Integral of squared Bessel function

Question

I have been trying to find the solution to the integral

\begin{equation} \int J_0^2(x) dx \end{equation} I tried a lot of stuff with known formulas for integrals of Bessel functions (see Abramowitz & Stegun (1972)) but haven't been successful...

Answer 1

The answer can easily be given in Wolfram Alpha, but that would feel almost like cheating. Instead, let’s use @Gary’s link on DLMF 1.8.3 or 16.12.1 and the Generalized Hypergeometric function $\,_2F_3$. Note how the bottom of the DLMF page gives a hypergeometric representation comment. In theory, integration of a hypergeometric function is mainly another hypergeometric function:

$$\int J_0^2(x)dx=\int\,_0F_1(a,x)^2dx=\int\,_0F_1^2\left(1,-\frac{x^2}{4}\right)dx=\int\,_2F_3\left(1,\frac12,1,1,1,-x^2\right)dx=x\,_2F_3\left(\frac12,\frac12,1,1,\frac32,-x^2\right)+C$$

Note there are many alternate form for the answer like series and related functions. Try to simplify the solution if you like using the bolded link, but I have found nothing that has simplified for now. Please also see this Antiderivative of the $\,_2F_3$ function used here. Unfortunately, the area under the entire integrand, over the reals, is infinite. The takeaway is to put the function into hypergeometric form and integrate. Also note that there may be a typo. Please correct me and give me feedback