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?

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


Substitute $t = \sqrt{\tan x}$
\begin{align}
I=&\int_{0}^{\pi/2}x\sqrt{\tan{x}}\ln({\sin x})\ dx\\
= &\int_0^\infty \frac{t^2 \ln\frac{t^4}{1+t^4}\ \tan^{-1}t^2}{1+t^4}dt\\
=&\int_0^\infty \int_0^1 \frac{2y \ t^4 \ln\frac{t^4} {1+t^4}}{(1+t^4)(1+y^4 t^4)}dy \ dt\\
=&\int_0^1 \frac{2y}{1-y^4} \int_0^\infty\bigg(\frac{\ln\frac{t^4} {1+t^4}}{1+y^2t^4}- \frac{\ln\frac{t^4} {1+t^4}}{1+t^4} \bigg)dt \ dy
\end{align}
Utilize
$$J(a)= \int_0^\infty \frac{a \ln\frac{t^4} {1+t^4}}{1+a^4 t^4}dt
= -\frac\pi{\sqrt2}\left[\tan^{-1}a+\ln(1+a)+\frac12\ln(1+a^2) \right]
$$
to obtain
\begin{align}
I= &\int_0^1 \frac{2 [J(y)-yJ(1)] }{1-y^4}dy \\
=&\ \sqrt2\pi \int_0^1 \frac{\frac\pi4y-\tan^{-1}y}{1-y^4}
+\frac{y\ln2 -\ln(1+y)}{1-y^4}
+\frac{y\ln2 -\ln(1+y^2)}{2(1-y^4)}\ dy\\
 =&\ \sqrt2\pi \bigg[ \left(G-\frac{\pi^2}{16}-\frac\pi4\ln2\right)+\left( \frac{\pi^2}{12} -\frac\pi4\ln2 -\ln^22\right)\\
&\>\>\>\>\>\>\>\>\>\>\>\>\>
+\frac12\left( 2G+ \frac{\pi^2}{8} -\pi\ln2-\ln^22\right)\bigg]\\
=& \ \frac{\pi}{\sqrt2}\left( G+ \frac{\pi^2}{24}-\frac\pi2\ln2
-\frac34\ln^22\right)
\end{align}
