[VA] Short & Sweet Math Challenge #25 "San Valentin's Special: Weird Math"

[J-F Garnier]

I don't know the reason but I'm particularly attracted by the integral problems (like this previous challenge).
So I tested the "weird" integral:

(02-14-2021 08:58 PM)Valentin Albillo Wrote:  

where Γ is the Gamma function, ln is the natural logarithm (i.e., base e) and φ is the Golden Ratio = (1+ √5)/2.

The expression to integrate looks complicate, although without any trap, it is not defined at the integral boundaries but that's not a problem with the Romberg algorithm.
I took my old trusted HP-32S and entered the program:

LBL I
RCL X
LN
1
-
x!
RCL F
x²
RCL X
-
LN
1
-
x!
+
LASTx
x<>y
/
RTN

Then:
FIX 11
put the golden ratio in F: 5 SQRT 1 + 2 / STO F
FN= I
1
RCL F
∫FN dX
and quickly got the answer up to at least 11 places.

So what's special? It was fast and easy, without any problem.
Oh wait, it was too easy, too fast. That's weird.

[...Investigating a bit...]

Now that I think I found what is "weird", I can even do the calculation by hand, and get the symbolic result without having to identify the numeric result:
[... hidden for now ...]

J-F