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

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

Complete the square (for steps, visit the completing the square calculator): $$$x^{2} + x + 1 = \left(x + \frac{1}{2}\right)^{2} + \frac{3}{4}$$$:

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

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

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

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

Thus,

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

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:

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

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