When used with no article, and in its major uses (subject/object), it can normally only be equivalent to either a nominative or an accusative case; used with the article, it may be in any case (nominative, genitive, dative and accusative).

Complementary Infinitive: These infinitives are often described as completing the meaning of verbs of ability, desire, intention, will, and the like.

Articular:  form called the GERUND. Like infinitives, gerunds function as nouns, including serving as subjects or objects of a verb, or as objects of a preposition. English adds –ing to a verb to form this part of speech.

Indirect Statement: The construction used depends upon the verb of mental activity– saying, thinking, perceiving – that introduces the indirect statement.

Dynamic infinitive: The dynamic infinitive may be governed by verbs of will or desire to do something (ἐθέλω or βούλομαι "to be willing, wish to", εὔχομαι, etc. and verbs or verbal expressions denoting ability, fitness, necessity, capacity, etc. (δύναμαι, etc.)

Until/Before"The conjunction πρίν means until or before. To distinguish between the two meanings, Greek uses two different constructions. "Until" takss the finite verb. "Before" takes the infinitive.

Result Clause with ὥστε: A result clause indicates the result of the action of the main clause. If the clause shows the actual result, it takes a finite verb. If the clause shows an INTENDED/EXPECTED/PROBABLE/NATURAL result, it takes an infinitive.

SPEAKING: After a verb of speaking it can be translated either as an active verb ("be" or "am") and an infinite, "to be."

Infinitive Clause Noun Forms: These are those that have a direct article before the infinitive.  Usually, the Greek SUBJECT of the infinitive is rendered in the ACCUSATIVE case. The object is also accusative. If the infinitive has the SAME SUBJECT as that of the main verb it can leave it out or  the subject of the infinitive, or renders it in the NOMINATIVE for emphasis. The infinitive verb usually comes last.