Demand, claim, require, exact are comparable not as close synonyms but as sharing the basic meaning to ask or call for something as due or as necessary or as strongly desired.
Demand strongly implies peremptoriness or insistency; if the subject is a person or sometimes an expression of his will (as a law), it usually implies that he possesses or believes he possesses the right or the authority not only to issue a peremptory request but also to expect its being regarded as a command.
If the subject of the verb is a thing, the verb implies the call of necessity or of imperative need.
Claim implies a demanding either of the delivery or concession of something due one as one's own, one's right, or one's prerogative or of the admission or recognition of something which one asserts or affirms; thus, one who claims a piece of property demands its delivery to him as his own; one who claims that he has solved a problem demands recognition of the truth of his assertion.
Require is often used interchangeably with demand, but it usually distinctively implies imperativeness such as arises from inner necessity or the compulsion of law or regulation or the exigencies of the situation.
Exact implies not only demanding something but getting what one demands.