Nontrivial integral representations for $e$
Solution 1:
If you are asking whether e is a period, the question remains, technically, still open, but its answer is not expected to be affirmative. As to the non-algebraic integrands, we have $$\int_{-\infty}^\infty\frac{\cos(ax)}{1+x^2}~dx~=~\frac\pi{e^{|a|}}$$