Suppose that we have an integrable function $f(x)$ which is expressed in terms of elementary functions. By integrable, I mean that we can find its anti-derivative in terms of elementary functions, and by elementary function I mean a function of one variable built from a finite number of exponentials, logarithms, polynomials, trigs, inverses of trigs and roots of other elementary function through composition and combinations of the four elementary operations ($+, -, \times, \div$). So $\sqrt{\sin(x)}$ in this definition is elementary function.
Is it necessarily true that we can calculate its integral (anti-derivative) using integration by parts, partial fractions, substitutions, trigonometric & hyperbolic substitutions?
Of course there are functions whose anti-derivatives we can't find in terms of elementary functions (for example, $f(x)= \frac{\sin(x)}{x}$ or $f(x)=e^x \ln(x)$), but my question is about those for which we can find their anti-derivative in terms of elementary functions.
The reason behind specifying only those methods is that those methods are taught in every class in calculus when the instructor talk about methods of integration (also, another reason is that finding the derivative for any function can be calculated using the inverse of those integration methods such as the product rule or chain rule - except quotient rule!)
My own guess is: Suppose that $A$ is the set of functions $F(x)$ whose derivatives are calculated using the quotient rule on some level, and $D$ is the set of derivatives of functions of $A$ , then by definition all the functions in $D$ are integrable but there exists a function in $D$ which can't be integrated using ordinary integration methods (remember, this is just a guess!) .
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