Consider the following series:
$$S=\sum_{n=1}^\infty\frac{(-1)^n\ \Gamma\left(\frac{5}{4}+n\right)}{n^2\ \Gamma(n)}.$$
It can be expressed in terms of a hypergeometric function:
$$S=-\frac{5}{16}\Gamma\left(\frac{1}{4}\right)\ { _3F_2}\left(1,1,\frac{9}{4};2,2;-1\right).$$
I tried to find an expression of $S$ using elementary functions and ended up with this conjecture:
$$S\stackrel{?}{=}\frac{\Gamma\left(\frac{1}{4}\right)}{16}\Bigg(8\sqrt[4]{8}-16-\ln\frac{24\sqrt{69708+49291\sqrt{2}}+3168\sqrt{2}+4481}{4096}\\\\+\arctan\frac{24\sqrt{49291\sqrt{2}-13260}}{6913}\Bigg).$$My derivation of this formula is quite long and uses a non-rigorous, heuristic approach involving heavy use of techniques like guessing sequence formulas using *Mathematica* command FindSequenceFunction and OEIS superseeker, and recognizing approximate numeric quantities using RootApproximant, TranscendentalRecognize, Inverse Symbolic Calculator and *Mathematica* command

```
WolframAlpha[ToString[value], IncludePods -> "PossibleClosedForm", TimeConstraint -> ∞]
```

The formula holds with at least $1800$ decimal digits of precision.

Now I am looking for a rigorous way to prove this formula. Can you suggest any ideas how to do that?