I want to prove that these two functions $f(x)$ and $g(x)$ do not intersect for $x>1$: $$f(x)=\cosh \left(\frac{2 \sqrt{2} \pi x \left(x^2-1\right) \cosh (\pi x)}{\sqrt{x^4+6 x^2+\left(x^2-1\right)^2 \cosh (2 \pi x)+1}}\right)$$ $$g(x)=\frac{4 x^2+\left(x^2-1\right)^2 \cosh (2 \pi x)}{\left(x^2+1\right)^2}$$ Both functions are greater than one and strictly increasing. Subtraction and taking derivative do not work, the problem becomes more complicated.
Does anyone have an idea to prove it by assuming false assumption? Or to prove that $f(x)-g(x)$ has no real root?
Any hints or suggestions are really appreciated.
Since you said that both functions are greater than one and strictly increasing, they vary so fast that I should instead consider $$F(x)=\log(f(x)) \qquad \text{and} \qquad G(x)=\log(g(x))$$ If you plot them, you should notice that, as soon as $x > 3$, $F(x)$ and $G(x)$ looks very linear with $F(x) > G(x)$; I agree that this does not prove anything.
Now, consider a Taylor expansion of $F(x)-G(x)$ around $x=1$. This gives $$H(x)=F(x)-G(x)=(x-1)^2 \left((\pi ^2+1)+\left(\pi ^2-1\right) \cosh (2 \pi )\right)+O\left((x-1)^3\right)$$ So, starting from $0$ at $x=1$, $H(x)$ is an increasing function, which, according to Murphy's principle, will go through a maximum value. This maximum is located very close to $\frac 98$ and its nature is confirmed by the second derivative test.
For sure, now the problem is : what happens for infinitely large values of $x$ ?
For the asymptotics analysis, I shall assume that, for large $t$, $\cosh(t) \sim \frac 12 e^t$. Factoring the exponentials and ignoring the terms which are divided by $e^t$, we end with
$$f(x) \sim \frac 12 e^{2\pi x} \quad \text{and} \quad g(x) \sim \frac 12 e^{2\pi x} \frac{\left(x^2+1\right)^2}{\left(x^2-1\right)^2}\implies H(x) \sim \frac{4}{x^2}+O\left(\frac{1}{x^6}\right)$$ In other words, when $x$ is large $$f(x) \sim g(x) \, \exp \left({\frac{4}{x^2}}\right)$$
Computed exactly for $x=10$, the result is $H(10)=0.0400013$ !!
Edit
Looking at the last result, I computed as few values $$\left( \begin{array}{cc} x & x^2\, H(x) \\ 2 & 4.0859321 \\ 3 & 4.0165826 \\ 4 & 4.0052206 \\ 5 & 4.0021354 \\ 6 & 4.0010293 \\ 7 & 4.0005555 \\ 8 & 4.0003256 \\ 9 & 4.0002032 \\ 10 & 4.0001333 \\ 11 & 4.0000911 \\ 12 & 4.0000643 \\ 13 & 4.0000467 \\ 14 & 4.0000347 \\ 15 & 4.0000263 \\ 16 & 4.0000203 \\ 17 & 4.0000160 \\ 18 & 4.0000127 \\ 19 & 4.0000102 \\ 20 & 4.0000083 \end{array} \right)$$