The NASA calculations were state of the art for computation in 1969. Fifty years later, gigantic improvements in computing make it possible to do much larger and more complex calculations.
Our group at UCLA used super-computers to simulate a model of the activation of the heart, in order to understand the condition called ventricular fibrillation, the most frequent cause of sudden cardiac death.
Although this calculation was, literally, billions of times more complex than the orbits at NASA, what the supercomputers were doing is essentially Euler’s method, for approximately 50 million variables and a million time steps.