$\int\limits \frac{1}{sin^4x+cos^4x}dx=\int\limits \frac{2}{2-sin^{2}2x}dx$ $=\int\limits \frac{2}{sin^{2}2x+2cos^{2}2x}dx$
$=\int\limits \frac{1}{tan^{2}2x+2}.\frac{2}{cos^{2}2x}dx$
$=\int\limits \frac{1}{tan^{2}2x+2}d(tan2x)$
$=\frac{1}{\sqrt{2}}arctan(\frac{tan2x}{\sqrt{2}})+C$