Let $f: X \to B$ be a smooth fiber bundle with fiber $F$, and assume that $B$ is simply connected. Then higher direct images $R^p f_*(\mathbb R_X)$ are naturally isomorphic to the constant sheaves with fiber $H^p(F, \mathbb R)$.
I would like to consider also total derived functors, where leads to the question: is there a nice quasi-isomorphism representative of the total direct image $Rf_*(\mathbb R_X)\in D^+(Sh(B))$? Can we somehow explicity compute if we know enough information about $f$?
I think, I can prove that in general, $Rf_* (\mathbb R_X)$ might not be quasi-isomorphic to a complex of constant sheaves, because any such complex is also quasi-isomorphic to its cohomology, which is not true for total direct image in general. This (in a way) corresponds to Leray-Serre spectral sequence having non-trivial differentials.