How to calculate the indefinite integral of x ^ 4 / x ^ 2-1?

How to calculate the indefinite integral of x ^ 4 / x ^ 2-1?

x^4/(x^2- 1) = 1 +x^2 + (1/2) [ 1/(x-1) - 1/ (x+1) ]
I= x+ x^3 /3 + (1/2) ln| (x-1)/(x+1) | + C

Finding the indefinite integral of ∫ [1 / √ (2x + 1)] DX
The situation is urgent

Original formula = (1 / 2) ∫ D (2x + 1) / √ (2x + 1)
=(1 / 2) * 2 * √ (2x + 1) + C (C is any constant)
=√(2x+1)+C.

∫ 1 / (1 + 2x) ^ 2 DX for indefinite integral

Solution
∫1/(1+2x)²dx
=1/2∫1/(1+2x)²d(1+2x)
=1/2∫1/u²du
=1/2×(-1/u)+C
=-1/[2(1+2x)]+C

Indefinite integral ∫ (2x-1) ^ 10 DX
I want the process and the formula

A:
∫(2x-1)^10 dx
=(1/2) ∫(2x-1)^10 d(2x-1)
=(1/2)*(1/11)*(2x-1)^11+C
=(1/22)(2x-1)^11+C
The main difference is DX = (1 / 2) d (2x) = (1 / 2) d (2x-1)