The value of the integral \({\int }_{-{\log }_{e}2}^{{\log }_{e}2}{e}^{x}\left({\log }_{e}\left({e}^{x}+\sqrt{1+{e}^{2x}…

\({\log }_{e}\left(\frac{2(2+\sqrt{5})}{\sqrt{1+\sqrt{5}}}\right)-\frac{\sqrt{5}}{2}\)

\({\log }_{e}\left(\frac{\sqrt{2}(3-\sqrt{5}{)}^{2}}{\sqrt{1+\sqrt{5}}}\right)+\frac{\sqrt{5}}{2}\)

\({\log }_{e}\left(\frac{(2+\sqrt{5}{)}^{2}}{\sqrt{1+\sqrt{5}}}\right)+\frac{\sqrt{5}}{2}\)

\({\log }_{e}\left(\frac{\sqrt{2}(2+\sqrt{5}{)}^{2}}{\sqrt{1+\sqrt{5}}}\right)-\frac{\sqrt{5}}{2}\)