mystery regarding power series of $\frac{1}{\sqrt{1+x^{x}}}$

Well, $$\frac1{\sqrt{1+x^x}}=\frac1{\sqrt{1+e^y}}$$ where $y=x\log x$ so it's unsurprising you get a series in terms of $y=x\log x$. Then $$\frac1{\sqrt{1+e^y}}=\frac1{\sqrt2}\frac1{\sqrt{1+u}}$$ where $u=\frac12(e^y-1)=o(y)$. This explains the fact that the coefficients are rationals over $\sqrt2$.

Now consider $f(x)=\exp(\tanh^{-1}(\tan x))$. Now $$\tanh z=\frac{e^{2z}-1}{e^{2z}+1}$$ so that $$e^z=\sqrt{\frac{1+\tanh z}{1-\tanh z}}.$$ Hence, $$\exp(\tanh^{-1} t)=\sqrt{\frac{1+t}{1-t}}.$$ Putting $t=\tan x$ gives $$f(x)=\sqrt{\frac{1+\tan x}{1-\tan x}}=\sum_{n=0}^\infty a_n x^n.$$ Then \begin{eqnarray*} &&(1+i)f(ix)+(1-i)f(-ix)\\ &=& 2\sum_{m=0}^\infty (a_{4m} x^{4m}-a_{4m+1} x^{4m+1}-a_{4m+2} x^{4m+2}+a_{4m+3} x^{4m+3}). \end{eqnarray*} But $$f(ix)=\sqrt{\frac{1+i\tanh x}{1-i\tanh x}} =\frac{1+i\tanh x}{\sqrt{1+\tanh^2x}}$$ and so \begin{eqnarray*} &&(1+i)f(ix)+(1-i)f(-ix)\\ &=& \frac{2-2\tanh x}{\sqrt{1+\tanh^2 x}} =\frac{2\cosh x-2\sinh x}{\sqrt{\cosh^2x+\sinh^2x}}\\ &=&\frac{2e^{-x}}{\sqrt{(e^{2x}+e^{-2x})/2}}=\frac{2\sqrt2}{\sqrt{1+e^{4x}}} \end{eqnarray*} which explains why the coefficients in the two series are the same up to signs and powers of 4.

It's simple: $\rm\quad\quad f(x)\ =\ \cos x - \sin x \ =\ \frac{1+i}2 e^{ix} + \frac{1-i}2 e^{-ix}$

$\rm\quad\displaystyle \Rightarrow\quad\quad\quad\quad\ f(\tan^{-1} x)\ =\ \frac{1-x}{\sqrt{x^2+1}}\quad $ via $\rm\displaystyle\quad e^{\:i\:tan^{-1} x}\ =\ \frac{1+ x\: i}{\sqrt{x^2 + 1}}$

$\rm\quad\displaystyle \Rightarrow\quad f(\tan^{-1}\tanh x)\ =\ \ \frac{\sqrt 2}{e^{4x}+1}\quad\ $ via $\rm\displaystyle\quad\ \tanh{x}\ =\ 1 - \frac{2}{e^{2x}+1}$