What's wrong with $cos(t) /t$ and $e^{-t^2}$?

Question

Usually whenever these functions pop up in the computation of a proper integral, one is stucked finding their antiderivative.

Has it been proved that their antiderivative has no closed form in terms of a polynomial in $sin, cos, tan, exp, ln,... $?

EDIT: the question is indeed a duplicate.

For French speakers, this issue is completely assessed here 1995 Competitive exam