This interesting paper derives a possibly better and simpler approximation for integrating the optical depth and attenuation of (presumably) sunlight through an atmosphere.  The mathematics look clean, but it is still an approximation, it has errors, it is not a closed-form analytic solution.

The real problem is that the method relies on too many approximations (g = constant, H = constant, no refraction!).  In real-use situations, the model atmosphere has many distinct layers with highly varying composition and opacity (think clouds).  Current methods of calculating spherical atmospheres are decades beyond this research – see references below. 

I do not see how this paper can be published here.  It does not contribute to the current science of calculating the attenuation of sunlight in atmospheres.  Revisions would not improve this rating.

K.E. Trenberth, L. Smith (2005) The mass of the atmosphere: A constraint on global analyses, J. Clim. 18, 864–875.

M.J. Prather, J.C. Hsu (2019) A round Earth for climate models, Proc Natl Acad Sci, 116 (39) 19330-19335.

The manuscript "Analytical approximation of the definite Chapman integral for arbitrary zenith angle" by Dongxiao Yue is an interesting analytical approximation of the Chapman mapping function. The Chapman function arises naturally when one studies atmospheres in hydrostatic equilibrium characterized by spherical symmetry. Although both conditions are not met when one is dealing with realistic planetary atmospheres, the Chapman function remains a handy mapping function in several branches of atmospheric physics, such as ionospheric physics, radiative transfer theory, radiation absorption by planetary atmospheres, to name just a few. Compared to the simpler secant mapping function (which is also widely used), the Chapman mapping function is not divergent when the zenith angle argument tends to pi/2.

On the other hand, the Chapman mapping function is represented in the form of the integral, so the exact analytic approximation for it is desirable. 

Dongxiao Yue proposes one of the possible approximate expressions in the manuscript. This is actually done for a more general mapping function as given in the form of Eq. (6), with what I would call the incomplete Chapman mapping function. The results obtained are scientifically sound and the derived approximation works quite well. The author discusses in detail the asymptotic behavior of the proposed approximation and proposes the asymptotic expression suitable for computer computations. In summary, the manuscript presents an elegant analytical result that approximates the incomplete Chapman mapping function, and I recommend the manuscript for publication.

Some minor comments are listed below:

1. Line 40. The variable y is not shown in Fig. 1, and its actual introduction does not occur until Eq. (9). The sentence "As illustrated..." is therefore confusing.

2. Line 52. I would move the sentence "Since the thickness..." to line 55. This is logically connected to the next statement "Since $\lambda$ is large...".

3. Line 53. One more explanation of why rewriting (9) in the approximate form of (10) is necessary. Probably the author can add some words why the expression with -sin z term remains under the integral sign and the expression with +sin z term is simplified as given in (10).

4. Line 91. In the sentence "At the same time, the exponential factor in the equation..." it is necessary to specify "At the same time, the exponential factor in the second term inside the brackets of Eq. (11)...".