<s>[INST] 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...
 [/INST] $$
\begin{aligned}
\int \frac{1}{\sin ^4 x+\cos ^4 x} d x = & \int \frac{1}{1-\frac{\sin ^2 2 x}{2}} d x \\
= & 2 \int \frac{1}{2-\sin ^2 2 x} d x \\
= & \int \frac{\sec ^2 2 x}{2 \sec ^2 2 x-\tan ^2 2 x} d(2 x) \\
= & \int \frac{d\left(\tan ^2 x\right)}{2\left(1+\tan ^2 2 x\right)-\tan ^2 2 x}\\=& \int \frac{d(\tan 2 x)}{2+\tan ^2 2 x}
\\=&\int \frac{d(\tan 2 x)}{(\sqrt{2})^2+(\tan 2 x)^2}\\=&\frac{1}{\sqrt{2}}\arctan \left(\frac{\tan 2 x}{\sqrt{2}}\right)+C
\end{aligned}
$$
</s>