On $\int_{-\pi/2}^{\pi/2}\operatorname{Li}_3(\sin x)dx$ and its derivative

If $n$ is odd the integral $\int_{-\pi/2}^{\pi/2}(\sin\theta)^{n}\,d\theta$ equals zero, so the computation of $\int_{-\pi/2}^{\pi/2}\text{Li}_3(\sin\theta)\,d\theta$ boils down to the computation of $$ \sum_{n\geq 1}\left[\frac{1}{4^n}\binom{2n}{n}\right]\frac{1}{n^3}=\frac{1}{2}\int_{0}^{1}\sum_{n\geq 1}\left[\frac{1}{4^n}\binom{2n}{n}\right]x^{n-1}\log^2(x)\,dx $$ or (by integration by parts) $$\int_{0}^{1}\frac{\log^3(x)}{(1-x)^{3/2}}\,dx.$$ This can be easily tackled through the partial derivatives of the Beta function.