Firstly, by analytically integrate, I mean, is there an integration rule to solve this as opposed to numerical analyses (such as trapezoid, gauss-legendre or simpsons rules)?

I have a function $f(x) = x*lognormal(mu, sigma, x)$. Mu and sigma remain constant, while x varies.

I need to integrate f(x) over a positive range (a, b).

Currently, I'm doing this with numerical analyses using the gauss-legendre method. Because I need to run this a large number of times, performance is important. Before I look into optimizing the numerical analyses/other pieces, I would like to know if there are any integration rules to solve this.

I tried applying the integration by parts rule, and I got to this, where I'm stuck again -

1. $\int udv = uv - \int(vdu)$

2. $u=x => du = dx$

3. $dv = lognormal(x)dx => v = lognormalCDF(x)$

4. $uv - \int vdx = x*lognormalCDF(x) - \int lognormalCDF(x)dx$

I'm stuck, as I can't evaluate the $\int lognormalCDF(x)dx$.

This is for a software package I'm building.