Integrating Modified Bessel Function of Second Kind

Question

I am trying to show that $$ \int t(K_s(t))^2dt = \frac{1}{2}t^2(K_s(t))^2-\frac{1}{2}t^2K_{s-1}(t)K_{s+1}(t). $$ It is taken from this link https://functions.wolfram.com/Bessel-TypeFunctions/BesselK/21/01/04/01/01/01/0004/. I am not even sure that this is correct. I tried integration by parts to get \begin{align*} \int t(K_s(t))^2dt &= \frac{1}{2}t^2(K_s(t))^2-\frac{1}{2}\int t^2\cdot 2K_s(t)\cdot \frac{-1}{2}(K_{s-1}(t)+K_{s+1}(t))dt\\ &=\frac{1}{2}t^2(K_s(t))^2 +\frac{1}{2}\int t^2K_s(t)(K_{s-1}(t)+K_{s+1}(t))dt. \end{align*} I am stuck here and still figuring out how to proceed.