You're asking how to verify a definite integration? As far as I know, the only way to verify definite integration (you've thrown away information) is to redo it in a way that you trust. If you don't trust Mathematica's symbolic integration, you can always try to do it by hand, use a different solver, or integrate various controlled approximations to it.
Let's see what you're doing. Translating your text into an expression:
Xm1[nu_, delta_] :=
TransformedDistribution[x^(-1),
x \[Distributed] NoncentralChiSquareDistribution[nu, delta]]
To calculate the Mean we're doing something like:
Integrate[ x PDF[Xm1[nu, delta]][x], {x, 0, \[Infinity]},
Assumptions -> delta >= 0 && nu > 0]
Which indeed lands on your answer conditional upon nu>2
.
Of course one way of disproving it is to have it fail consistency checks, like take $\nu$ and $\delta$ to regions where the mean should vanish or become singular. If your expression doesn't vanish / become singular in these limits, you should be suspicious.
As a verification game you can play using mathematica (or by hand): you can integrate a series expansion around vanishing delta to say third order:
approxIntegrand=Assuming[x > 0,
x * Normal[Series[PDF[Xm1[nu, delta]][x], {delta, 0, 3}]] //
ExpandAll // Collect[#, delta, FullSimplify] &]
(2^(-2 - nu/2) delta E^(-(1/2)/x) x^(-1 - nu/2) (1 - nu x))/
Gamma[1 + nu/2] + (
2^(-5 - nu/2) delta^2 E^(-(1/2)/x)
x^(-2 - nu/2) (1 + (2 + nu) x (-2 + nu x)))/Gamma[2 + nu/2] + (
2^(-7 - nu/2) delta^3 E^(-(1/2)/x)
x^(-3 - nu/2) (1 + (4 + nu) x (-3 + (2 + nu) x (3 - nu x))))/(
3 Gamma[3 + nu/2]) + (2^(-nu/2) E^(-(1/2)/x) x^(-nu/2))/Gamma[nu/2]
and match it against the same series expansion of your result.
Of course this isn't a proof -- it merely demonstrates consistency in a region you have control over.
Also I'm not sure if typing: Mean[Xm1[mu,delta]]
precisely counts as intense symbolic computations ;-). But, as an aside, in Version 11.1 OS X at least, when I evaluate:
Assuming[delta >= 0 && nu > 2, Mean[Xm1[nu, delta]]]
I don't get your explicit complex dependence, instead it resolves happily to:
-2^(-2 + nu/2) (-delta)^(-nu/2) delta E^(-delta/2) (Gamma[-1 + nu/2] - Gamma[-1 + nu/2, -(delta/2)])
Update: to further discuss "how to justify" as per comment.
I'm not so sure this adjunct is really a Mathematica answer as much as it is "good practices."
As a scientist you are justified to disseminate a result you believe in as much as you track and communicate any qualifications. Numerical support is sufficient for certain problems in certain fields. Professional symbolic software integration can generally be relied upon, but of course even Mathematica has had fairly spectacular bugs, so it's a good idea to cross-check and rederive in a number of ways to the point you're convinced.
My field is full of types that would be easily embarrassed when there's a nice closed form for an integration (in one variable!!) we couldn't eventually work out on our own (especially once we know the answer), even if the original integrand was intimidating and full of special functions. The nice thing is that almost everything intimidating is itself a solution to a differential equation, so if we know there's a closed path there's a hair to pull on. So we'll put extra energy into tracking it down -- relearning high-school math sometimes in the process. Then we'll put extra energy into making sure the end form is as compact/symmetric/etc as possible.
Often times however (and I'm thinking multi-loop Feynman integrals) we do rely on fairly sophisticated technology (like that of the Smirnovs ) and then we have to come up with our own consistency checks which we absolutely discuss and disclose.
The most important thing to do is convince yourself the expression is valid and simplified to a point you won't find embarrassing. Once you're there, depending upon pride/shame/dignity etc, it's possible to cite in following form:
"It is easy to see\footnote{e.g. using Mathematica, \tt{ Mean[xxx]}} that blah blah blah."
In general if you would feel comfortable trusting a table of definite integrals (do people feel comfortable trusting a table of definite integrals??) then you can cite as you would that.
If the only way you know how to get at a result is via code, you should make the algorithm clear, and for the Good of Science$^{\rm TM}$ consider sharing the code directly, and if you have any doubts, consider disclosing what you checked and how you convinced yourself that things were ok in the end. If the algorithm is a black box like Mathematica's symbolic integration then reference the black box. In general for 1D integrals, as a professional, it's probably worthwhile figuring out how to at least sketch out the integration -- convincing yourself it's true. When I get stuck, I ask my friends.
Aside: I had a collaborator with the following strategy: when he was blocking on something possibly silly (math/conceptual/etc) he would start by asking undergrads, then grad students, then postdocs, and eventually if the problem was tricky enough he was having brilliant conversations with his senior colleagues. This is not (merely) as cynically careerist as it might appear; it created a pretty thermal environment. Oftentimes people up and down would learn something they otherwise wouldn't have had he just sat in his room banging his head against the wall.
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