$\int \frac{{\sin }^{8}x-{\cos }^{8}x}{1-2{\sin }^{2}x{\cos }^{2}x}dx$

 $\int \frac{{\sin }^{8}x-{\cos }^{8}x}{1-2{\sin }^{2}x{\cos }^{2}x}dx$
use ${\sin }^{8}x-{\cos }^{8}x=({\sin }^{4}x-{\cos }^{4}x)({\sin }^{4}x+{\cos }^{4}x)$   
$=({\sin }^{4}x-{\cos }^{4}x)(1-2{\sin }^{2}x{\cos }^{2}x)$
${\cos }^{4}x-{\sin }^{4}x=({\cos }^{2}x-{\sin }^{2}x)({\cos }^{2}x+{\sin }^{2}x)=\cos 2x$