Maximise $$\large \left(\cos x\ln (x^{\frac1x})-\frac{\cos x\log_{10}x}{\ln(\ln x)}\right)^{\cos x}$$
I wonder why Mathematica says infinity as answer, but on desmos graph it is clear that it is less than 2
$\color{blue}{\text{Even WolframAlpha gave correct answer}}\color{red}{\text{ local maximum value as 1.461}}$
My try:
I used the command Maximise with and without constraints. I tried both using constaint as $\ln x >1$ But nothing correct come up.
Next I also tried FindMaximum it didn't show anything either, but WolframAlpha command gave local maximum correctly.
I wonder is there any analytical way to find $\color{green}{\text{Local Maximum Value ?}}$