$\int \frac{\tan x}{\sqrt{{\sin }^{4}x+{\cos }^{4}x}}dx$ is equal to

 

Solution:

Let

$\begin{array}{l}I=\int \frac{\tan x}{\sqrt{{\sin }^{4}x+{\cos }^{4}x}}dx\\ =l=\int \frac{\tan x{\sec }^{2}x}{\sqrt{{\tan }^{4}x+1}}dx=\frac{1}{2}\int \frac{1}{\sqrt{{\left({\tan }^{2}x\right)}^2+1}}d\left({\tan }^{2}x\right)\\ \Rightarrow I=\frac{1}{2}{\log }_e\left\{{\tan }^{2}x+\sqrt{1+{\tan }^{4}x}+C\right.\end{array}$