I posted an answer to the question An Integral Involving The Inverse Of $f(x)$ and my answer depends on knowing where the function $f(x)$ is $0$ or $1$. The function itself is
$$f(x) = \log x - \log \cos x + x \tan x$$
on $x \in (0,\pi/2)$. On this range it can clearly be written as
$$f(x) = \log\left(\frac{x}{\cos x}\right)+ x \tan x$$
Is there a closed for $x_0$ and $x_1$ where $f(x_0) = 0$ and $f(x_1)=1$? Numerically these two values are
$$x_0 = 0.576412723\ldots$$ $$x_1 = 0.807626251\ldots$$
I would accept an interesting series form for either one as "closed enough".