Answer
$$\int \frac{sin2x}{1+cos^{4}x}dx=-arctan(cos^{2}x)+C$$
Work Step by Step
Observe that:
$$sin2x=2sinxcosx={(-cos^{2}x)}'$$
Hence:
$$\int \frac{sin2x}{1+cos^{4}x}dx=-\int \frac{1}{1+(cos^{2}x)^{2}}d(cos^{2}x)$$
$$=-arctan(cos^{2}x)+C$$