"At a loose end" refers to a state of idleness, boredom, and having nothing to do. The antonyms for this phrase that denote opposite meanings include being occupied, engaged in activity, or being productive. A person who is not at a loose end is busy, involved, or committed to a task or project. They are not wasting time or feeling directionless. Alternatively, they may be relaxed, but actively engaged in leisure activities or work that brings them satisfaction. Therefore, those antonyms can be interpreted as diligent, productive, active, or purposeful.