Expectation of maximum of two lognormal random variables (another view)

Following Expected Value of Maximum of Two Lognormal Random Variables . I made an additional assumption, that $$X,Y$$ are jointly normal with parameters given above.

I tried to derive the simmilar result by first finding the density function of $$max(X,Y)$$, and I have the following CDF: $$P(\max(X,Y) < m ) = P(X < m, Y < m) = \Phi(\frac{(\log(m) - \mu)} {\sigma}, \frac{(\log(m) - \nu)} {\tau})$$, where $$\Phi$$ is the CDF of multivariate standard normal.

Then, I computed the density function which is $$\phi_{max} = \frac{1}{\sigma \times m}\phi_1(\frac{(\log(m) - \mu)} {\sigma}) + \frac{1}{\tau \times m} \phi_2(\frac{(\log(m) - \nu)} {\tau})$$, where $$\phi_1$$ and $$\phi_2$$ are marginal distribution functions of joint multivariate normal, which in turn are just densities of related normal.

Since I know, the distribution function at each $$m$$, then I can compute its expectation. I form $$E(max(X,Y)) = \int_0^{\infty}(m \times \phi_{max} dm) = \int_0^{\infty} m \times \frac{1}{\sigma \times m}\phi_1(\frac{(\log(m) - \mu)} {\sigma}) dm + \int_0^{\infty} m \times \frac{1}{\tau \times m} \phi_2(\frac{(\log(m) - \nu)} {\tau}) dm = \int_0^{\infty} \frac{1}{\sigma}\phi_1(\frac{(\log(m) - \mu)} {\sigma}) dm + \int_0^{\infty} \frac{1}{\tau} \phi_2(\frac{(\log(m) - \nu)} {\tau}) dm = \exp(\mu + \frac{\sigma^2}{2}) + \exp(\nu + \frac{\tau^2}{2})$$. Seemingly, I have made mistake when considering marginal distributions but cannot see where exactly.

Where I have got a flaw in my derivations?