Math Gravity Offset Circle

In this calculation we compute the gravitational attraction on a point mass above a circle, where the point mass is *NOT* on the axis of the circle.  It results in an elliptical integral that cannot be integrated.

Instead we use Desmos graphing calculator to estimate the integral and understand the forces as the point is set along different offsets, both in the X-Y plane of the circle, and slightly above the circle. 

Set up some basic equations.  

Get an expression for the infinitesimal Vector force.  

Do the integration/summation of the forces along the path of the circle.  

Compute the force in the x direction Fx when the point mass is offset from the Z-axis and moved to a position along the X-axis.   Position along the X-axis is (Xnot, 0, Znot).  This is a tricky integration. 
Note: the force in the y direction is zero by symmetry.

Compute the force in the Z direction - when the point mass is along the Z-axis (the axis of the circle).  

Below is a graphic of the force experienced by the point mass as its distance various along the Z-axis. 
The point mass has both x and y as zero.  Created by Desmos calculator.  

Below is a graphic of the force experienced by the point mass if the point mass is in the X-Y plane.
The horizontal axis is X. The vertical axis is the force experienced in the X direction.
Graphic created by Desmos integration calculator. 

Below is a graphic of the force experienced by the point mass if the point mass is at a distance of 0.5 units above the X-Y plane.  The horizontal axis is X. The vertical axis is the force experienced in the X direction.

This is the most fascinating graphic, and may be somewhat counterintuitive.
Notice how the direction of the force in the X direction swings back and forth.
I thought the calculations were wrong at first.  Maybe there is a mistake somewhere?
Graphic created by Desmos integration calculator.