This questions is related to my previous one.
I am interested in a explicit evaluation in terms of Euler sums for $$ \int_{0}^{\pi/4}\text{Li}_3(\cos^2\theta)\,d\theta = \frac{1}{2}\int_{0}^{1/2}\frac{\text{Li}_3(1-z)}{\sqrt{z(1-z)}}\,dz.$$
It is not difficult to show that $$ \int_{0}^{\color{red}{1}}\frac{\text{Li}_3(1-z)}{\sqrt{z(1-z)}}\,dz =-\frac{\pi^3}{3}\log(2)+\frac{4\pi}{3}\log^3(2)+2\pi\zeta(3)\tag{A}$$ but I have not managed to make a wise use of the trilogarithm functional identities for computing $$ \int_{0}^{1/2}\frac{\text{Li}_3(z)}{\sqrt{z(1-z)}}\,dz\stackrel{\text{IBP}}{\longrightarrow}\int_{0}^{\pi/4}\theta\cot(\theta)\text{Li}_2(\sin^2\theta)\,d\theta \quad\text{or}\quad\int_{0}^{1/2}\frac{\text{Li}_3(z)-\text{Li}_3(1-z)}{\sqrt{z(1-z)}}\,dz ,$$ which would have solved the problem. One might need substantial extensions of the result about $\mathcal{I}(a,b)$ proved here by nospoon. I am expecting the integral above to be related with Euler sums with (total) weight five. Maybe the shifted-Fourier-Chebyshev expansion of $\text{Li}_3(x)$ over $(0,1)$ is already known in the literature, but I have not been able to find it.