Originally Posted by
HallsofIvy Since the numerator has higher degree than the denominator, start by dividing: $\displaystyle \frac{x^4}{x^2+ 1}= x^2- 1- \frac{1}{x^2+ 1}$. The first two should be easy and $\displaystyle \int \frac{1}{x^2+ 1} dx= arctan(x)+ C$