Actually, in high difficulties, the optimal research rate at your rapid expansion phase is nothing else than a fat 0%.
Except having a stable cash income from city occupying, the research rate is set at the maximum rate where your turn income remains positive. For example,
Research rate 40%, turn income +23
Research rate 50%, turn income +10
Research rate 60%, turn income -3
Research rate 70%, turn income -16
Then you set the research rate between 50%-60%.
It certainly takes times to adjust to the game, learning handy skills such as:
1) Take/Found cities that will be useful
2) Make the most useful improvement at the time
3) Swapping harvest tiles based on need
4) Switch civ when most needed
etc.