\[\large \displaystyle \int_0^2 \text{Li}_3 (4^x) \, dx\]

Given the definite integral above, if its value can be represented in the closed form as

\[\dfrac{a}{b \log b} \bigg( \text{Li}_c (d) - \zeta (c) \bigg)\]

Find \(a+b+c+d\).

**Notations:**

- \(\text{Li}_n (a)\) denotes the polylogarithm function, that is \(\text{Li}_n (a) = \displaystyle \sum_{k=1}^{\infty} \dfrac{a^k}{k^n}\).
- \(\zeta (s)\) represents the Riemann zeta function, that is \(\zeta (s) = \displaystyle \sum_{n=1}^{\infty} \dfrac{1}{n^s}\).
- \(\log (\cdot)\) denotes the natural logarithm function, that is \(\log_{e}{(\cdot)}\) or \(\ln(\cdot)\).

Inspired by Hummus A.

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