A new analytical solution for the distance modulus in flat cosmology

A new analytical solution for the luminosity distance in flat $\mathrm{\Lambda }$CDM cosmology is derived in terms of elliptical integrals of first kind with real argument. The consequent derivation of the distance modulus allows evaluating the Hubble constant, $H_0=69.77\pm 0.33$, $\text{om}=0.295\pm 0.008$ and the cosmological constant, $\mathrm{\Lambda }=(1.194\pm 0.017){10}^{-52}\frac{1}{m^2}$.