Enraging Pythagoras 2.0

Calculus Level 2

True or False?

If $\int \frac{e^x}{1+x^2} \ dx = f(x)e^x + C,$ where $C$ is a constant, then $f$ is a rational function. That is, $f(x) = \frac{\mathrm{P}(x)}{\mathrm{Q}(x)},$ where $\mathrm{P}(x)$ and $\mathrm{Q}(x)$ are real polynomials in $x.$

Note: This problem was adapted from a question in the 2016 STEP III exam.