1. Identidad hiperbólica:
$\sinh^2 x = \cosh^2 x - 1$.
$I = \int (\cosh^2 x - 1) \cosh^2 x \sinh x \, dx$

2. Cambio de variable:
Sea $u = \cosh x \implies du = \sinh x dx$.
$I = \int (u^2 - 1) u^2 du = \int (u^4 - u^2) du$
$I = \frac{u^5}{5} - \frac{u^3}{3} + C = \frac{1}{5} \cosh^5 x - \frac{1}{3} \cosh^3 x + C$

3. Factorización alternativa:
$I = \frac{1}{5} \cosh^3 x (\cosh^2 x - \frac{5}{3}) = \frac{1}{5} \cosh^3 x (\sinh^2 x + 1 - \frac{5}{3})$
$$I = \frac{1}{5} \left( \sinh^2 x - \frac{2}{3} \right) \cosh^3 x + C$$