On the fourth moment in the Rankin-Selberg problem

If $$\Delta(x) := \sum_{n\le x}c_n - Cx$$ denotes the error term in the classical Rankin-Selberg problem, then it is proved that $$\int_0^X \Delta^4(x)\d x \ll_\epsilon X^{3+\epsilon},\quad \int_0^X \Delta_1^4(x)\d x \ll_\epsilon X^{11/2+\epsilon},$$ where $\Delta_1(x) = \int_0^x\Delta(u) du$. The latter bound is, up to `$\epsilon$', best possible.