It is known that the anti derivative of $\sin(x^2)$ is not an elementary function, and one can represent it using a power series by term-by-term integration of its Taylor series.
However, is there any way to show that it's antiderivative is transcendental and cannot be represented by an elementary function?
My thoughts are to assume that there is an elementary function $f$ such that $f'=\sin(x^2)$, and show a contradiction. However, are there any special properties that only elementary or only transcendental functions have (so I can show that $f$ does not satisfy one or satisfies one)?
As people pointed out in comment, the opposite of transcendental is algebraic, (and therefore $\int \sin(x^2)$ is certainly transcendental)... Maybe ignore transcendental and imagine it to mean "non elementary".