Let's say we have two tubes with charges $q_1$, $q_2$, radii $b_1$, $b_2$ and lengths $l_1$ and $l_2$. They are placed along the surface of each other like in this figure:
To calculate the electrostatic force between these tubes, we first have to find the electrostatic force between two rings since tubes are made out of infinite infinitesimal rings. The force between two infinitesimal rings is; $$ F= \frac {dq_1 q_2 E[\frac{-(4b_1 b_2)}{(d^2+(b_1-b_2)^2 })]}{2π^2 ε_0 (d^2+(b_1+b_2)^2) \sqrt{(d^2+(b_1-b_2)^2 )}} $$ where $d$ is the distance between the centres of the rings, $b_1$ is the radius of the first ring and $b_2$ is the radius of the other ring. $E$ is the complete elliptic integral of the second kind. How should I modify and integrate this equation to find the force between two tubes? I just want to know what would be the equation if we use the integral. You don't have to solve the integral. Thanks.