come up with (an idea, plan, explanation, theory, or principle) after a mental effort
Excogitate a way to measure the speed of light.
source: WordNet 3.0
reflect deeply on a subject
I mulled over the events of the afternoon. Philosophers have speculated on the question of God for thousands of years. The scientist must stop to observe and start to excogitate.
source: WordNet 3.0
To think out; to find out or discover by thinking; to devise; to contrive.
