Integral of $$$\frac{\operatorname{atan}{\left(x \right)}}{4 x^{2}}$$$

Find $$$\int \frac{\operatorname{atan}{\left(x \right)}}{4 x^{2}}\, dx$$$.

Apply the constant multiple rule $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ with $$$c=\frac{1}{4}$$$ and $$$f{\left(x \right)} = \frac{\operatorname{atan}{\left(x \right)}}{x^{2}}$$$:

$${\color{red}{\int{\frac{\operatorname{atan}{\left(x \right)}}{4 x^{2}} d x}}} = {\color{red}{\left(\frac{\int{\frac{\operatorname{atan}{\left(x \right)}}{x^{2}} d x}}{4}\right)}}$$

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 becomes

$$\frac{{\color{red}{\int{\frac{\operatorname{atan}{\left(x \right)}}{x^{2}} d x}}}}{4} = \frac{{\color{red}{\int{\left(- \operatorname{acot}{\left(u \right)}\right)d u}}}}{4}$$

Apply the constant multiple rule $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ with $$$c=-1$$$ and $$$f{\left(u \right)} = \operatorname{acot}{\left(u \right)}$$$:

$$\frac{{\color{red}{\int{\left(- \operatorname{acot}{\left(u \right)}\right)d u}}}}{4} = \frac{{\color{red}{\left(- \int{\operatorname{acot}{\left(u \right)} d u}\right)}}}{4}$$

For the integral $$$\int{\operatorname{acot}{\left(u \right)} d u}$$$, use integration by parts $$$\int \operatorname{\omega} \operatorname{dv} = \operatorname{\omega}\operatorname{v} - \int \operatorname{v} \operatorname{d\omega}$$$.

Let $$$\operatorname{\omega}=\operatorname{acot}{\left(u \right)}$$$ and $$$\operatorname{dv}=du$$$.

Then $$$\operatorname{d\omega}=\left(\operatorname{acot}{\left(u \right)}\right)^{\prime }du=- \frac{1}{u^{2} + 1} du$$$ (steps can be seen ») and $$$\operatorname{v}=\int{1 d u}=u$$$ (steps can be seen »).

The integral becomes

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

Let $$$v=u^{2} + 1$$$.

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

Thus,

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

Apply the constant multiple rule $$$\int c f{\left(v \right)}\, dv = c \int f{\left(v \right)}\, dv$$$ with $$$c=- \frac{1}{2}$$$ and $$$f{\left(v \right)} = \frac{1}{v}$$$:

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

The integral of $$$\frac{1}{v}$$$ is $$$\int{\frac{1}{v} d v} = \ln{\left(\left|{v}\right| \right)}$$$:

$$- \frac{u \operatorname{acot}{\left(u \right)}}{4} - \frac{{\color{red}{\int{\frac{1}{v} d v}}}}{8} = - \frac{u \operatorname{acot}{\left(u \right)}}{4} - \frac{{\color{red}{\ln{\left(\left|{v}\right| \right)}}}}{8}$$

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

$$- \frac{u \operatorname{acot}{\left(u \right)}}{4} - \frac{\ln{\left(\left|{{\color{red}{v}}}\right| \right)}}{8} = - \frac{u \operatorname{acot}{\left(u \right)}}{4} - \frac{\ln{\left(\left|{{\color{red}{\left(u^{2} + 1\right)}}}\right| \right)}}{8}$$

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

$$- \frac{\ln{\left(1 + {\color{red}{u}}^{2} \right)}}{8} - \frac{{\color{red}{u}} \operatorname{acot}{\left({\color{red}{u}} \right)}}{4} = - \frac{\ln{\left(1 + {\color{red}{\frac{1}{x}}}^{2} \right)}}{8} - \frac{{\color{red}{\frac{1}{x}}} \operatorname{acot}{\left({\color{red}{\frac{1}{x}}} \right)}}{4}$$

Therefore,

$$\int{\frac{\operatorname{atan}{\left(x \right)}}{4 x^{2}} d x} = - \frac{\ln{\left(1 + \frac{1}{x^{2}} \right)}}{8} - \frac{\operatorname{acot}{\left(\frac{1}{x} \right)}}{4 x}$$

Simplify:

$$\int{\frac{\operatorname{atan}{\left(x \right)}}{4 x^{2}} d x} = \frac{- x \left(- 2 \ln{\left(x \right)} + \ln{\left(x^{2} + 1 \right)}\right) - 2 \operatorname{atan}{\left(x \right)}}{8 x}$$

Add the constant of integration:

$$\int{\frac{\operatorname{atan}{\left(x \right)}}{4 x^{2}} d x} = \frac{- x \left(- 2 \ln{\left(x \right)} + \ln{\left(x^{2} + 1 \right)}\right) - 2 \operatorname{atan}{\left(x \right)}}{8 x}+C$$

$$$\int \frac{\operatorname{atan}{\left(x \right)}}{4 x^{2}}\, dx = \frac{- x \left(- 2 \ln\left(x\right) + \ln\left(x^{2} + 1\right)\right) - 2 \operatorname{atan}{\left(x \right)}}{8 x} + C$$$A