Question:

Let $I = \int \frac{1+x^2}{1+x^4} dx = \int \frac{1+\frac{1}{x^2}}{x^2+\frac{1}{x^2}} dx$<br>Put $x-\frac{1}{x}=t \Rightarrow x^2+\frac{1}{x^2}=t^2+2$ and $\left(1+\frac{1}{x^2}\right)dx=dt$<br>$\therefore I = \int \frac{dt}{t^2+2} = \frac{1}{\sqrt{2}}\tan^{-1}\left(\frac{t}{\sqrt{2}}\right)+C$<br>$= \frac{1}{\sqrt{2}}\tan^{-1}\left(\frac{x^2-1}{x\sqrt{2}}\right)+C$