$\int \frac{{\sin }^{4}x}{{\sin }^{4}x+{\cos }^{4}x}dx=$

$\int \frac{{\sin }^{4}x}{{\sin }^{4}x+{\cos }^{4}x}dx$

$=\frac{1}{2}\int \frac{\left({\sin }^{4}x+{\cos }^{4}x\right)+\left({\sin }^{4}x-{\cos }^{4}x\right)}{{\sin }^{4}x+{\cos }^{4}x}dx$

$\left.=\frac{1}{2}\int \frac{\left({\sin }^{4}x+{\cos }^{4}x\right)+\left({\sin }^{2}x-{\cos }^{2}x\right)}{{\sin }^{4}x+{\cos }^{4}x}\right]dx$

$=\frac{1}{2}\int \left(1-\frac{\cos 2x}{1-2{\sin }^{2}x{\cos }^{2}x}\right)dx$

$=\frac{1}{2}\left[x-\int \frac{\cos 2x}{1-\frac{1}{2}{\sin }^{2}2x}dx\right]$

$$\begin{gathered} =\frac{1}{2}\left[x-\int \frac{2\cos 2x}{2-{\sin }^{2}2x}dx\right. \\ put \sin 2x=t\Rightarrow 2\cos 2x dx=dt \end{gathered}$$

$=\frac{1}{2}\left[x-\int \frac{dt}{2-t^2}\right]$

$=\frac{1}{2}\left[x+\int \frac{dt}{t^2-(\sqrt{2}{)}^2}\right]$

$=\frac{1}{2}\left[x+\frac{1}{2\sqrt{2}}\log \left|\frac{\sin 2x-\sqrt{2}}{\sin 2x+\sqrt{2}}\right|\right]+c$

$=\frac{1}{2}\left[x-\frac{1}{2\sqrt{2}}\log \left|\frac{\sqrt{2}+\sin 2x}{\sqrt{2}-\sin 2x}\right|\right]+C$