Inspired by Mr. Olivier Oloa in this question. Does the following integral admit a closed form?
\begin{align}
\mathscr{R}=\int_0^{\Large\frac{\pi}{2}}\sin^2x\,\ln\big(\sin^2(\tan x)\big)\,\,dx
\end{align}
It will be my last question before I take a long break from my activity on Mathematics StackExchange. So, please be nice. No more downvotes for no reason because this is a challenge problem.
Edit :
I am also interested in knowing the numerical value of $\mathscr{R}$ to the precision of at least $50$ digits. If you use Mathematica to find its numerical value, please share your method & the code.
Best Answer
Here's a slightly different approach. Using the power reduction formula and getting rid of the second exponent of the sine using the properties of the logarithm the integral becomes:
$$I=\int_0^{\frac{\pi}{2}} (1-\cos 2x) \log (\sin (\tan x))dx$$
Using the substitution $2x=u$ we have:
$$I = \frac{1}{2}\int_0^\pi (1-\cos u) \log\bigg(\sin \bigg(\tan \bigg(\frac{u}{2}\bigg)\bigg)\bigg)du$$
We have the integral ready for a Weierstrass substitution, after which it becomes:
$$I = \int_0^\infty\bigg(1-\frac{1-t^2}{1+t^2}\bigg) \log(\sin (t)) \frac{1}{1+t^2}dt$$
Or:
$$I = \int_0^\infty \frac{2t^2}{(1+t^2)^2} \log(\sin (t))dt$$
From now on I'll use $x$ again. The Fourier series of $\log (\sin x)$ is well known and it is:
$$\log(\sin x)= -\log 2 -\sum_{n=1}^\infty \frac{\cos(2nx)}{n}$$
So the integral, exchanging integration and summation, becomes:
$$I=-2\log 2 \int_0^\infty \frac{x^2}{(1+x^2)^2}dx-2\sum_{n=1}^\infty \frac{1}{n} \int_0^\infty \frac{x^2\cos(2nx)}{(1+x^2)^2}$$
These are both easy integrals from the point of view of residue calculus. The final result is:
$$I=-\frac{\pi \log 2}{2} -\frac{\pi}{2}\sum_{n=1}^\infty \frac{e^{-2n}}{n}+\pi \sum_{n=1}^\infty e^{-2n}$$
These sums can be evaluated using the geometric series and its integral. So we have:
$$I=-\frac{\pi \log 2}{2}+\frac{\pi}{e^2-1}-\frac{\pi}{2}(2-\log(e^2-1))$$
Simplifying:
$$I=\frac{\pi}{2}\log \bigg( \frac{e^2-1}{2} \bigg) +\pi\bigg(\frac{2-e^2}{e^2-1}\bigg)$$