We have,
$$\phi=\sqrt{1+\sqrt{1+\sqrt{1+\cdots}}}$$
$$P=\sqrt[3]{1+\sqrt[3]{1+\sqrt[3]{1+\cdots}}}$$
with golden ratio $\phi$ and plastic constant $P$. If,
$$\int_0^1 \frac{dx}{\sqrt{x+\sqrt{x+\sqrt{x+\cdots}}}}=\int_0^1 \frac{2}{1+\sqrt{1+4x}}dx=\frac{2}{\phi}-\ln \phi$$
given in this answer, would it follow that the integral,
$$\int_0^1 \frac{dx}{\sqrt[3]{x+\sqrt[3]{x+\sqrt[3]{x+\cdots}}}}=\,?$$
is expressible in terms of the plastic constant?