Is this closed-form of $\int_0^1 \operatorname{Li}_3^2(x)\,dx$ correct?

Question

According to Freitas' paper at page $11$.

$$\int_0^1 \operatorname{Li}_3^2(x)\,dx = 20-8\zeta(2)-10\zeta(3)-\frac{15}{2}\zeta(4)-2\zeta(2)\zeta(3)+\zeta^2(3).$$

I evaluated the LHS and it is $0.427714784290824$ to me, but the RHS is $-15.8071337213762487846272$. Where am I wrong? Is this closed-form correct? If not, what is the correct closed-form?

Edit. The correct closed-form is $$\int_0^1 \operatorname{Li}_3^2(x)\,dx = 20-8\zeta(2)-10\zeta(3)+\frac{15}{2}\zeta(4)-2\zeta(2)\zeta(3)+\zeta^2(3).$$

Answer 1

The trilogarithm has the antiderivative,

$$\int\operatorname{Li}_{3}{\left(x\right)}\,\mathrm{d}x=x\operatorname{Li}_{3}{\left(x\right)}-x\operatorname{Li}_{2}{\left(x\right)}+x-x\ln{\left(1-x\right)}+\ln{\left(1-x\right)}+\color{grey}{constant}.$$

So, integrating by parts we find after integrating all the terms with simple antiderivatives:

$$\begin{align} \int_{0}^{1}\operatorname{Li}_{3}^2{\left(x\right)}\,\mathrm{d}x &=\left[\left(x\operatorname{Li}_{3}{\left(x\right)}-x\operatorname{Li}_{2}{\left(x\right)}+x-x\ln{\left(1-x\right)}+\ln{\left(1-x\right)}\right)\operatorname{Li}_{3}{\left(x\right)}\right]_{0}^{1}\\ &~~~~~ -\int_{0}^{1}\left(x\operatorname{Li}_{3}{\left(x\right)}-x\operatorname{Li}_{2}{\left(x\right)}+x-x\ln{\left(1-x\right)}+\ln{\left(1-x\right)}\right)\frac{\operatorname{Li}_{2}{\left(x\right)}}{x}\,\mathrm{d}x\\ &=\zeta{(3)}-\zeta{(2)}\zeta{(3)}+\zeta{(3)}^2-\int_{0}^{1}\operatorname{Li}_{3}{\left(x\right)}\operatorname{Li}_{2}{\left(x\right)}\,\mathrm{d}x+\int_{0}^{1}\operatorname{Li}_{2}^2{\left(x\right)}\,\mathrm{d}x\\ &~~~~ -\int_{0}^{1}\operatorname{Li}_{2}{\left(x\right)}\,\mathrm{d}x+\int_{0}^{1}\ln{\left(1-x\right)}\operatorname{Li}_{2}{\left(x\right)}\,\mathrm{d}x-\int_{0}^{1}\frac{\ln{\left(1-x\right)}\operatorname{Li}_{2}{\left(x\right)}}{x}\,\mathrm{d}x\\ &=4-2\zeta{(2)}-\zeta{(3)}+\frac54\zeta{(4)}-\zeta{(2)}\zeta{(3)}+\zeta{(3)}^2\\ &~~~~ +\int_{0}^{1}\operatorname{Li}_{2}^2{\left(x\right)}\,\mathrm{d}x-\int_{0}^{1}\operatorname{Li}_{3}{\left(x\right)}\operatorname{Li}_{2}{\left(x\right)}\,\mathrm{d}x.\\ \end{align}$$

For the remaining two integrals, you may substitute the values reported in the paper you cited.

Answer 2

There is a wrong sign, the correct closed form is

$$20-8\,\zeta \left( 2 \right) -10\,\zeta \left( 3 \right) +{\frac {15}{2}}\,\zeta \left( 4 \right) -2\,\zeta \left( 2 \right) \zeta \left( 3 \right) + \left( \zeta \left( 3 \right) \right) ^{2} $$