I do not know what you imply by the terms "energy efficient" and "energy densed."
In terms of the amount of energy that nutrients contain, there are three nutrients that can provide energy. These are carbohydrates, proteins and fats.
Carbohydrates contain the least amount of energy at 3.75 kilocalories per gram, proteins have a higher amount of energy at 4 kilocalories per gram and fats have the maximum at 9 kilocalories per gram.
The body, though, is able to extract energy from carbohydrates in the easiest manner as compared to proteins and fats. This is the reason why carbohydrates are first used as a source of energy and only when they get exhausted does the body move to extracting energy from proteins and fats.
No, fats. It contains more energy than carbohydrates/proteins when burnt by the body.