Nonelementary integral

Integrals not expressible in closed-form from elementary functions In mathematics, a nonelementary antiderivative of a given elementary function is an antiderivative (or indefinite integral) that is, itself, not an elementary function. A theorem by Liouville in 1835 provided the first proof that nonelementary antiderivatives exist. This theorem also provides a basis for the Risch algorithm for determining (with difficulty) which elementary functions have elementary antiderivatives. Examples Examples of functions with nonelementary antiderivatives include: 1 − x 4 {\displaystyle {\sqrt {1-x^{4}}}} (elliptic integral) 1 ln ⁡ x {\displaystyle {\frac {1}{\ln x}}} (logarithmic integral) e − x 2 {\displaystyle e^{-x^{2}}} (error function, Gaussian integral) sin ⁡ ( x 2 ) {\displaystyle \sin(x^{2})} and cos ⁡ ( x 2 ) {\displaystyle \cos(x^{2})} (Fresnel integral) sin ⁡ ( x ) x = sinc ⁡ ( x ) {\displaystyle {\frac {\sin(x)}{x}}=\operatorname {sinc} (x)} (sine integral, Dirichlet integral) e − x x {\displaystyle {\frac {e^{-x}}{x}}} (exponential integral) e e x {\displaystyle e^{e^{x}}\,} (in terms of the exponential integral) ln ⁡ ( ln ⁡ x ) {\displaystyle \ln(\ln x)\,}