You may view my related problem: A definite integral with trigonometric functions: $\int_{0}^{\pi/2} x^{2} \sqrt{\tan x} \sin(2x) \, \mathrm{d}x$
Show that : $$\int_0^{\frac{\pi}{2}}x\sqrt{\tan x}\text{e}^{2ix}\text{d}x=\frac{\pi \text{e}^{\frac{i\pi}{4}}}{8i}(\pi+2+2i\ln 2)$$ $$\int_0^{\frac{\pi}{2}}x^2\sqrt{\tan x}\text{e}^{2ix}\text{d}x=\frac{\pi \text{e}^{\frac{i\pi}{4}}}{48i}(24i+(12+4i)\pi+5\pi^2+i(-4\pi+12(-2+\pi)\ln 2)-12\ln ^22)$$
