How to compute this integral containing Bessel function and log function？

Question

Now I would like compute the integral $$\int_y^\infty\frac{1}{x^2}Y_{l+\frac{1}{2}}(x)Y_{l+\frac{1}{2}}(x)\ln x\,\, dx$$ where $y>0$, and $l$ is positive integer. Mathematica can give result, but it contains a divergent term, which contains ${}_2F_3(-l,-l;1-2l,1-l,\frac{3}{2}-l;-y^2)$. This hypergeometric is divergent when $l$ is integer. But numerical integral tells me that this integral is convergent. Therefore, at present, I don't know how to compute this integral or how to transform this divergent hypergeometric function into a convergent term.

Answer 1

Not an answer but too long for a comment.

$$I_k=\int_y^\infty \frac 1 {x^2}\Big[Y_{\frac{k}{2}+l}(x)\Big]^2 \log(x)\,dx$$ is convergent if $y>0$. The problem is, as you noticed, when the solution express in terms of this hypergeometric function.

For the first $$16 \pi y^4\,I_1=8 y^4 \text{Ci}(2 y)+4 y^2+8 y^2 \log (y)+2 y \left(-2 y^2+4 \log (y)+1\right) \sin (2 y)+$$ $$\left(2 y^2+4 \log (y)+1\right) \cos (2 y)+4 \log (y)+1$$

For the second $$48 \pi y^6\,I_2=8 y^6 \text{Ci}(2 y)+3 \left(4 y^4+3 y^2+4 \left(2 y^4+3 y^2+6\right) \log (y)+4\right)+$$ $$2 y \left(-2 y^4+y^2-12 \left(y^2-6\right) \log (y)+12\right) \sin (2 y)+$$ $$\left(2 y^4-15 y^2+36 \left(2-3 y^2\right) \log (y)+12\right) \cos (2 y)$$

I have been able to compute a few more but I never obtained the general form you give.