New Derivative Based Open Newton-Cotes Quadrature Rules

Table 17

Results and observed orders for the rules with order 8, for ∫¹₀(𝑥ln(1+𝑥)/(1+𝑥²))𝑑𝑥.

𝑁	1	2	4	8	16
Seven point ONC	0.162856790	0.162864940	0.162865006	0.162865006	0.162865006
𝑝	NA	6.96	Exact	Exact	Exact
Four point ONC1	0.162853419	0.1628649100	0.1628650058	0.1628650059	0.1628650059
𝑝	NA	6.92	9.41	Exact	Exact
Three point ONC2	0.162877298	0.1628650905	0.1628650060	0.1628650059	0.1628650059
𝑝	NA	7.18	9.59	Exact	Exact
Five point ONC3	0.162857063	0.162864944	0.162865006	0.162865006	0.162865006
𝑝	NA	7.01	Exact	Exact	Exact
Five point ONC4	0.162873175	0.162865061	0.162865006	0.162865006	0.162865006
𝑝	NA	7.21	Exact	Exact	Exact