Maximise $(\cos x\ln (x^{\frac1x})-\frac{\cos x\log_{10}x}{\ln(\ln x)})^{\cos x}$ 
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 $\color{green}{\text{Local Maximum Value ?}}$
 A: I have a variety of comments to share. Here is my graph.
1.There are endpoints at $$\left(\frac \pi2+\pi(n\in\Bbb N),1\right)$$
This part of the graph looks suspiciously similar to a transformed inverse (co)sine graph.
There is also (1 or e,0) as a solution.
2.The other critical points which may show where other extrema or discontinuities exist where  the derivative of your function is 0. I am going to be efficient and have software help as you did:$=0$
Please see the graph for the critical points. You can verify that they are extrema by comparison of the 0th derivative and 1st derivative graphs. Please give me feedback and correct me!
Taking the natural log, deriving, and using product property gives a complicated fraction, a tangent function, and a logarithm of the original function raised to just the power of 1 to show there may not be any solvable answer...
A: Using Mathematica NMaximize, I obtained an almost immediate result for $1,000$ exact figures.
$$1.460697870308468497370818908815784175460033519589233153539805\cdots$$
Computing the derivative and expanding it as a series around $x=\frac {7\pi}2$gives as an estimate the solution of
$$\log \left(\frac{1}{7} \left(x-\frac{7 \pi }{2}\right) \log
   \left(\frac{7 \pi }{2}\right) \left(\frac{2}{\pi
   }-\frac{7}{\log (10) \log \left(\log \left(\frac{7 \pi
   }{2}\right)\right)}\right)\right)+1=0$$ which is $x=10.6174$ while the exact solution is $x=10.6062$.
Using the approximate solution, the maximum value is $1.46049$ while the exact solution is $x=1.46070$.
