The following [equality](https://www.facebook.com/photo.php?fbid=3215866335176809&set=a.222846247812181&type=3&theater)

$$\sum\_{n=1}^\infty(-1)^{n-1}\frac{H\_n^2}{2n+1}=\frac{3}{16}\pi^3-\frac34\ln^2(2)\pi-8\Im\left\{\text{Li}\_3\left(\frac{1+i}{2}\right)\right\}$$

can be proved if we are allowed to use [the generating function](https://math.stackexchange.com/q/3366039) ( see Eq$(3)$)

$$\sum\_{n=1}^\infty x^nH\_n^2=\frac{\ln^2(1-x)+\text{Li}\_2(x)}{1-x}$$

But the problem-proposer mentioned that the sum to be calculated without using this generating function.

I have no clue how to approach it with such restriction. any idea ? Thanks in advance.