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...
I just thought about this one, suppose we have the classic double slit experiment setting, and we add near the holes in the slits a strong magnetic field, what will happen? what if the magnetic field was only on one slit? What if we put that magnetic field in place (2) instead of (1), would that...