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But if one applies multivariate elliptic or hypergeometric functions it was established in the late 1800s and early 1900s that one can in principle reach any algebraic number. … Multiple integrals of rational functions can be more complicated, as in
Integrate[1/(1 + x 2 + y 2 ), {x, 0, 1}, {y, 0, 1}] HypergeometricPFQ[{1/2, 1, 1}, {3/2, 3/2}, 1/9]/6 + 1/2 π ArcSinh[1] - Catalan
and presumably often cannot be expressed at all in terms of standard mathematical functions.

(Examples of more difficult cases include HypergeometricPFQ[a, b, 1] and StieltjesGamma[k] , where logarithmic series can require an exponential number of terms.

Note that for any j the 6j symbols can be given in terms of HypergeometricPFQ .)