Evaluate: $$\int_0^{\frac{\pi}{2}} \frac{\arctan{\left(\frac{2\sin{x}}{2\cos{x}-1}\right)}\sin{\left(\frac{x}{2}\right)}}{\sqrt{\cos{x}}} \, \mathrm{d}x$$
I believe there is a "nice" closed form solution but Wolfram is too weak. These arctan integrals are so tricky! I sense a substitution like $\sin{\frac{x}{2}}$ because of arctan argument and $\sqrt{\cos{x}}$ but I just cant get it. Any ideas or tips please.
Source: https://tieba.baidu.com/p/4794735082 (Exercise 3.1.22).