The normal conditional density's normalizing constant is not free of lambda, so Wolfram is integrating the wrong thing.
$$
 (2\pi)^{-1}\int_0^\infty (2\pi\tau^2\lambda^2)^{-1/2}\exp\left[ - \frac{\beta^2}{2\tau^2\lambda^2} \right](1 + \lambda^2)^{-1} d \lambda.
$$