Can $\tanh(\arctan(x))$ be simplified or re-expressed in terms of algebraic functions (or maybe a log here and there), kind of like the forward-inverse identities? (perhaps using the myriad of trig and hyperbolic formulas)
The only thing I found is the Gudermannian function, which in some sense yields the opposite of what I need
$$gd(x) = 2 \arctan\left(\tanh\left(\frac{x}{2}\right)\right)$$
Also, when I let Mathematica spit out the MacLaurin series, it seems that $\tan(\arctan(x))$ and $\tanh(\arctan(x))$ have similar expansions (differing only by the usual alternating sign). Is there a place where these kind of relationships between the trigonometric and hyperbolic functions are studied? Anything known about the properties of the coefficients of such series?
How do you feel about $$ \tanh \arctan x = \frac{(1-\mathrm{i}x)^\mathrm{i} - (1+\mathrm{i}x)^\mathrm{i}}{(1-\mathrm{i}x)^\mathrm{i} + (1+\mathrm{i}x)^\mathrm{i}} \text{?} $$
This is obtained from $\tanh y = \frac{\mathrm{e}^{2 y}-1}{\mathrm{e}^{2 y}+1}$ and $\arctan x = \frac{\mathrm{i}}{2} \log \frac{1-\mathrm{i} x}{1+\mathrm{i} x}$, followed by some simplifying.