【Anger】 , 【incense】 , 【enrage】 , 【infuriate】 and 【madden】 all these verbs carry in common with 【anger】 , their general term, the denotation to make angry or to rouse to 【anger】 .

• laugh then at any, but at fools or foes; these you but 【anger】 , and you mend not those
  —Pope
• angered by his son’s repeated disobedience

【Incense】 implies hotness of 【anger】 , especially as provoked by something excessively irritating and offensive

• Mr. Critchlow, aged and unaccustomed to interference, had to render accounts of his trusteeship to this young man, and was incensed
  —Bennett
• magistrates and populace were incensed at a refusal of customary marks of courtesy and respect for the laws
  —Inge

【Enrage】 suggests a violent display of wrath or fury.

• I pray you, speak not . . . question enrages him
  —Shak.

【Infuriate】 may imply a sense of being outraged or sometimes no more than of being thoroughly irritated or exasperated.

• how it infuriates a bigot, when he is forced to drag into the light his dark convictions!
  — L. P. Smith
• his colleagues and his subordinates had been alternately delighted and infuriated by his assumed reluctance to deal with any practical question
  —Sackville-West

【Madden】 is often not distinguishable from 【infuriate】 .

• can it be fancied that Deity ever vindictively made in his image a mannikin merely to 【madden】 it?
  —Poe

Like the former it may imply merely excessive annoyance or vexation.

• maddening delays