Rationally slice knots in $S^3$ are in abundance due to the collection of Kawauchi's theorems and the recent result of Kim and Wu:

**Theorem ([1] + [2]):** Any hyperbolic amphicheiral knot $K$ in $S^3$ is rationally slice.

**Theorem ([3]):** Any fibered, negative amphicheiral knot $K$ in $S^3$ with irreducible Alexander polynomial (called *Miyazaki knot*) is rationally slice.

Using these theorems, you may find plenty of rationally slice knots that are not smoothly slice.

**[1]:** Kawauchi, Akio. "The invertibility problem on amphicheiral excellent knots." *Proceedings of the Japan Academy, Series A, Mathematical Sciences* 55.10 (1979): 399-402.

**[2]:** Kawauchi, Akio. "Rational-slice knots via strongly negative-amphicheiral knots." *Commun. Math. Res.* 25.2 (2009): 177-192.

**[3]:** Kim, Min Hoon, and Zhongtao Wu. "On rational sliceness of Miyazaki's fibered,− amphicheiral knots." *Bulletin of the London Mathematical Society* 50.3 (2018): 462-476.