The Risch algorithm is used to find closed-form antiderivatives.
If I understand the article right, only heuristics are known. On the other hand, I came across the claim that it is known that $e^{-x^2}$ has no closed-form antiderivative.
Is the Risch-algorithm successful in the case $f(x)=e^{-x^2}$ ? Is it actually proven that no closed-form antiderivative exist or just very probable because no form has been found using the Risch algorithm ?
This question could well be a duplicate, but I am not sure whether the aspect of decidability has been asked.
If an elementary derivative exists then the Risch algorithm will construct it. Otherwise, the proof of the correctness of the algorithm implies that no elementary antiderivative exists. The algorithm could be modified to output witnesses that no solution exists, but that is rarely done in practice since there is no need to do so.
Most expositions of the algorithm include small worked examples (such as yours) which will help you gain intuition on how the algorithm works, e.g. see the links I gave in this prior answer, esp. max Rosenlicht's Monthly expsoition, which is one of the most readable introductions. There he proves the classical Liouville criterion, that for rational functions $\,f,g\,$ the integral $\int f e^g$ is elementary $\iff$ there is a rational function $\,h\,$ such that $\,f = h' + h g'$.