The following integral showed up in a physics problem:$$\frac{\mathrm d}{\mathrm dz}\int^1_z\mathrm dt\,\frac{1}{(t(t-z))^{1/6}}\bigg(\frac{1}{(1+\sqrt t )^{1/3}}+\frac{1}{(1-\sqrt t)^{1/3}}\bigg),$$ where $z\in[0,1]$.
I am interested in an analytical solution. The integral clearly diverges if the Leibniz rule is naively applied. I can get $z$ out of the integration limits by changing variables to $s=\dfrac{1-z}{1-t}-1$ and am left with an integral over $s\in[0,\infty)$. Unfortunately, I could not solve that integral either. Mathematica does not return a solution.
Can anyone see how to approach this integral?
Thanks.