Compute the following integral.

The denominator is already factored, so we write

This gives us the equation

First, we can evaluate at to obtain

Then, we multiply out and equate like powers of ,

Therefore, equating like powers of , we have the following equations

Solving this system we obtain

Therefore, evaluating the integral we have

Hint: We can avoid using partial fractions by noticing $x^4 + 1 = (x^2 + 1)^2 – 2x^2$

edit: x^4 + 1 = (x^2 + 1)^2 – 2x^2