My recent papers: Stability problems in symbolic integration (https://dl.acm.org/doi/abs/10.1145/3476446.3535502) and Stability Problems on D-finite Functions (https://dl.acm.org/doi/10.1145/3597066.3597085) may partially answer this question. For a self-map $f$ from $A$ to $A$, an element $a$ in $A$ is said to be stable if there exists a sequence $a_i$ in $A$ such that $a_0=a$ and $a_i = f(a_{i+1})$ for all $i\geq 0$. Now taking $A$ to be the field of elementary functions and $f$ the derivation on $A$. This paper shows that $\exp(f(x))$ with $f$ being a rational-function in $\mathbb{C}(x)$ is stable if and only if $f = ax + b$ with $a, b$ being constants in $\mathbb{C}$. Similarly, $\log(f(x))$ with $f$ being a rational-function in $\mathbb{C}(x)$ is stable if and only if $f$ is a Laurent polynomial in $\mathbb{C}[x, 1/x]$. In general, it is a challenging problem to characterize all possible stable elementary functions, especially the case of algebraic functions.

My recent papers

Chen, Shaoshi, Stability Problems in Symbolic Integration, MR4488863.

Chen, Shaoshi; Feng, Ruyong; Guo, Zewang; Lu, Wei, Stability problems on D-finite functions, ZBL07760759 MR4618424.

may partially answer this question. For a self-map $f$ from $A$ to $A$, an element $a$ in $A$ is said to be stable if there exists a sequence $a_i$ in $A$ such that $a_0=a$ and $a_i = f(a_{i+1})$ for all $i\geq 0$. Now taking $A$ to be the field of elementary functions and $f$ the derivation on $A$. This paper shows that $\exp(f(x))$ with $f$ being a rational-function in $\mathbb{C}(x)$ is stable if and only if $f = ax + b$ with $a, b$ being constants in $\mathbb{C}$. Similarly, $\log(f(x))$ with $f$ being a rational-function in $\mathbb{C}(x)$ is stable if and only if $f$ is a Laurent polynomial in $\mathbb{C}[x, 1/x]$. In general, it is a challenging problem to characterize all possible stable elementary functions, especially the case of algebraic functions.

My recent papers: Stability problems in symbolic integration (https://dl.acm.org/doi/abs/10.1145/3476446.3535502) and Stability Problems on D-finite Functions (https://dl.acm.org/doi/10.1145/3597066.3597085) may partially answer this question. For a self-map f from A to A, an element a in A is said to be stable if there exists a sequence a[i] in A such that a[0]=a and a[i] = f(a[i+1]) for all i>=0. Now taking A to be the field of elementary functions and f the derivation on A. This paper shows that exp(f(x)) with f being a rational-function in C(x) is stable if and only if f = ax + b with a, b being constants in C. Similarly, log(f(x)) with f being a rational-function in C(x) is stable if and only if f is a Laurent polynomial in C[x, 1/x]. In general, it is a challenging problem to characterize all possible stable elementary functions, especially the case of algebraic functions.

My recent papers: Stability problems in symbolic integration (https://dl.acm.org/doi/abs/10.1145/3476446.3535502) and Stability Problems on D-finite Functions (https://dl.acm.org/doi/10.1145/3597066.3597085) may partially answer this question. For a self-map $f$ from $A$ to $A$, an element $a$ in $A$ is said to be stable if there exists a sequence $a_i$ in $A$ such that $a_0=a$ and $a_i = f(a_{i+1})$ for all $i\geq 0$. Now taking $A$ to be the field of elementary functions and $f$ the derivation on $A$. This paper shows that $\exp(f(x))$ with $f$ being a rational-function in $\mathbb{C}(x)$ is stable if and only if $f = ax + b$ with $a, b$ being constants in $\mathbb{C}$. Similarly, $\log(f(x))$ with $f$ being a rational-function in $\mathbb{C}(x)$ is stable if and only if $f$ is a Laurent polynomial in $\mathbb{C}[x, 1/x]$. In general, it is a challenging problem to characterize all possible stable elementary functions, especially the case of algebraic functions.