$\int \frac{x^4+1}{x^6+1}dx$ is equal to

Solution:

Let

$\begin{array}{l}I=\int \frac{x^4+1}{x^6+1}dx=\int \frac{\left(x^4-x^2+1\right)+x^2}{x^6+1}dx\\ \Rightarrow I=\int \frac{x^4-x^2+1}{{\left(x^2\right)}^3+1}dx+\int \frac{x^2}{x^6+1}dx\\ \Rightarrow I=\int \frac{1}{x^2+1}dx+\frac{1}{3}\int \frac{1}{{\left(x^3\right)}^2+1}d\left(x^3\right)\\ \Rightarrow I={\tan }^{-1}x+\frac{1}{3}{\tan }^{-1}\left(x^3\right)+C\end{array}$