[1] A theorem by Liouville in 1835 provided the first proof that nonelementary antiderivatives exist.
The examples above name the corresponding special functions in parentheses.
Nonelementary antiderivatives can often be evaluated using Taylor series.
Even if a function has no elementary antiderivative, its Taylor series can always be integrated term-by-term like a polynomial, giving the antiderivative function as a Taylor series with the same radius of convergence.
However, even if the integrand has a convergent Taylor series, its sequence of coefficients often has no elementary formula and must be evaluated term by term, with the same limitation for the integral Taylor series.