Hi Tumeni,
Great question. Computing space craft trajectories mainly requires the 2 body problem, as the sun is usually not relevant (of course, when we launch satellites to make measurements of the sun then it is).
What makes space travel so complicated (among many, many other reasons) is the level of precision we need to solve the equations. A rough estimate simply will not do! We need to have the craft flight plans computed very carefully, otherwise we will not achieve lunar orbit (for example), or we’ll skip that craft right off the atmosphere on its return trip (another example).
What is impressive to me is that NASA accomplished this before modern computers. The way this managed this was to have rooms full of “human computers” that calculated the pieces of it as a full-time job!