【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.