While doing research in statistics, sometimes I came across complex integration that I can't or don't want to solve analytically.
Is it okay to use Mathematica to solve the integration and use the result in my research? Do I need to report this in the academic article? My qualm is that perhaps I am supposed to be able to solve these things by hand.
(Some context: I'm from the social sciences and thus unfamiliar with what's permitted regarding these tools.)
Is it okay to use Mathematica to solve the integration and use the result in my research?
Of course it is.
It might be useful, though, to make the Mathematica code employed for the calculations available to others, either through a public repository or by a note in the paper suggesting to contact the author(s). This will allow reviewers or other readers to check your calculations, or, possibly, reuse or extend them.
In a paper where I made extensive use of Mathematica, I wrote the following note at the end:
To those interested, the authors can provide the Mathematica notebooks of the full calculations developed in this work.