Intent, engrossed, absorbed, rapt mean having one’s mind or attention deeply fixed on something.
Intent implies that one’s mind, one’s desires, or one’s energies are eagerly bent on something; it therefore suggests a directing of the entire attention toward a particular end or thing.
Engrossed implies monopolization of one’s attention either by a driving purpose or emotion or an eager interest or by the force or urgency of circumstances beyond one’s control.
Absorbed often differs little from engrossed in this sense, but it may carry a stronger suggestion of the power of the thing on which the attention is fixed to capture one’s attention and to hold it firmly so that there is difficulty in distracting it.
Rapt implies both extreme intentness and complete absorption, as though one were taken out of oneself or were in an ecstatic trance.