  We prove a local higher integrability result for the spatial gradient of weak
solutions to doubly nonlinear parabolic systems whose prototype is $$
\partial_t \left(|u|^{q-1}u \right) -\operatorname{div} \left( |Du|^{p-2} Du
\right) = \operatorname{div} \left( |F|^{p-2} F \right) \quad \text{ in }
\Omega_T := \Omega \times (0,T) $$ with parameters $p>1$ and $q>0$ and
$\Omega\subset\mathbb{R}^n$. In this paper, we are concerned with the ranges
$q>1$ and $p>\frac{n(q+1)}{n+q+1}$. A key ingredient in the proof is an
intrinsic geometry that takes both the solution $u$ and its spatial gradient
$Du$ into account.
