Integral of $\frac{1}{\sqrt{x^{2} + x + 1}}$

Find $\int \frac{1}{\sqrt{x^{2} + x + 1}}\, dx$.

Complete the square (steps can be seen »): $x^{2} + x + 1 = \left(x + \frac{1}{2}\right)^{2} + \frac{3}{4}$:

$${\color{red}{\int{\frac{1}{\sqrt{x^{2} + x + 1}} d x}}} = {\color{red}{\int{\frac{1}{\sqrt{\left(x + \frac{1}{2}\right)^{2} + \frac{3}{4}}} d x}}}$$

Let $u=x + \frac{1}{2}$.

Then $du=\left(x + \frac{1}{2}\right)^{\prime }dx = 1 dx$ (steps can be seen »), and we have that $dx = du$.

So,

$${\color{red}{\int{\frac{1}{\sqrt{\left(x + \frac{1}{2}\right)^{2} + \frac{3}{4}}} d x}}} = {\color{red}{\int{\frac{1}{\sqrt{u^{2} + \frac{3}{4}}} d u}}}$$

Let $u=\frac{\sqrt{3} \sinh{\left(v \right)}}{2}$.

Then $du=\left(\frac{\sqrt{3} \sinh{\left(v \right)}}{2}\right)^{\prime }dv = \frac{\sqrt{3} \cosh{\left(v \right)}}{2} dv$ (steps can be seen »).

Also, it follows that $v=\operatorname{asinh}{\left(\frac{2 \sqrt{3} u}{3} \right)}$.

Integrand becomes

$\frac{1}{\sqrt{ u ^{2} + \frac{3}{4}}} = \frac{1}{\sqrt{\frac{3 \sinh^{2}{\left( v \right)}}{4} + \frac{3}{4}}}$

Use the identity $\sinh^{2}{\left( v \right)} + 1 = \cosh^{2}{\left( v \right)}$:

$\frac{1}{\sqrt{\frac{3 \sinh^{2}{\left( v \right)}}{4} + \frac{3}{4}}}=\frac{2 \sqrt{3}}{3 \sqrt{\sinh^{2}{\left( v \right)} + 1}}=\frac{2 \sqrt{3}}{3 \sqrt{\cosh^{2}{\left( v \right)}}}$

$\frac{2 \sqrt{3}}{3 \sqrt{\cosh^{2}{\left( v \right)}}} = \frac{2 \sqrt{3}}{3 \cosh{\left( v \right)}}$

So,

$${\color{red}{\int{\frac{1}{\sqrt{u^{2} + \frac{3}{4}}} d u}}} = {\color{red}{\int{1 d v}}}$$

Apply the constant rule $\int c\, dv = c v$ with $c=1$:

$${\color{red}{\int{1 d v}}} = {\color{red}{v}}$$

Recall that $v=\operatorname{asinh}{\left(\frac{2 \sqrt{3} u}{3} \right)}$:

$${\color{red}{v}} = {\color{red}{\operatorname{asinh}{\left(\frac{2 \sqrt{3} u}{3} \right)}}}$$

Recall that $u=x + \frac{1}{2}$:

$$\operatorname{asinh}{\left(\frac{2 \sqrt{3} {\color{red}{u}}}{3} \right)} = \operatorname{asinh}{\left(\frac{2 \sqrt{3} {\color{red}{\left(x + \frac{1}{2}\right)}}}{3} \right)}$$

Therefore,

$$\int{\frac{1}{\sqrt{x^{2} + x + 1}} d x} = \operatorname{asinh}{\left(\frac{2 \sqrt{3} \left(x + \frac{1}{2}\right)}{3} \right)}$$

Simplify:

$$\int{\frac{1}{\sqrt{x^{2} + x + 1}} d x} = \operatorname{asinh}{\left(\frac{\sqrt{3} \left(2 x + 1\right)}{3} \right)}$$

Add the constant of integration:

$$\int{\frac{1}{\sqrt{x^{2} + x + 1}} d x} = \operatorname{asinh}{\left(\frac{\sqrt{3} \left(2 x + 1\right)}{3} \right)}+C$$

$\int \frac{1}{\sqrt{x^{2} + x + 1}}\, dx = \operatorname{asinh}{\left(\frac{\sqrt{3} \left(2 x + 1\right)}{3} \right)} + C$A