t = var('t')
s = var('s')
a = var('a')
w = var('w')

show(laplace(exp(- a * t), t, s))

show(laplace(sin(w * t), t, s))

show(laplace(cos(w * t), t, s))

inverse_laplace(s / (s^2 + 1), s, t)

inverse_laplace(1 / (s^2 + 1), s, t)

inverse_laplace(1 / s, s, t)

inverse_laplace(1 / s^2, s, t)

inverse_laplace(1 / (s + 1), s, t)

inverse_laplace(1, s, t)

ex = (s^3 + 4 * s^2 + 6 * s + 4) / (s^3 + 3 * s^2 + 3 * s + 1)
show(ex)

show(inverse_laplace(ex * 1 / s, s, t))
