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

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

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

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

The integral can be rewritten as

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

Apply the constant multiple rule $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ with $$$c=-1$$$ and $$$f{\left(u \right)} = \frac{1}{\sqrt{u^{2} + u + 1}}$$$:

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

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

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

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

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

So,

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

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

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

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

Therefore,

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

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

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

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

Therefore,

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

Apply the constant rule $$$\int c\, dw = c w$$$ with $$$c=1$$$:

$$- {\color{red}{\int{1 d w}}} = - {\color{red}{w}}$$

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

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

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

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

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

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

Therefore,

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

Simplify:

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

Add the constant of integration:

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

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