We can solve the integral $\int\mathrm{coth}\left(3x\right)dx$ by applying integration by substitution method (also called U-Substitution). First, we must identify a section within the integral with a new variable (let's call it $u$), which when substituted makes the integral easier. We see that $3x$ it's a good candidate for substitution. Let's define a variable $u$ and assign it to the choosen part

Substituting $u$ and $dx$ in the integral and simplify

Take the constant $\frac{1}{3}$ out of the integral

We can solve the integral $\int\mathrm{coth}\left(u\right)du$ by applying integration by parts method to calculate the integral of the product of two functions, using the following formula

Now, identify $dv$ and calculate $v$

Solve the integral to find $v$

The integral of a constant is equal to the constant times the integral's variable

Multiply the single term $\frac{1}{3}$ by each term of the polynomial $\left(u\mathrm{coth}\left(u\right)+\int u\mathrm{csch}\left(u\right)^2du\right)$

Multiply the fraction and term in $3\cdot \frac{1}{3}x\mathrm{coth}\left(u\right)$

Gather the results of all integrals

Cancel like terms $x\mathrm{coth}\left(3x\right)$ and $-x\mathrm{coth}\left(3x\right)$

As the integral that we are solving is an indefinite integral, when we finish integrating we must add the constant of integration $C$