## 2 A Brief Review of the Shapiro Delay Formulae and the Aims of This Paper
### Comparison with Some Definitions from Standard Textbooks  
In the textbook [41], a statement is expressed (although without any proof) that \"higher-order elliptic integrals in the Legendre form can be expressed by means of the three standard integrals  
\[\int\frac{dy}{\sqrt{(1-y^{2})(1-q^{2}y^{2})}}\,\ \ \int\frac{y^{2}dy}{\sqrt{(1-y^{2})(1-q^{2}y^{2})}},\]  
\[\int\frac{dy}{(1+hy^{2})\sqrt{(1-y^{2})(1-q^{2}y^{2})}}, \tag{52}\]  
where the parameter \(h\) can be also a complex number\". As asserted in [41], these integrals \"cannot be expressed through elementary functions\". This assertion is not precise, because in the preceding sections, an example was found, when the second integral in (52) in fact can be expressed through an elementary function--for this case, by the logarithmic term in Equation (73). A similar statement to the one in [41] is given also in the monograph [38]: \"Integrals of the type (48) \(\int R(y,\sqrt{G(y)})dy\) and \(\int R(x,\sqrt{P(x)})dx\) cannot be expressed by elementary functions in their final form, even if an extended understanding of this notion is considered\".  
In the contemporary monograph [34], a statement is expressed in a correct and precise manner, namely, \"Every elliptic integral can be represented in the form of a linear combination of a rational function of the variables \(y\) and \(\widetilde{y}\), integrals of a rational function of the variable \(y\) and also the integrals  
\[\int\frac{dy}{\sqrt{G(y)}},\int\frac{ydy}{\sqrt{G(y)}},\]  
\[\int\frac{y^{2}dy}{\sqrt{G(y)}},\int\frac{dy}{(y-c)\sqrt{G(y)}}^{n}. \tag{53}\]  
In this statement, it is not mentioned that elliptic integrals cannot be expressed by elementary functions. One new analytical algorithm for expressing elliptic integrals in the Legendre form will be proposed in the next sections. Since the monographs [38; 41] are older monographs, this example clearly shows that some basic facts about elliptic functionsand integrals cannot be considered to be established forever and the theory may change in time. One basic and contemporary monograph on elliptic functions is [42] (see, especially, ch. 8, dedicated to elliptic integrals). A short summary of the various types of elliptic integrals and functions is given in Appendix A of the book [43], as well as applications of elliptic functions in some problems of biophysics.  
Definitions of Elliptic Integrals of the First, of the Second and of the Third Kind and of Zero Order  
These integrals will be defined, because they were already encountered in the calculations of the propagation time of the signal, emitted by a satellite moving along a plane elliptical orbit.  
The elliptic integrals of the first, second and third kind and of the zero order (see ch. 8 of the monograph by Armitage and Eberlein [42] and also [34]) are partial cases of the more generally defined integrals (50) and (51) The three types of integrals are  
\[\int\frac{dx}{\sqrt{(1-x^{2})(1-q^{2}x^{2})}}=\int\limits_{0}^{\phi}\frac{d\phi }{\sqrt{1-q^{2}\sin^{2}\phi}}=F(q,\phi), \tag{54}\]  
\[\int\frac{\sqrt{1-q^{2}x^{2}}}{\sqrt{1-x^{2}}}dx=\int\sqrt{1-q^{2}\sin^{2}\phi }d\phi=E(q,\phi), \tag{55}\]  
\[\int\frac{dx}{(1+nx^{2})\sqrt{(1-x^{2})(1-q^{2}x^{2})}}=\int\frac{d\phi}{(1+n \sin^{2}\phi)\sqrt{1-q^{2}\sin^{2}\phi}}=\Pi(q,\phi). \tag{56}\]  
In (54), \(F(q,\phi)\) is the commonly accepted notation for an elliptic integral of the first kind, in (55) \(E(q,\phi)\) is the notation for an elliptic integral of the second kind and, in (56), \(\Pi(q,\phi)\) is the notation for the elliptic integral of the third kind. In the above equalities, the transformation \(x=\sin\phi\) has been used.  
The contemporary definition of the elliptic integrals (50) for \(J_{n}^{(4)}(y)\) and \(J_{n}^{(3)}(x)\) and (51) for \(H_{n}^{(4)}(y)\) and \(H_{n}^{(3)}(x)\) in the monograph [34] is fully consistent with the definition in older monographs, such as [44].