$\int \frac{\sin 2t}{{\sin }^{4}t+{\cos }^{4}t}dt\text{ is equal to }$

$\text{ Let }I=\int \frac{\sin 2t}{{\sin }^{4}t+{\cos }^{4}t}dt=\int \frac{2\sin t\cos t}{{\cos }^{4}t\left[{\tan }^{4}t+1\right]}dt$

$I=\int \frac{2\left(\frac{\sin t}{\cos t}\right){\sec }^{2}tdt}{\left[{\left(\tan t\right)}^4+1\right]}$

$\text{ Put }{\tan }^{2}t=x\Rightarrow 2\tan t{\sec }^{2}tdt=dx$

$\begin{array}{l}I=\int \frac{dx}{x^2+1}\\ ={\tan }^{-1}\left(x\right)+C\\ ={\tan }^{-1}\left({\tan }^{2}t\right)+C\end{array}$