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

$\frac{1}{2} \log _e\left(\tan ^2 x+\sqrt{1+\tan ^4 x}\right)+C$

Let

$I =\int \frac{\tan x}{\sqrt{\sin ^4 x+\cos ^4 x}} d x$

$\Rightarrow I =\int \frac{\tan x \sec ^2 x}{\sqrt{\tan ^4 x+1}} d x=\frac{1}{2} \int \frac{1}{\sqrt{\left(\tan ^2 x\right)^2+1}} d\left(\tan ^2 x\right)$

$\Rightarrow I =\frac{1}{2} \log _e\left\{\tan ^2 x+\sqrt{1+\tan ^4 x}+C\right\}$