I've been reading through Stewart's calculus, and it mentions how most elementary functions don't have elementary antiderivatives. I don't understand why the derivative of an elementary function is an elementary function, but the inverse operation(antiderivative) isn't. Below is a snippet of the text,
For instance, the function $$f(x)=\sqrt{\frac{x^2-1}{x^3+2x-1}}+\ln(\cosh(x))-xe^{\sin(2x)}$$ is an elementary function.
If $f$ is an elementary function, then $f’$ is an elementary function but $\int f(x)dx$ need not be an elementary function. Consider $f(x)=e^{x^2}$. Since $f$ is continuous, its integral exists, $\ldots$