Answer
$$
\int \frac{\operatorname{coth}(1 / y)}{y^{2}} d y=-\ln |\sinh (1 / y)|+C
$$
Work Step by Step
$$
\int \frac{\operatorname{coth}(1 / y)}{y^{2}} d y
$$
If we make the substitution
$$
u=1 / y, \quad d u=-1 / y^{2} d y
$$
and we look at the Table of Integrals , we see that the closest entry is number $106$ :
$$
\begin{aligned}
\int \frac{\operatorname{coth}(1 / y)}{y^{2}} d y &=\int \operatorname{coth} u(-d u) \quad\left[\begin{array}{c}
u=1 / y, \\
d u=-1 / y^{2} d y
\end{array}\right] \\
& \stackrel{106}{=}-\ln |\sinh u|+C \\
&=-\ln |\sinh (1 / y)|+C
\end{aligned}
$$
