This question recently popped up on one of my tests and I still have no idea on how to even begin.
I tried assuming $\sin(x+2A)+\sin(A)=t^2$ which did not help at all.
Then I went on to multiply and divide by $\tan x$ to create a derivative on the numerator, still does not help.
I even tried to expand $\sin(x+2A)$ which too does not seem to follow.
I have also tried Approach$0$ and Wolfram Alpha but they do not seem to help.
and many more...
Now I am stumped I have no idea as to how to even begin. I have tried everything that I could think of.
Note that we are only familiar with basics of integration(no contour integrals ; no special functions etc..)
$\bullet~\textbf{Question:}~$
"Can the above integral be evaluated in elementary functions?" If yes then how? If not , can we prove that it cannot be solved using elementary funcions?