Your intuition is right. There is a general algorithm for finding the antiderivatives: https://en.wikipedia.org/wiki/Risch_algorithm Its simplified form can solve pretty much all the undergrad antiderivation problems.
I'm a math major, but I consider the time spent learning the tricks for antiderivation to be kinda useless.
It doesn't. But if there is an elementary antiderivative, the Risch algorithm will find it (given the caveats listed in the Wikipedia article). But it might require a lot of substitutions, making its manual application impractical.
Another caveat is that Risch algorithm applies only to antiderivatives, not to the definite integrals. Some definite integrals can be computed without finding the antiderivative, often with the help of Feynman's trick.
I'm a math major, but I consider the time spent learning the tricks for antiderivation to be kinda useless.