How to evaluate $\int \frac { \sin x+\cos x }{ \sin^4 x+\cos^4x}\, dx$? How can one find $$\int  \frac { \sin x+\cos x }{ \sin^4 x+\cos^4x}\, dx?$$

Split the integral as
$$
\int\frac {\cos x}{\sin^4 x + (1 - \sin^2 x)^2} dx 
+ \int \frac{\sin x}{(1 - \cos^2 x)^2 + \cos^4 x} dx
$$
Compute the integrals with the substitutions $u = \sin x$ and $u = \cos x$ respectively.
