Deriving this "familiar arccosine form" of integral

[Odious Suspect]

I posted a question about this yesterday, but realized I had made a stupid mistake in my derivation.
Orbital dynamics: "The familiar arc-cosine form"
That error has been corrected. I still have a deeper question. I believe this expression can be developed using the geometry of an ellipse in a way which adds insight into the original physics where it is being applied.

Some 30 odd years ago I saw something of this nature done in an introductory physics course.

Is anybody here familiar with such a development?

$$-\int \frac{1}{\sqrt{a+2 bx-hx^2}} \, dx=\frac{1}{\sqrt{h}}\arccos \left[\frac{b-hx}{\sqrt{a+b^2h}}\right]$$

 

[blue_leaf77]

I am not sure what to do with your post, do you want to prove the above equation or what?

 

[Odious Suspect]

No. I can prove it. I want a geometric development. That is to say, I believe the constants and variables in the expression $cos \theta = \frac{b-hx}{\sqrt{a+b^2h}}$ have some geometric significance pertaining to an ellipse.

 

[robphy]

possibly useful:
https://math.berkeley.edu/~peyam/Math1AFa10/Arccos.pdf

 

[Odious Suspect]

That only addresses the form I really am familiar with. Notice that the form I am asking about is more complicated.

 

[robphy]

That only addresses the form I really am familiar with. Notice that the form I am asking about is more complicated.

Yes, I know that you are asking about something more complicated.
Might it be the case that the method used in the simple case can be generalized? I don't know. I offered a possible starting point.
Sorry, if I didn't provide you with exactly what you seek.

 

[Odious Suspect]

Sorry about being so abrupt. I get grouchy when I can't figure something out.

I'm guessing this might be something to do with rotation of coordinates to eliminate the "cross product" term.

In some universe, the denominator in the cosine expression represents a radius.