nonelementary

The evaluation of nonelementary antiderivatives can often be done using Taylor series.

Many nonelementary integrals can be expanded in a Taylor series and integrated term by term.

A theorem by Liouville in 1835 provided the first proof that nonelementary antiderivatives exist.

These response-model equations are quite complicated and some involve a nonelementary integral; the exact solutions can be found here.

This "solution" generally does not have many uses because most of the nature of the solution is hidden in the nonelementary integral (nonelementary even if ).

Second and higher order ordinary differential equations (more generally, systems of nonlinear equations) rarely yield closed form solutions, though implicit solutions and solutions involving nonelementary integrals are encountered.

In mathematics, a nonelementary antiderivative is an antiderivative for which it can be shown that there exists no formula in terms of elementary functions (i.e. involving polynomials, and the standard functions sin, cos, exp, and so on).