When is $\int_a^b \frac{1}{x}\ln\bigg(\frac{x^3+1}{x^2+1}\bigg)dx=0$?

Question

I would like to find positive, distinct, algebraic real numbers $a,b\in \mathbb R^+\cap\mathbb A$ satisfying $$\int_a^b \frac{1}{x}\ln\bigg(\frac{x^3+1}{x^2+1}\bigg)dx=0$$ Does anyone know of a systematic way to go about solving this problem? Calculating a definite integral is one thing, but solving for the values of its bounds is something that I have no experience with. If we let $a,b$ be numbers satisfying the above relation, then we know that $$\frac{db}{b}\ln\bigg(\frac{b^3+1}{b^2+1}\bigg)=\frac{da}{a}\ln\bigg(\frac{a^3+1}{a^2+1}\bigg)$$ ...but this is not useful since the chance of an antiderivative existing is slim.

Can anyone find such $a,b$?

Inspired by this question.

NOTE: Because the antiderivative of the integrand can be expressed in terms of dilogarithms, the problem is equivalent to finding distinct real algebraic numbers $a,b$ satisfying $$\frac{\text{Li}_2(-b^3)}{3}+\frac{\text{Li}_2(-b^2)}{2}=\frac{\text{Li}_2(-a^3)}{3}+\frac{\text{Li}_2(-a^2)}{2}$$

Answer 1

Just an extended comment...

A figure might be helpful. Using the Rubi package in Mathematica one finds the following:

Get["Rubi`"]
integral = Int[(Log[x^3 + 1] - Log[x^2 + 1])/x, {x, a, b}]

$$\frac{1}{6} \left(2 \text{Li}_2\left(-a^3\right)-3 \text{Li}_2\left(-a^2\right)\right)+\frac{1}{6} \left(3 \text{Li}_2\left(-b^2\right)-2 \text{Li}_2\left(-b^3\right)\right)$$

as shown by the OP. A contour plot shows the contours of zero:

ContourPlot[integral, {a, -1, 2}, {b, -1, 2}, Contours -> {0},
 PlotPoints -> 100, ContourShading -> None, AspectRatio -> 1]