I'm interested in computing eta invariants of Dirac operators (on spinor bundles tensored with some vector bundles) on the total space of $S^{2n}$ bundles over odd-dimensional manifolds. I found the papers of [Bismut-Cheeger](https://www.jstor.org/stable/1990912) and [Dai](https://www.jstor.org/stable/2939276) where a formula is given in terms of Bismut-Cheeger eta forms, but I'm having a hard time actually computing them. 

Are there places where they are computed explicitly for some $S^{2n}$-bundles? [This MO answer](https://mathoverflow.net/a/237960/) gives various explicit computations of eta invariants but I didn't find $S^{2n}$ bundles there.