How to find the integration of $\sin2\theta\sqrt{\tan\theta} \ d\theta$

Question

I am trying to find the area between the line with slope $=.5$ and a circle with center $(5,5)$ and radius $=5$ but using the polar integration I changed the circle equation to polar form you can see here

I had difficulty integrating both
$$\int\sin2\theta\sqrt{\tan\theta} \ d\theta$$

And

$$\int\sin2\theta\sqrt{cot\theta} \ d\theta$$

Answer 1

$$I=\int \sin 2\theta \sqrt{\tan \theta}d\theta=4\int \frac{t^4}{1+t^4}dt $$ where $t^2=\tan \theta$