5.76 How to simplify hyperbolic expression ?

How to convert

\[ \frac {3+2\sinh (x)^2}{\sinh (x)^2\tanh (x)} \]

to

\[ 3 \coth ^3(x)-\coth (x) \]

restart; 
e := (3+2*sinh(x)^2)/(sinh(x)^2*tanh(x)); 
expand(student[changevar](sinh(x)^2=tanh(x)^2/(1-tanh(x)^2),e));