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