# Let's do some calculus! (39)

Calculus Level 5

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

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.