A curious relation between the flat cosmological model and the elliptic integral of the first kind

Context. The dependence of the luminosity distance on the redshift has a key importance in the cosmology. This dependence can well be given by standard functions for the zero cosmological constant. Aims. The purpose of this article is to present a similar relation also for the non-zero cosmological constant, if the universe is spatially flat. Methods. A definite integral was used. Results. The integration ends in the elliptic integral of the first kind. Conclusions. The result shows that no numerical integration is needed for the non-zero cosmological constant if the universe is spatially flat