Assuming that random variables X and Y are independent, what is $\displaystyle Var((1+X)(1+Y)-1)=Var(X+Y+XY)$?

Should I start as follows
\begin{equation}
Var((1+X)(1+Y)-1)\\
=Var((1+X)(1+Y))\\
=(E[(1+X)])^2 Var(1+Y)+(E[(1+Y])^2 Var(1+X)+Var(1+X)Var(1+Y)
\end{equation}

or maybe as follows

\begin{equation}
\\
Var((1+X)(1+Y)-1)\\
=Var(1+Y+X+XY-1)\\
=Var(X+Y+XY)\\
=Var(X)+Var(Y)+Var(XY)+2Cov(X,Y)+2Cov(X,XY)+2Cov(Y,XY)
\end{equation}

I'm considering could I express the problem in terms of covariances (and variances) between individual random variables. I would like to forecast the variance by individual covariances in my model if its possible. Does to solution simplify if expected values of the variables are zero?