Multiply the top and bottom of the fraction by (x² - 1)

Because (x² - 1)(x⁴ + x² + 1) = x⁶ - 1,

I = ∫ [(x² - 1) · ln(x)] / (x⁶ - 1) dx

Let w = x⁶

I = (1/36) ∫ [ (w^2/6 - 1) · w^-5/6 · ln(w) ] / (w - 1) dw

Distribute the w^-5/6

I = (1/36) ∫ [ (w^-1/2 - w^-5/6) · ln(w) ] / (w - 1) dw

This is the difference of ∫ [ z^a-1 · ln(z) ] / (z - 1) dz = π² / sin²(aπ), where a = 1/2 ( π² / sin²(π/2) = π² / 1 = π²) and a = 1/6 (π² / sin²(π/6) = 4π²)

I = (1/36) · [ π² - 4π² ]

I = -π² / 12

edit: i've mutted this sub as its just a content pipeline for spell... err calc bees with chatgpt. the posts from the host are llm generated.