Which functions have a compositional inverse which is the antiderivative? Which functions $f$ of one variable have a compositional inverse which is the antiderivative of $f$?

Following up Arthur's suggestion and assuming a form $a x^{b}$ for $f$:
Let $g(x) = f^{-1}(x)$ and let $g'(x) = f(x)$
$$
f = a x^b \Rightarrow g = \frac{a}{b+1} x^{b+1}
$$
I accept I've ignored a possible integration constant for $g$ above.
$$
f (g (x)) = x 
\Rightarrow
f (\frac{a}{b+1} x^{b+1}) = x
$$
$$
a \left(
\frac{a}{b+1} x^{b+1}
\right)^{b}
=
x
$$
$$
\frac{a^{b+1}}{(b+1)^{b}}
x^{b(b+1)}
=
x
$$
Which can be solved with two assumptions:
$$
b(b+1) = 1
\Rightarrow
b^{2} + b - 1 = 0
$$
and
$$
\frac{a^{b+1}}{(b+1)^{b}}
=1
\Rightarrow
a = (b+1)^{\frac{b}{b+1}}
$$
The first gives
$$
b =
\frac{-1 \pm \sqrt{5}}{2}
$$
which can be used to obtain $a$.
So this form for $f$ seems solvable.
I wonder what other function forms are possible.
