@Chris'ssis The best I can get as follows
Using the identity $$ \sum_{n=0}^{\infty} a^{n} \cos(nx) = \frac{1- a \cos x}{1-2a \cos x + a^{2}} \ ,\ \mbox{for}\ |a| <1 ,$$
one finds that $$ 1 + 2 \sum_{n=1}^{\infty} a^{n} \cos(2nx) = \frac{1-a^{2}}{1-2a \cos2 x +a^{2}}=\frac{1-a^{2}}{1+a^{2}}\cdot\frac{1}{1-\frac{2a}{1+a^{2}} \cos x }$$
and
\begin{align}
\int_0^{\pi/2}\frac{x\sin2x}{3+\cos4x}\,dx&=\frac{1}{12}\int_0^{\pi}\frac{x\sin x}{1+\frac{1}{3}\cos2x}\,dx
\end{align}
For $-\frac{2a}{1+a^{2}}=\frac{1}{3}$, we obtain $a=2\sqrt{2}-3$, then