If you could find a closed form for ##a(x)## then you'd be able to find any closed form formula for any primitive of any function, by putting ##f(x)=\ln g(x)## in your formula, but that's impossible by Louisville's theorem.
I started reading Baby Rudin with @VKnopp for some weeks, but unfortunately, I was confiscated any access to the Internet for many months, I'm only back now. Is anyone still interested in such a project?
Also anhttp://www.uni-graz.at/~gronau/TMCS_1_2003.pdf that attempts to answer: "Why is ##\Gamma(n)=(n-1)!## and not ##\Gamma(n)=n!##?"
Edit: Some other papers:
Wladimir de Azevedo Pribitkin, Laplace's Integral, the Gamma Function, and beyond, American Mathematical Monthly.
Gopala Krishna...
That's because $$\frac{1}{\cos x+i\sin x}=\frac{\cos x-i\sin x}{(\cos x +i\sin x)(\cos x-i\sin x)}=\ldots$$ Then use the Pythagorean identity.
I would say: it isn't safe.
Here are some references I found out on reddit: (I don't know if that's advanced enough for your level)
Irresistible Integrals: Symbolics, Analysis and Experiments in the Evaluation of Integrals, by George Boros and Victor Moll.
Chapter 11 deals with the Gamma and Beta functions.
Paul Nahin...