Derivative of function containing parametric integral

Question

I have the following function:

$$p_{D}(t_{D}) = \frac{4}{\pi^2}\int\limits_0^\infty\frac{1-\exp\bigl(-u^2t_{D}\bigr)}{u^3\bigl(J_{1}^2(u) + Y_{1}^2(u)\bigr)}\,du $$

for which I need the derivative $p_{D}'(t_{D})$. I am stumped so would appreciate any help anyone could give. Thanks!

Answer 1

After some internet searching I came across this link

It seems that Liebnitz got here some time ago, so following the info in the link, the analytical derivative of my function is
.
I checked the resulting derivative by comparing it to the derivative of the spline of the pD function and the two match pretty closely: