Indefinite Integrals

For \(\alpha, \beta, \gamma, \delta \in \mathbb{N}\), if \(\int\left(\left(\frac{x}{e}\right)^{2 x}+\left(\frac{e}{x}\right)^{2 x}\right) \log _{e} x d x=\frac{1}{\alpha}\left(\frac{x}{e}\right)^{\beta x}-\frac{1}{\gamma}\left(\frac{e}{x}\right)^{\delta x}+C\)

, where \(e=\sum_\limits{n=0}^{\infty} \frac{1}{n !}\) and \(\mathrm{C}\) is constant of integration, then \(\alpha+2 \beta+3 \gamma-4 \delta\) is equal to :