# Existence of closed form for indefinite integral

During calculus studies, I tried to find a primitive for the following indefinite integral, in a simple form using standard functions:

$\int \sqrt{\sin x} \mathrm{d}x$

I always failed. It may be possible to prove that the primitive exists, or even to find an infinite series expansion for it. But I suspect that it is not possible to find a simple closed-form expression for it.

Am I right? If so, how to prove it?

• I think there is no elementary function whose derivative is your function. Elliptic functions will do it. – André Nicolas Jul 19 '13 at 4:35

Depends on what functions you call "standard". It has no elementary antiderivative, but does have one in terms of elliptic functions.

• Elliptic functions are not the same as elliptic integrals, although they are related. – Robert Israel Jul 19 '13 at 4:40

Maple and Mathematica both give answers in terms of elliptic integral functions, which are not elementary functions and so probably don't qualify as "simple closed form". A proof that the antiderivative is not elementary would involve the Risch algorithm, but since this is the mixed transcendental-algebraic case (which isn't even implemented in Maple) I think this would be very complicated.