I was not sure how to go about finding a closed form for $$\int\log\left(\cos\left(\frac{a}{x}\right)\right) \ dx\,\,\,\,(1)$$
so I was wondering if there is some clever integration trick that would work?
(Perhaps using $a$ as an integral upper limit and lower limit $>=0$ might produce some result as a definite integral?)
Some work done so far by me...
Trying an asymptotic expansion using a integration by parts with a=1 seems to look like things will get rather complicated quick or at least I cannot see an easily identifieble series emerging for (1) How can one tell in advance? $$-\frac{1}{2} x^2 \left(-\frac{2 \tan ^2\left(\frac{1}{x}\right)}{x^3 \log ^3\left(\cos \left(\frac{1}{x}\right)\right)}-\frac{\sec ^2\left(\frac{1}{x}\right)}{x^3 \log ^2\left(\cos \left(\frac{1}{x}\right)\right)}-\frac{\tan \left(\frac{1}{x}\right)}{x^2 \log ^2\left(\cos \left(\frac{1}{x}\right)\right)}\right)+\frac{x}{\log \left(\cos \left(\frac{1}{x}\right)\right)}+\frac{\tan \left(\frac{1}{x}\right)}{\log ^2\left(\cos \left(\frac{1}{x}\right)\right)}+\frac{\tan \left(\frac{1}{x}\right)}{x \log ^2\left(\cos \left(\frac{1}{x}\right)\right)}$$