Integrate $ \int \frac{1}{\sin^{4}x+\cos^{4}x}dx $ Show that$$ \int \frac{1}{\sin^{4}(x)+\cos^{4}(x)}dx \ = \frac{1}{\sqrt{2}}\arctan\left(\frac{\tan2x}{\sqrt{2}}\right)+C$$
I have tried using Weierstrass substitution but I can't seem to get to the answer... Should I be using the said method or is there another way I can approach the question? Since the integrand evaluates into an arctangent function I am assuming there is some trickery in the manipulation that can get me there. But I just can't seem to see it...

\begin{align}
\int \frac{dx}{\sin^{4} x+\cos^{4} x}
&= \int \frac{dx}{\frac14(1-\cos2x)^2+\frac14(1+\cos2x)^2}\\
 &= \int \frac{2dx}{1+\cos^22x}=\int\frac{2\sec^22xdx}{2+\tan^22x}\\
&=\frac1{\sqrt2}\int\frac{d(\frac{\tan2x}{\sqrt2})}{1+(\frac{\tan2x}{\sqrt2})^2}=\frac{1}{\sqrt{2}}\arctan\frac{\tan2x}{\sqrt{2}}+C
\end{align}
