Evaluation of $\int\frac{dx}{x+ \sqrt{x^2-x+1}}$ Evaluate :
$$\frac{dx}{x+ \sqrt{x^2-x+1}}$$
After dividing and multiply with $x-\sqrt{x^2-x+1}$, I get
$x+\ln |x-1|-\int \frac{\sqrt{x^2-x+1}}{x-1}dx$.
Is $\int \frac{\sqrt{x^2-x+1}}{x-1}dx$ is integrable in terms of elementary functions?

We have $$I =\int \frac {\sqrt {x^2-x+1}}{x-1} dx =\frac {1}{2 }\int \frac {\sqrt {(2x-1)^2+3}}{x-1} dx $$ Substituting  $u =2x-1$, we get, $$I =\frac {1}{2} \int \frac {\sqrt{u^2+3}}{u-1} du $$ Now perform hyperbolic substitution $u =\sqrt {3} \operatorname {sinh}(v)$ and simplifying gives us $$I =\frac {3}{2} \int \frac {(\operatorname {cosh }(v))^2}{\sqrt {3}\operatorname{sinh}(v) -1} dv $$ which can be solved by Weierstrass substitution. Hope it helps. 
