Calculate in closed form: $\int_{0}^{\pi/2}\arctan\left(\frac{1}{\sqrt{2\sin x}}\right)dx$

Question

I did the replacement, $u = \sqrt{2\sin x}$, but I did not succeed. $u = \sqrt{2\sin x}$. I found, $$ \int_{0}^{\pi/2}\arctan(x)\cot(x)\,dx, \quad \int_{0}^{\pi/2}\arctan(\sin x)\,dx $$ But the techniques used in these integrals did not help much.

Thank you for any help.