In a German forum, a user asked how the "Feynman"-trick works. The example was $$ f(x)=x e^{x} $$ Another user mentioned that the Risch algorithm should be taught. Therefore, I wonder whether the Risch algorithm is useful for calculating antiderivates by hand. My questions:
Risch algorithm is rather complicated. Wikipedia page has few cute example, but you will see they degenerate fast. The function you wrote down is easily integrable even without Feynman's trick (a by parts integration is faster, to me). If you want a good example via Feynman's trick, use this
$$\int \ln(2 + \text{tg}^2(x))\ \text{d}x$$
W. Mathematica makes use of billions of lines of codes, which space from numerical computations, to symbolic calculations, to approximations, asymptotic, infinite series, infinite products and so on.
It can return the solution of some integral in terms of special functions, which are generally defined by infinite series, or by other integrals (which can be represented as infinite series too, ish) or numerical evaluations (with high precision, but still not analytical).
In any case, I have seen over here people who could solve analytically exactly o numerically exactly integrals by hand and mind that W. Mathematical couldn't, so it's all a matter of fun, time, energy and skills.