A reachable set computation method for dynamical systems with an integral constraint over the input set is proposed. These models are typical in robustness analysis when studying the impact of bounded energy noises over a system response and can also model a large family of complex systems. The reachable set is over-approximated using a guaranteed set-based integration method within the interval arithmetic framework. A Runge-Kutta guaranteed integration scheme with pessimistic bounds over the input provides a first conservative bound over the reachable tube. Then, the integral constraint is used to define a contractor over the reachable tube. This contractor and a propagation step are successively applied on the over-approximation until a fixed point is reached. We evaluated our algorithm with DynIbex library to simulate a delayed system, i.e., an infinite dimensional system that can be modeled as a linear time-invariant system subject to an integral quadratic constraint. Our approach is shown to be tractable and enables the use of interval arithmetic and guaranteed integration for a richer set of dynamical systems.