"Integration in finite terms" deals with integration of function terms. Its theorems yield therefore only formal antiderivatives. Clearly, the concrete antiderivative depends on the concrete domain of the function in the integrand. If you integrate function terms by applying other methods, you have also make decisions about the domain of the functions.

"Integration in finite terms" uses an exact definition of the class of elementary functions. According to J. F. Ritt, $\exp$, $\ln$ and the algebraic functions are analytic almost everywhere, and therefore the elementary functions.

"Integration in finite terms" treats only formal antiderivatives. Clearly, the concrete antiderivative depends on the concrete domain of the function in the integrand. If you integrate functions by applying other methods, you have also make decisions about the domain of the functions.