Integral of $$$\frac{1}{\ln\left(n^{3}\right)}$$$

Find $$$\int \frac{1}{3 \ln\left(n\right)}\, dn$$$.

The input is rewritten: $$$\int{\frac{1}{\ln{\left(n^{3} \right)}} d n}=\int{\frac{1}{3 \ln{\left(n \right)}} d n}$$$.

Apply the constant multiple rule $$$\int c f{\left(n \right)}\, dn = c \int f{\left(n \right)}\, dn$$$ with $$$c=\frac{1}{3}$$$ and $$$f{\left(n \right)} = \frac{1}{\ln{\left(n \right)}}$$$:

$${\color{red}{\int{\frac{1}{3 \ln{\left(n \right)}} d n}}} = {\color{red}{\left(\frac{\int{\frac{1}{\ln{\left(n \right)}} d n}}{3}\right)}}$$

This integral (Logarithmic Integral) does not have a closed form:

$$\frac{{\color{red}{\int{\frac{1}{\ln{\left(n \right)}} d n}}}}{3} = \frac{{\color{red}{\operatorname{li}{\left(n \right)}}}}{3}$$

Therefore,

$$\int{\frac{1}{3 \ln{\left(n \right)}} d n} = \frac{\operatorname{li}{\left(n \right)}}{3}$$

Add the constant of integration:

$$\int{\frac{1}{3 \ln{\left(n \right)}} d n} = \frac{\operatorname{li}{\left(n \right)}}{3}+C$$

$$$\int \frac{1}{3 \ln\left(n\right)}\, dn = \frac{\operatorname{li}{\left(n \right)}}{3} + C$$$A