Which rational functions over $\mathbb{R}$ have rational functions as antiderivatives?
I can think of a few trivial examples. Polynomials over $\mathbb{R}$ are rational functions and of course have rational functions as antiderivatives.
If $n \geq 2$ and $a \mathbb{R}$, then $r(x) =\frac{a}{x^n}$ is a rational function with an antiderivative $$ \int r(x) dx = - \frac{a}{(n-1)x^{n -1}} $$ which is a rational function.
I can also construct examples like $$ \frac{1}{x^2} + \frac{1}{(x + 1)^2} = \frac{2x^2 + 2x + 1}{x^4 + 2x^3 + x^2} $$ The left hand side clearly has a rational antiderivative, and the right hand side loks like a nontrivial rational function. Essentially I've just added two different functions like $r(x)$ above.
To construct even more examples, one can simply differentiate rational functions. For example, $\frac{2x}{(x^2 + 1)^2}$ has a rational function antiderivative. Importantly, this example shows that we can't simply identify which rational functions will have rational function antiderivatives by looking at the partial fraction decomposition.
With these two constructions in hand (the results of which can themselves be added together to yield still more examples), it seems the set of rational functions with rational function antiderivatives is quite diverse. Is there a general way to determine whether a rational function will have a rational function antiderivative?