miscalculation
correct
https://gyazo.com/167c7aba561c269e5e0955882aebbcd8
error
https://gyazo.com/090b285af1e307f38b1cb3556e2fa31a
I see!
$ J = \frac{2x^2}{(1-x)^3}
Did (1-x)^3 correspond to taking the difference three times?
$ 4F = G + J = \frac{1}{1-x^2} + \frac{2x^2}{(1-x)^3} = \frac{-(2 x^3 + 3 x^2 - 2 x + 1)}{((x - 1)^3 (x + 1))}
Hmmm, what's with the much different shape?
---
This page is auto-translated from /nishio/計算ミス. If you looks something interesting but the auto-translated English is not good enough to understand it, feel free to let me know at @nishio_en. I'm very happy to spread my thought to non-Japanese readers.