how do you find the integral of sec^2 [ (e^2x) + 1]? Does a solution exists?

Answer 1

The solution involves so called "Hypergeometric functions"
So a solution in elementary terms does not exist

Answer 2

If you let u = (e^2x) + 1 , then du = 2e^(2x) dx.

since there is not such an expression outside of the secant squared, that expression adds to the degree of difficulty of the question.
the new integral is 1/2(sec^2u)/(u-1) du
unfortunately no further technique will aid in arriving at a closed form for this question.
Integration by parts, and other miscellaneous substitutions will not work now.

A solution does not exist: meaning there is no closed form for the answer. ©