I am looking for a rational bijection from the reals to the open unit interval $(0,1)$ in such a way that the inverse is also rational.
The main purpose is to change the interval of integration from an infinite one into a finite one while also keeping the integrand as simple as possible, and the change of variable as simple as possible both back and forth.
So, for example, $\frac{1}{1-x}-\frac{1}{x}$ maps (0,1) to the reals and is rational, but the inverse could be a nicer function.
Being for integration, it would be okay to define the map piece-wise.
For example, using $\frac{x}{1-x}$ and $\frac{-x}{1-x}$ for mapping $[0,1)$ to each half of the line. Then with convenient linear transformations send the two copies of $[0,1)$ to $(0,1/2]$ and to $[1/2,1)$ respectively.
Am I going to be able to find a single rational function to do this job?
No. If $r:\Bbb R\to\Bbb R$ is rational and $r$ has finite limits at $\pm\infty$ then the limit at $\infty$ equals the limit at $-\infty$.