Integrate $\int_{0}^{\pi/2}x\sqrt{\tan{x}}\log{(\sin{x})}\,\mathrm dx$ When trying to solve this problem: How to Integrate $ \int^{\pi/2}_{0} x \ln(\cos x) \sqrt{\tan x}\,dx$
I found his sister integral has an interesting closed form provided my calculation is correct. I use an ugly series to find it. Can you use other methods to evaluate the integral? Such as Gamma function or residue method?

$$\int_{0}^{\pi/2}x\sqrt{\tan{x}}\log{\sin{x}}\,\mathrm dx=-\frac{\pi\sqrt{2}}{48}\big(\pi^2+12\pi \log{2}+24\log^2{2}\big) $$


For first:
$$\int_{0}^{\pi/2}x\sqrt{\tan x}\log\sin x\,dx = \frac{1}{2}\int_{0}^{+\infty}\frac{\sqrt{t}\arctan t\log\frac{t^2}{1+t^2}}{1+t^2}\,dt. $$
Now differentiation under the integral plus the residue theorem give:
$$\begin{eqnarray*}\int_{0}^{+\infty}\frac{t^\alpha \arctan t}{1+t^2}\,dt &=&\int_{0}^{1}\int_{0}^{+\infty}\frac{t^{\alpha+1}}{(1+t^2)(1+\beta^2 t^2)}\,dt\,d\beta\\&=&\frac{\pi}{2\sin\left(\frac{\pi\alpha}{2}\right)}\int_{0}^{1}\frac{1-\beta^{-\alpha}}{\beta^2-1}d\beta\\&=&-\frac{\pi}{2\sin\left(\frac{\pi\alpha}{2}\right)}\left(\log 2+\frac{1}{2} H_{-\frac{\alpha+1}{2}}\right)\end{eqnarray*}$$
so $\int_{0}^{+\infty}\frac{\sqrt{t}\arctan t\log t}{1+t^2}\,dt$, that is the derivative of the previous expression at $\alpha=\frac{1}{2}$, just depends on the values of $\psi(z)$ and $\psi'(z)$ at $z=\frac{1}{4}$. Luckily, they are both not so difficult to compute:
$$ \psi\left(\frac{1}{4}\right)=-\frac{\pi}{2}-3\log 2-\gamma,\qquad \psi'\left(\frac{1}{4}\right)= \pi^2+8K,$$
where $K$ is the Catalan constant. 
The other piece, $\int_{0}^{+\infty}\frac{\sqrt{t}\arctan t\log(1+t^2)}{1+t^2}\,dt$, can be computed in a similar way.
It is worth mentioning that this approach applies to the other question, too.
