The Hulse-Taylor binary provides possibly the best test of GR to date. We find the modified quadrupole formula for infinite derivative gravity (IDG). We investigate the backreaction formula for propagation of gravitational waves, found previously for effective quantum gravity (EQG) for a flat background and extend this calculation to a de Sitter background for both EQG and IDG. We put tighter constraints on EQG using new LIGO data. We also find the power emitted by a binary system within the IDG framework for both circular and elliptical orbits and use the example of the Hulse-Taylor binary to show that IDG is consistent with GR.