Damped harmonic Oscillator Lagrangian equivalence

The objective is to prove that the Lagrangian:
$$L’=frac{2dot x+lambda x}{2Omega x}tan^{-1}(frac{2dot x+lambda x}{2Omega x})-frac{1}{2}ln(dot x^2+lambda dot{x } x + omega^2x^2), qquad Omega=sqrt{omega^2-lambda^2/4},$$
is equivalent to the lagrangian of the damped harmonic oscillator:
$$L=e^{lambda t}(frac{m}{2}dot x^2 -frac{momega^2}{2}x^2), $$
but I dont know how to show that there is a time derivative of a function that differs from one Lagrangian to the other;

(It’s exercise 2.14 from Nivaldo Lemos, Analytical Mechanics, 2018.)