How to solve the double integration then gradient of this function?

I want to double integrate this function and then gradient (finding H) the answer from double integration and run the range value of ρ and z by fixing the value of ϕto see the characteristic of H as a graph. I'm new to this and have no idea what to do whether it's possible or not. Thank you very much.

[Torsten]

You can differentiate under the integral.