This paper deals with the problem of estimating expectations of sums of additive functionals under the joint smoothing distribution in general hidden Markov models. Computing such expectations is a key ingredient in any kind of expectation-maximization-based parameter inference in models of this sort. The paper presents a computationally efficient algorithm for online estimation of these expectations in a forward manner. The proposed algorithm has a linear computational complexity in the number of particles and does not require old particles and weights to be stored during the computations. The algorithm avoids completely the well-known particle path degeneracy problem of the standard forward smoother. This makes it highly applicable within the framework of online expectation-maximization methods. The simulations show that the proposed algorithm provides the same precision as existing algorithms at a considerably lower computational cost.