Indeed there is such an expression, which is dependent on the value of the <PERSON> metric at various values of the coordinate $r$ and does not actually involve complicated integrals at all! We need to be careful in how we talk about distances here, because the radial coordinate does not, strictly speaking, correspond to the distance from some "center point" of the black hole. Recall that the <PERSON> metric, which describes the spacetime outside of any spherically symmetric mass distribution, is given by
$$g_{\mu \nu} = \text{diag}\left(-\left(1-\frac{r_s}{r}\right), \left(1-\frac{r_s}{r}\right)^{-1}, r^2, r^2\sin^2(\theta)\right)$$
Just to be clear, I am using a metric with signature (-+++), and $r_s$ is the Schwarzschild radius, $r_s \equiv \frac{2GM}{c^2}$ (with $G$ Newton's gravitational constant, $M$ the mass of the star/black hole/whathaveyou, $c$ the speed of light). Now we can consider the proper times experienced by a stationary observer at infinity and an observer orbiting the black hole. When we take the radial coordinate to infinity, this metric reduces to that for a flat (<PERSON>) spacetime, and the observer here will have a proper time equivalent to the coordinate time. To find how an observer in orbit around the black hole, at some finite radius, will measure time, we need to consider that observer's 4-velocity.
The 4-velocity is given by $$u^{\mu} = \left(\gamma c, \gamma \vec{v}\right)$$ where $\gamma$ is the Lorentz coefficient defined by $\gamma = \frac{1}{\sqrt{1-\frac{v^2}{c^2}}}$ and $\vec{v}$ is the familiar velocity vector from Newtonian mechanics. We can also rewrite this 4-velocity as a derivative of the particle's position with respect to a new variable which we call proper time, $\tau$. This proper time is the time experienced by the particle whose motion we are considering. 483
In this form $u^{\mu}$ is simply $\frac{d}{d\tau} x^{\mu}$ where $x^{\mu} = (t, x, y, z)$ or $(t, r, \theta, \phi)$, etc. depending on your coordinates, is the position vector of the particle. It is this aforementioned $\gamma$ coefficient which will allow us to compare the passage of an observer's proper time to the coordinate time that a distant observer will measure, through the formula $dt = \gamma d\tau$. What is particularly nice about this 4-velocity is that we can form a relativistic invariant by taking the inner product between this vector and itself $$u_{\mu} u^{\mu} = g_{\mu \nu} u^{\mu} u^{\nu} \equiv -c^2 $$
Here I am using the <PERSON> summation convention, which allows us to omit the summation symbol with the implied understanding that whenever we find a pair of upper and lower indices, we sum over those values from 0 to 3. What we really want is to write this out in an explicit coordinate form so that we can solve for the time dilation factor.
$$-\gamma^2 c^2 \left(1-\frac{r_s}{r}\right) + \left(\frac{dr}{d\tau}\right)^2 \left(1 - \frac{r_s}{r}\right)^{-1} + r^2 \left(\frac{d\theta}{d\tau}\right)^2 + r^2 \sin^2(\theta)\left(\frac{d\phi}{d\tau}\right)^2 = -c^2 $$
Now this may look daunting, but we can simplify this expression. All particle orbits in the Schwarzschild metric occur in a plane, and we are free to choose the orientation of our coordinate axes so that the orbits we are considering occur in the $\theta = \pi/2$ plane. Moreover, this value of $\theta$ is constant and therefore independent of the proper time $\tau$, so we can throw out the term $r^2 \left(\frac{d\theta}{d\tau}\right)^2$. At the moment I am not willing to jump in and perform the necessary work for orbits where $r$ may vary (non-circular orbits). 818