Calculate the integral $\int\frac{dx}{(x+1)\sqrt{x^2+x+1}}$

Question

I would like to solve the integral:
$$\int\frac{1}{(x+1)\sqrt{x^2+x+1}}dx$$ How I do this kind of integrals?

Can you give me any tips?

Answer 1

just a hint

$$x^2+x+1=(x+1/2)^2+3/4$$

put $$x+1/2=\sqrt {3}/2\sinh (t) $$

and then $$u=\tanh (t/2) $$

Answer 2

If you are not comfortable with hyperbolic functions you could always use trigonometric ones.

Following @Salahaman_ Fatima hint one would have, on completing the square first

$$x^2 + x + 1 = \left (x + \frac{1}{2} \right )^2 + \frac{3}{4},$$ setting $$x + \frac{1}{2} = \frac{\sqrt{3}}{2} \tan \theta,$$ then $$t = \tan (\theta/2).$$