Compute $\sum_{n=1}^{\infty} \frac{ H_{n/2}}{(2n+1)^3}$ How to prove that 

$$S=\displaystyle \sum_{n=1}^{\infty} \frac{ H_{n/2}}{(2n+1)^3} \quad=\quad \frac{\pi^2G}{4}-\frac{21\zeta(3)\ln(2)}{8}+\frac{\pi^4}{64}+\frac{\Psi^{(3)}(\frac{1}{4})}{512}- \frac{\Psi^{(3)}(\frac{3}{4})}
{512}$$

This problem was proposed by @Ahmad Bow but unfortunately it was closed as off-topic and you can find it here.

Any way, I tried hard on this one but no success yet. here is what I did:
Using the identity  
$$H_{n/2}=H_n-n\int_0^1 x^{n-1}\ln(1+x)\ dx, \quad x\mapsto x^2$$
$$H_{n/2}=H_n-2n\int_0^1 x^{2n-1}\ln(1+x^2)\ dx$$
We can write 
$$S=\sum_{n=0}^\infty\frac{H_n}{(2n+1)^3}-\int_0^1\frac{\ln(1+x^2)}{x}\sum_{n=0}^\infty \frac{2nx^{2n}}{(2n+1)^3}\ dx$$
where
\begin{align}
\sum_{n=0}^\infty \frac{2nx^{2n}}{(2n+1)^3}&=\frac1x\sum_{n=0}^\infty \frac{x^{2n+1}}{(2n+1)^2}-\frac1x\sum_{n=0}^\infty \frac{x^{2n+1}}{(2n+1)^3}\\
&=\frac1{2x}\sum_{n=0}^\infty \frac{x^{n+1}}{(n+1)^2}(1+(-1)^n-\frac1{2x}\sum_{n=0}^\infty \frac{x^{n+1}}{(n+1)^3}(1+(-1)^n\\
&=\frac1{2x}\sum_{n=1}^\infty \frac{x^{n}}{n^2}(1-(-1)^n-\frac1{2x}\sum_{n=1}^\infty \frac{x^{n}}{n^3}(1-(-1)^n\\
&=\frac1{2x}\left(\operatorname{Li}_2(x)-\operatorname{Li}_2(-x)-\operatorname{Li}_3(x)+\operatorname{Li}_3(-x)\right)
\end{align}
Therefore 
$$S=\sum_{n=0}^\infty\frac{H_n}{(2n+1)^3}-\frac12\int_0^1\frac{\ln(1+x^2)}{x^2}\left(\operatorname{Li}_2(x)-\operatorname{Li}_2(-x)-\operatorname{Li}_3(x)+\operatorname{Li}_3(-x)\right)\ dx$$
The sum can be done using the following identity
$$ \sum_{n=1}^{\infty} \frac{H_{n}}{ (n+a)^{2}}= \left(\gamma + \psi(a) \right) \psi_{1}(a) - \frac{\psi_{2}(a)}{2} \, , \quad a >0.$$
Differentiate both sides with respect to $a$ then set $a=1/2$ we get 
$$\sum_{n=0}^\infty\frac{H_n}{(2n+1)^3}=\frac{45}{32}\zeta(4)-\frac74\ln2\zeta(3)$$

and the question here is how to calculate the the remaining integral or a different way to tackle the sum $S$ ? Thanks


Another proof besides Cornel's one:
Let
\begin{align}
S&=\sum_{n=0}^\infty(-1)^n(H_{n/2}-H_n+\ln2)e^{ix(2n+1)}\\
&=\sum_{n=0}^\infty(-1)^n\int_0^1\frac{y^n \ dy}{1+y}e^{ix(2n+1)}\\
&=\int_0^1\frac{e^{ix}\ dy}{1+y}\sum_{n=0}^\infty\left(-ye^{2ix}\right)^n\\
&=\int_0^1\frac{e^{ix}\ dy}{(1+y)(1+e^{2ix}y)}\\
&=\frac{e^{ix}}{1-e^{2ix}}\left(\ln2-\ln(1+e^{2ix})\right)\\
&=\frac{i}{2\sin x}\left(-\ln(\cos x)-ix\right)\\
&=\frac{x}{2\sin x}-i\frac{\ln(\cos x)}{2\sin x}
\end{align}
substituting $e^{ix(2n+1)}=\cos(x(2n+1))+i\sin(x(2n+1))$ and comparing the real and imaginary parts we get:
$$\sum_{n=0}^\infty(-1)^n(H_{n/2}-H_n+\ln2)\cos(x(2n+1))=\frac{x}{2\sin x}\tag1$$
$$\sum_{n=0}^\infty(-1)^n(H_{n/2}-H_n+\ln2)\sin(x(2n+1))=-\frac{\ln(\cos x)}{2\sin x}\tag2$$

If we follow the same approach, we get 
$$\sum_{n=0}^\infty(H_{n/2}-H_n+\ln2)\cos(x(2n+1))=-\frac{\ln(\sin x)}{2\cos x}\tag3$$
$$\sum_{n=0}^\infty(H_{n/2}-H_n+\ln2)\sin(x(2n+1))=\frac{\pi/2-x}{2\cos x}\tag4$$

The identity in (1) is discovered by @Ahmad Bow.
