[Math] Closed form solution to $\int_0^1\arctan^2(x)\,\sqrt{x}\,dx$

Question

I need to compute this integral:
$$\int_0^1\arctan^2(x)\,\sqrt{x}\,dx$$
I tried integration by parts, and also introducing a parameter $\arctan(a\,x)$ and differentiation wrt it, but these approaches did not lead to anything useful. Please help.

Answer 1

$$\frac{\pi^2}{24}-\frac{2\pi}3+\frac1{36\sqrt{2}}\left[5\pi^2+12\left(4+\ln\left(\frac{1+\sqrt2}2\right)\right)\left(\pi-2\ln\left(1+\sqrt2\right)\right)-48\operatorname{Li}_2\left(\sqrt2-1\right)\right]$$