In this paper, we consider a time-slotted cognitive radio (CR) setting with buffered and energy harvesting primary and CR users. At the beginning of each time slot, the CR user probabilistically chooses the spectrum sensing duration from a predefined set. If the primary user (PU) is sensed to be inactive, the CR user accesses the channel immediately. The CR user optimizes the sensing duration probabilities in order to maximize its mean data service rate with constraints on the stability of the primary and cognitive queues. The optimization problem is split into two subproblems. The first is a linear-fractional program, and the other is a linear program. Both subproblems can be solved efficiently.