how WolframAlpha define these non elementary integrals

8 special functions for your non-elementary integration needs, as seen on WolframAlpha! Shop non-elementary t-shirts: https://blackpenredpen.creator-spring.com/listing/nonelementary?product=2 1. Error function erf(x) https://en.wikipedia.org/wiki/Error_function http://mathworld.wolfram.com/Erf.html 2. Imaginary error function erfi(x) http://mathworld.wolfram.com/Erfi.html 3. Exponential integral function Ei(x) https://en.wikipedia.org/wiki/Exponential_integral http://mathworld.wolfram.com/ExponentialIntegral.html 4. Logarithmic integral function li(x) very important in number theory/prime numbers https://en.wikipedia.org/wiki/Logarithmic_integral_function http://mathworld.wolfram.com/LogarithmicIntegral.html 5. Sine and Cosine integrals Si(x) and Ci(x) https://en.wikipedia.org/wiki/Trigonometric_integral 6. Fresnel's integral S(x) and C(x), https://en.wikipedia.org/wiki/Fresnel_integral http://mathworld.wolfram.com/FresnelIntegrals.html @blackpenredpen