Need help in indefinite integral elimination

Solution 1:

Motivated by completing the square in $x^2+x+1$, the change of variables $u = \sqrt{4/3} (x + 1/2)$ makes the integral $$ \int \dfrac{\sqrt{u^2+1}}{(u + 1/\sqrt{3})^2}\; du $$ Then another substitution $v = \sqrt{u^2+1} - u$ transforms this to the integral of a rational function, which can be done using partial fractions.

EDIT: More generally, the change of variables $v = \sqrt{x^2 + a x + b} - x$ (and thus $x = (v^2-b)/(a-2v)$) is often useful for integrals involving $\sqrt{x^2+ax+b}$. It transforms the integral $$ \int \sqrt{x^2+ax+b}\; R(x)\; dx$$ to $$ \int \dfrac{2(v^2-av+b)^2}{(a-2v)^3} R\left(\frac{v^2-b}{a-2v}\right)\; dv$$