I'm struggling to find the inverse here. I've tried using Wolfram Alpha, and it keeps having errors. I tried computing the integral of the inverse of v(t), but it doesn't give the inverse of x(t).
$$ x\left(t\right)=\int_{0}^{t}v_{x}\left(x\right)dx .$$ $$ v_{x}\left(t\right)=\frac{2}{c_{2}+C_{d}rt} $$
I don't know where to begin. I just need a good starting point, and I think I could figure it out. The derivative of the indefinite integral of v(t) is not the same as x(t), but if I add 3.19962 they become quite similar, but not exact. $$ C_d=0.5, r=1.1455, c_2=0.4 $$
How does the number 3.2 appear, and how can I calculate the inverse without using magic numbers?