How do I understand if an indefinite integral is not solvable in terms of elementary functions?

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Many indefinite integrals cant be solved in terms of elementary functions.e.g.-$$\int \frac{\sin(x)}{x}dx$$ But many hard looking ones are still solvable by weird substitutions and other tricks. Is there a test(like there are tests of solvability/convergence.etc in mathematics) of whether it is solvable or not by which I can see beforehand without wasting hours of time after which I come to know that it is not even solvable?