I=\int \frac{2\sqrt{2}\sin ^3(t)}{2-2\sin ^2(t)}\sqrt{2}\cos (t)\text{dt}=2\int \frac{\sin ^2(t)\sin (t)}{1-\sin ^2(t)}\cos (t)\text{dt}=2\int \frac{1-\cos ^2(t)}{\cos (t)}\sin (t)dt=2\int \frac{\sin (t)}{\cos (t)}\text{dt}-\int \sin (2t)dt=-2\ln |\cos (t)|+\frac{\cos (2t)}{2}