Finding indefinite integral ∫ (e ^ 2x) / (2 + e ^ x) DX Such as the title

Finding indefinite integral ∫ (e ^ 2x) / (2 + e ^ x) DX Such as the title

A:
∫ [(e^x)^2/(2+e^x)] dx
= ∫ [e^x /(2+e^x)] d(e^x)
=∫ [(e^x+2-2)/ (2+e^x) ] d(e^x)
=∫ [1-2 / (e^x+2) ] d(e^x+2)
=e^x-2ln(e^x+2)+C

Seeking indefinite integral ∫ (x ^ 2 + x) * e ^ (2x)

∫(x^2+x )* e^(2x)dx
=1/2∫(x^2+x )*de^(2x)
=1/2(x^2+x )*e^(2x)-1/2∫e^(2x)*(2x+1)dx
=1/2(x^2+x )*e^(2x)-1/4∫(2x+1)de^(2x)
=1/2(x^2+x )*e^(2x)-1/4(2x+1)*e^(2x)+1/4∫2e^(2x)dx
=1/2(x^2+x )*e^(2x)-1/4(2x+1)*e^(2x)+1/4e^(2x)+C
= 1/2x^2e^(2x)+C

For example, find e ^ (2x) / (e ^ x + 1) ^ 3 indefinite integral,

∫e^2x dx/(e^x+1)^3
=∫e^(-x)dx/(e^(-x)+1)^3
=-(1/2)(1/(e^(-x)+1)^2)+C

Indefinite integral solution: ∫ (2x ^ 2 + 2x + 20) / [(x ^ 2 + 2x + 5) (x-1)] DX

∫{ (2x^2+2x+20)/[(x^2+2x+5)(x-1)] }dx= 2∫{ (x^2+2x+5)/[(x^2+2x+5)(x-1)] }dx - ∫ (2x-10)/[(x^2+2x+5)(x-1)] dx= 2∫ [1/(x-1) ]dx - ∫ (2x-10)/[(x^2+2x+5)(x-1)] dxlet(2x-10)/[(x^2+2x+5)(x-1) ≡ A/(x-1...