$\int_{0}^{\ln \sqrt{2}} \frac{1+\cos \left(e^{-2 x}\right)}{e^{2 x}} d x=?$

$\frac{1}{8}+\frac{1}{2}\left(\sin 1-\sin \frac{1}{4}\right)$

$\frac{1}{4}+\frac{1}{2}\left(\sin 1-\sin \frac{1}{2}\right)$

$\frac{3}{8}+\frac{1}{2}\left(\sin 1-\sin \frac{1}{4}\right)$

$\frac{3}{4}+\frac{1}{2}\left(\sin 1-\sin \frac{1}{2}\right)$