堪える: こたえる vs. こらえる vs. たえる
First of all, the trio of words can easily be divided into two groups for their meanings and that would be:
Group 1: こたえる
Group 2: こらえる and たえる
The meaning of こたえる is quite different from those of the other two. It means "to get to" in the sense of annoying a person physically and/or psychologically. You can say:
「インフルはマジでこたえた。」 = "The flu really got to me."
It is naturally Group 2 that is more complex as the two words are "similar" in meaning.
「こらえる」 is transitive -- 「～～をこらえる」, and
「たえる」 is intransitive -- 「～～にたえる」.
Next, the meanings in my own words (as the bilingual dictionary definitions would be of little help).
「こらえる」is most often used to mean "to control or suppress the outburst of one's own strong emotions".
「たえる」 is most often used to mean "to persevere through the adversity/pressure coming from outside that could affect you in a negative way".
In real life, however, the two words are occasionally used interchangeably by many of us native speakers, to be completely honest.