All forms of integrate possible variants of integrate(exp(-x**2)) is extracted from the differentiation of diff(exp(-x**2)) // Per Lindholm 2018
I think this applies to any function. So from the differentiation and compare I think you could figure out what functions to use.
From this I see that
-60480*x**2*exp(-x**2) + 30240*exp(-x**2)
is a possible integrate possible variant of exp(-x**2). I wonder if you can iterate the first part of the function. Could the integration solution succesfullness be like a function that has only one solution. So the two parts belong together in a higher equation?