A sharp inequality involving hyperbolic and inverse hyperbolic functions

We prove that the inequality $$\cosh \left( \mathrm{arcosh}(2 \cosh u) \cdot \tanh u \right) < \exp \left( u \cdot \tanh u \right)$$ holds for all $u > 0$. We check with the computation program Mathematica that the ratio between the left-hand and the right-hand side is greater than 0,97 for all $u \ge 0$, so this is a quite sharp inequality. It is also equivalent to any of the two inequalities: $$\cosh \left( \sqrt{1 - \frac{1}{t^2}} \cdot \mathrm{arcosh}\,{2t} \right) < \exp \left( \sqrt{1 - \frac{1}{t^2}} \cdot \mathrm{arcosh}\,{t} \right)$$ for all $t > 1$, and $$\cosh \left( c \cdot \mathrm{arcosh}{\frac{2}{\sqrt{1-c^2}}} \right) < \exp \left( c \cdot \mathrm{arcosh}{\frac{1}{\sqrt{1-c^2}}} \right)$$ for all $c \in (0,1)$.