2} \sum_{n=0}^{\infty} \frac{1}{(6n+5)^{2}} \Bigg) \\ &+ \ 2 \Bigg( \frac{1}{2} \sum_{n=0}^{\infty} \frac{1}{(6n+1)^{3}} – \frac{1}{2} \sum_{n=0}^{\infty} \frac{1}{(6n+2)^{3}} – \sum_{n=0}^{\infty} \frac{1}{(6n+3)^{3}} – \frac{1}{2} \sum_{n=0}^{\infty} \frac{1}{(6n+4)^{3}} \\ &+ \frac{1}{2} \sum_{n=0}^{\infty} \frac{1}{(6n+5)^{3}} + \sum_{n=1}^{\infty} \frac{1}{(6n)^{3}} \Bigg) – 2 \zeta(3) \\ &= \frac{\pi \sqrt{3}}{108} \Bigg( \psi_{1} \left(\frac{1}{6} \right) + \psi_{1} \left(\frac{1}{3} \right) – \psi_{1} \left(\frac{2}{3} \right) – \psi_{1}\left(\frac{5}{6} \right) \Bigg) + \frac{1}{432} \Bigg( – \psi_{2} \left(\frac{1}{6} \right) + \psi_{2} \left(\frac{1}{3} \right) \\ &-28 \zeta(3) + \psi_{2} \left(\frac{2}{3} \right) – \psi_{2} \left(\frac{5}{6} \right) + 4 \zeta(3)\Bigg) – 2 \zeta (3) .\end{align}$$
But from here I've been going in circles trying to get the result in the form given on the Wolfram MathWorld site.
EDIT:
Using the duplication formula for the trigamma function (i.e., $ \displaystyle 4 \psi_{1}(2x) = \psi_{1}(x) + \psi_{1} \left(x + \frac{1}{2} \right) $), I get
$$ \begin{align} &\psi_{1} \left(\frac{1}{6} \right) + \psi_{1} \left(\frac{1}{3} \right) – \psi_{1} \left(\frac{2}{3} \right) – \psi_{1}\left(\frac{5}{6} \right) \\ &= 4 \psi_{1} \left(\frac{1}{3} \right) -\psi_{1} \left(\frac{2}{3} \right) + \psi_{1} \left(\frac{1}{3} \right) -\psi_{1} \left(\frac{2}{3} \right) – 4 \psi_{1} \left(\frac{2}{3} \right) + \psi_{1} \left(\frac{1}{3} \right) \\ &= 6 \psi_{1} \left(\frac{1}{3} \right) – 6 \psi_{1} \left(\frac{2}{3} \right). \end{align}$$
Therefore,
$$ \begin{align} \sum_{n=1}^{\infty} \frac{1}{n^{3} \binom{2n}{n}} &= \frac{\sqrt{3} \pi}{18} \Bigg( \psi_{1} \left(\frac{1}{3} \right) – \psi_{1} \left(\frac{2}{3} \right) \Bigg) +
\frac{1}{432} \Bigg( – \psi_{2} \left(\frac{1}{6} \right) + \psi_{2} \left(\frac{1}{3} \right) \\ &+ \psi_{2} \left(\frac{2}{3} \right) – \psi_{2} \left(\frac{5}{6} \right) – 2 \zeta(3)\Bigg) – 2 \zeta (3). \end{align}$$
So it comes down to somehow showing that $$-\psi_{2} \left(\frac{1}{6} \right) + \psi_{2} \left(\frac{1}{3} \right) + \psi_{2} \left(\frac{2}{3} \right) – \psi_{2} \left(\frac{5}{6} \right) = 312 \zeta(3) .$$
SECOND EDIT:
Using the duplication formula for $\psi_{2}(x)$, I get
$$ \begin{align} &-\psi_{2} \left(\frac{1}{6} \right) + \psi_{2} \left(\frac{1}{3} \right) + \psi_{2} \left(\frac{2}{3} \right) – \psi_{2} \left(\frac{5}{6} \right) \\ &= -8 \psi_{2} \left(\frac{1}{3} \right) +\psi_{2} \left(\frac{2}{3} \right) + \psi_{2} \left(\frac{1}{3} \right) +\psi_{2} \left(\frac{2}{3} \right) – 8 \psi_{2} \left(\frac{2}{3} \right) + \psi_{2} \left(\frac{1}{3} \right) \\ &= -6 \psi_{2} \left(\frac{1}{3} \right) – 6 \psi_{2} \left(\frac{2}{3} \right) . \end{align}$$
And then using the triplication forumula , I get
$$ \psi_{2} \left(\frac{1}{3} \right) + \psi_{2} \left( \frac{2}{3} \right) + \psi_{2}(1) = 27 \psi_{2} (1) .$$
Therefore,
$$ -6 \psi_{2} \left(\frac{1}{3} \right) – 6 \psi_{2} \left(\frac{2}{3} \right) = -156 \psi_{2} (1) = 312 \zeta(3) .$$
Best Answer
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 $$ \begin{align} \sum_{n=1}^{\infty} \frac{1}{n^{3} \binom{2n}{n}} &=\frac{2 \pi}{3}\operatorname{Cl}_2\left(\frac{\pi}{3}\right)+2\operatorname{Cl}_3\left(\frac{\pi}{3}\right)-2 \zeta(3)\\ &=\frac{2 \pi}{3}\frac{\sqrt 3}{12}\left[\psi_1\left(\frac{1}{3}\right)-\psi_1\left(\frac{2}{3}\right)\right]+2\frac{1}{3}\zeta(3)-2\zeta(3)\\ &=\frac{\pi\sqrt 3}{18}\left[\psi_1\left(\frac{1}{3}\right)-\psi_1\left(\frac{2}{3}\right)\right]-\frac{4}{3}\zeta(3). \end{align} $$