Mathematics Question from JEE Main 2023 · 10 April, Shift 2

For $\alpha, \beta, \gamma, \delta \in \mathbb{N}$, if

$$\int \left( \frac{x}{e} \right)^{2x} + \left( \frac{e}{x} \right)^{2x} \log x \, dx = \frac{1}{\alpha} \left( \frac{x}{e} \right)^{\beta x} - \frac{1}{\gamma} \left( \frac{e}{x} \right)^{\delta x} + C$$

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