How to perform this integration symbolically?

Hello, I need to evaluate the following integral expression symbolically using Integrate:

-(Sqrt[Pi]*Integrate[E^u^2*(-4*Erf[u]+4*Erf[u]^2)*(Erfc[u]-(2*u*(E^(-u^2)-Sqrt[Pi]*u*Erfc[u]))/Sqrt[Pi]), {u, 0,Infinity}])/2

I have reasons to expect that the result of this integration is equal 4/Pi - 1, which can be verified numerically after replacing Integrate by NIntegrate. Unfortunately, Integrate seems unable to obtain this result symbolically. Is there any way to change this situation? Leslaw

Version 13.3 gives the result in an instant.