How to elegantly prove that $$\sum\_{n=1}^\infty\frac{\overline{H}\_n-H\_{n/2}}{n^3}=2\text{Li}\_4\left(\frac12\right)-\frac{49}{16}\zeta(4)+\frac72\ln2\zeta(3)-\frac12\ln^22\zeta(2)+\frac1{12}\ln^42$$

where $\overline{H}\_n=\sum\_{k=1}^n\frac{(-1)^{k-1}}{k}$ is the alternating harmonic number, $H\_{n/2}=\int\_0^1\frac{1-x^{n/2}}{1-x}\ dx$ is the harmonic number, $\text{Li}\_r$ is the polylogarithm function and $\zeta$ is the Riemann zeta function. 

What I mean by elegant solutions is solutions involving cancellation of challenging integrals/ sums , symmetry , manipulations and new ideas that save us tedious calculations. However, all solutions are appreciated. 

Thank you