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Dynamics of a Discrete-Time Predator–Prey System with Holling II Functional Response

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Abstract

The dynamics behavior of a discrete-time predator–prey system, with Holling II functional response, is analyzed. The model shows a rich dynamical behavior in the feasible region. Some invariant sets are found and parameter conditions for the existence and stability of the fixed points are given. A parameter region where the system exhibits either a period-doubling or a Neimark–Sacker bifurcation is shown. In addition, conditions are provided on parameters that lead to chaotic dynamics. Finally, to illustrate our theoretical analysis some numerical simulations are shown.

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Acknowledgements

The first author thanks CONACYT for the scholarship granted. We thank to PAPIIT IA203922. We thank the referees for their valuable comments to improve this manuscript.

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Authors and Affiliations

  1. División Académica de Ciencias Básicas, UJAT, Km 1 Carretera Cunduacán-Jalpa de Méndez, C.P. 86690, Cunduacán, Tabasco, Mexico

    Carlos F. Arias & Gamaliel Blé

  2. Departamento de Matemáticas, Facultad de Ciencias, UNAM, C.P. 04510, C. Universitaria, C. de México, Mexico

    Manuel Falconi

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  1. Carlos F. Arias
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  2. Gamaliel Blé
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Appendices

Expressions \({f_1 (u,\overline{e},v)}\) and \({f_2(u,\overline{e},v)}\) of Map (21)

$$\begin{aligned} f_1 (u,\overline{e},v)= & {} (a (-2 (-3 + a) (1 + c)^2 + (1 + c) (-5 + a + (-1 + a) c) d - (3 + a\\&+ (-1 + a) c) d^2) u^2)/( d (-(-1 + a) (-1 + c^2)\\&+ (3 + a + (-1 + a) c) d)) - ( a (1 + c) (1 + c - d) (-1 + c^2\\&- (3 + c) d) u v)/( d^2 (-(-1 + a) (-1 + c^2)\\&+ (3 + a + (-1 + a) c) d)) - ( a^2 (1 + c)^2 (1 + c - d) (1 - c^2\\&+ (3 + c) d)^2 u^2 v)/( d^3 ((-1 + a) (-1 + c^2)\\&- (3 + a + (-1 + a) c) d)^2) - ( 2 (-3 + a) a (1 + c) (2\\&+ 2 c - d) u \overline{e})/( d (-(-1 + a) (-1 + c^2)\\&+ (3 + a + (-1 + a) c) d)) - ( 2 (-3 + a) a^2 (1 + c)^2 (3\\&+ 3 c - 2 d) (-1 + c^2 - (3 + c) d) u^2 \overline{e})/( d^2 ((-1 + a) (-1 + c^2)\\&- (3 + a + (-1 + a) c) d)^2) + (-(( a (1 + c)^2 (-1 + c^2 \\&- (3 + c) d))/(d^2 (-(-1 + a) (-1 + c^2)+ (3 + a + (-1 + a) c) d)))\\&- ( a^2 (1 + c)^2 (2 + 2 c - d) \\&\times (1 - c^2 + (3 + c) d)^2 u)/( d^3 ((-1 + a) (-1 + c^2)\\&- (3 + a + (-1 + a) c) d)^2) - ( a^3 (1 + c)^3 (3 + 3 c - 2 d) (-1 \\&+ c^2- (3 + c) d)^3 u^2)/( d^4 (-(-1 + a) (-1 + c^2)\\&+ (3 +a + (-1 + a) c) d)^3)) v \overline{e}\\&- ( 2 (-3 + a) a (1 + c)^2 \overline{e}^2)/( d (-(-1 + a) (-1 + c^2)\\&+ (3 + a + (-1 + a) c) d)) - ( 2 (-3 + a) a^2 (1 + c)^2 (3 + 3 c \\&- d) (-1 + c^2- (3 + c) d) u \overline{e}^2)/( d^2 ((-1 + a) (-1 + c^2)\\&- (3 + a + (-1 + a) c) d)^2) - ( 6 (-3 + a) a^3 (1 + c)^3 (2 + 2 c \\&- d) (1 - c^2+ (3 + c) d)^2 u^2 \overline{e}^2)/( d^3 (-(-1 + a) (-1 + c^2)\\&+ (3 + a + (-1 + a) c) d)^3) + (-(( a^2 (1 + c)^3 (1 - c^2\\&+ (3 + c) d)^2)/( d^3 ((-1 + a) (-1 + c^2)\\&- (3 + a + (-1 + a) c) d)^2)) - ( a^3 (1 + c)^3 (3 + 3 c - d) (-1 + c^2\\&- (3 + c) d)^3 u)/( d^4 (-(-1 + a) (-1 + c^2)\\&+ (3 + a + (-1 + a) c) d)^3) - ( 3 a^4 (1 + c)^4 (2 + 2 c - d) (1 - c^2\\&+ (3 + c) d)^4 u^2)/( d^5 ((-1 + a) (-1 + c^2)\\&- (3 + a + (-1 + a) c) d)^4)) v \overline{e}^2+ {{\varvec{O}}}((|X|+|\overline{e}|+|Y|)^{3}). \end{aligned}$$
$$\begin{aligned} f_2 (u,\overline{e},v)= & {} -((2 (-3 + a) a (1 + c) (1 + c - d) u^2)/((-1 + a) (-1 + c^2)\\&- (3 + a + (-1 + a) c) d)) + ( a (1 + c) (1 + c - d) (-1 + c^2 \\&- (3 + c) d) u v)/(d (-(-1 + a) (-1 + c^2) + (3 + a + (-1 + a) c) d)) \\&+ (a^2 (1 + c)^2 (1 + c - d)\\&\times (1- c^2 + (3 + c) d)^2 u^2 v)/(d^2 ((-1 + a) (-1 + c^2) \\&- (3 + a + (-1 + a) c) d)^2)- (2 (-3 + a) a (1 + c) (2 + 2 c\\&-d) u \overline{e})/((-1 + a) (-1 + c^2) - (3 + a + (-1 + a) c) d)\\&+ ( 2 (-3 + a) a^2 (1 + c)^2 (3 + 3 c - 2 d)\\&\times (-1 + c^2 - (3 + c) d) u^2 \overline{e})/( d ((-1 + a) (-1 + c^2)\\&- (3 + a + (-1 + a) c) d)^2) + (( a (1 + c)^2 (-1 + c^2\\&- (3 + c) d))/( d (-(-1 + a) (-1 + c^2) + (3 + a + (-1 + a) c) d))\\&+ ( a^2 (1 + c)^2 (2 + 2 c - d) \\&\times (1 - c^2 + (3 + c) d)^2 u)/( d^2 ((-1 + a) (-1 + c^2)\\&- (3 + a + (-1 + a) c) d)^2) + ( a^3 (1 + c)^3 (3\\&+ 3 c - 2 d) (-1 + c^2 - (3 + c) d)^3 u^2)/( d^3 (-(-1 + a) (-1 + c^2)\\&+ (3 + a + (-1 + a) c) d)^3)) v \overline{e} - ( 2 (-3 + a) a (1\\&+ c)^2 \overline{e}^2)/((-1 + a) (-1 + c^2) - (3 + a + (-1 + a) c) d)\\&+ ( 2 (-3 + a) a^2 (1 + c)^2 (3 + 3 c - d) \\&\times (-1 + c^2 - (3 + c) d) u \overline{e}^2)/( d ((-1 + a) (-1 + c^2)\\&- (3 + a + (-1 + a) c) d)^2) + ( 6 (-3 + a) a^3 (1 + c)^3\\&\times (2 + 2 c - d) (1 - c^2\\&+ (3 + c) d)^2 u^2 \overline{e}^2)/( d^2 (-(-1 + a) (-1 + c^2) \\&+ (3 + a + (-1 + a) c) d)^3) + (( a^2 (1 + c)^3 (1 - c^2\\&+ (3 + c) d)^2)/( d^2 ((-1 + a) (-1 + c^2) - (3 + a + (-1 + a) c) d)^2)\\&+ ( a^3 (1 + c)^3 (3 + 3 c - d)\\&\times (-1 + c^2 - (3 + c) d)^3 u)/( d^3 (-(-1 + a) (-1 + c^2)\\&+ (3 + a + (-1 + a) c) d)^3)\\&+ ( 3 a^4 (1 + c)^4 (2 + 2 c - d) \\&\times (1 - c^2 + (3 + c) d)^4 u^2)/( d^4 ((-1 + a) (-1 + c^2)\\&- (3 + a + (-1 + a) c) d)^4)) v \overline{e}^2+ {{\varvec{O}}}((|X|+|\overline{e}|+|Y|)^{3}). \end{aligned}$$

Expressions \({F_1(X, \mu ,Y)}\) and \({F_2(X, \mu ,Y)}\) of Map (22)

$$\begin{aligned} F_1(X, \mu ,Y)= & {} (a (1 - c^2 + (3 + c) d)^2 ((-3 + a) (-1 + c^2)^2\\&- 2 (-1 + a + (-3 + a) c) (-1 + c^2) d \\&+ (9 + a (1 + c)^2 - c (2 + 3 c)) d^2) X^2)/((-3 + a) (1 + c\\&- d) d^2 ((1 + c) (-1 + a + (-5 + a) c) \\&- (-9 + a + (-5 + a) c) d) ((-1 + a) (-1 + c^2)\\&- (3 + a + (-1 + a) c) d)) + ( a (1 + c)\\&\times ((7 + a (-3 + c) - c) (-1 + c)^2 (1 + c)^3\\&- (-1 + c) (1 + c)^2 (15 - 11 a + 12 c + 3 (-1 + a) c^2) d \\&+ 3 (-1 + c) (1 + c) (3 + c) (3 + a + (-1 + a) c) d^2\\&- (3 + c)^2 (3 + a + (-1 + a) c) d^3) X Y)/( 2 d^2 (-(1 + c) \\&\times (-1 + a + (-5 + a) c) \\&+ (-9 + a + (-5 + a) c) d) (-(-1 + a) (-1 + c^2)\\&+ (3 + a + (-1 + a) c) d)) - ( a^2 (-1 + c) (1 + c)^2\\&\times (-2 (2 + c (3 + c - d)) \\&+ a (1 + c) (1 + c - d)) (-1 + c^2 - (3 + c) d)^3 X^2 Y)/((-3 + a) d^4\\&\times (-(1 + c) (-1 + a + (-5 + a) c) \\&+ (-9 + a + (-5 + a) c) d) ((-1 + a) (-1 + c^2) \\&- (3 + a + (-1 + a) c) d)^2) + ((-3 + a) a (1 + c)^3 (1 \\&+ c - d) (-(-2 + a - c) (-1 + c^2)\\&- 2 (-1 + c) (2 + c) d + (a + c) d^2) Y^2)/( 2 d^2 (-(1 + c)\\&\times (-1 + a + (-5 + a) c) \\&+ (-9 + a + (-5 + a) c) d) (-(-1 + a) (-1 + c^2) \\&+ (3 + a + (-1 + a) c) d)) \\&- ( a^2 (-1 + c) (1 + c)^3 (1 + c - d) (1 - c^2\\&+ (3 + c) d)^2 ((1 + c) (-7 + a + c + a c) \\&- (9 + a + c + a c) d) X Y^2)/( 4 d^4 (-(1 + c) (-1 + a + (-5 + a) c) \\&+ (-9 + a + (-5 + a) c) d) ((-1 + a) (-1 + c^2) \\&- (3 + a + (-1 + a) c) d)^2) - ( a (1 + c) (-1 + c^2 \\&- (3 + c) d)^3 X \mu )/( d^2 (-(1 + c) (-1 + a + (-5 + a) c)\\&+ (-9 + a + (-5 + a) c) d) ((-1 + a) (-1 + c^2) \\&+ (-9 + a + c - a c) d)) - ( a^2 (-1 + c) (1 + c)^2 (1 - c^2\\&+ (3 + c) d)^4 ((1 + c) (-17 - c + a (5 + c))\\&- 2 (a + c) d - (-3 + a) d^2) X^2 \mu )/((-3 + a) d^4 (-(1 + c)\\&\times (-1 + a + (-5 + a) c) + (-9 + a\\&+ (-5 + a) c) d) ((-1 + a) (-1 + c^2) \\&- (3 + a + (-1 + a) c) d)^2 ((-1 + a) (-1 + c^2) \\&+ (-9 + a + c - a c) d))\\&- ((-3 + a) a (1 + c)^2 (1 + c - d) (1 - c^2\\&+ (3 + c) d)^2 Y \mu )/( 2 d^2 (-(1 + c) (-1 + a + (-5 + a) c) + (-9 + a\\&+ (-5 + a) c) d) ((-1 + a) (-1 + c^2) \\&+ (-9 + a + c - a c) d)) - ( a^2 (-1 + c) (1 + c)^2 (1 + c - d) (-1\\&+ c^2 - (3 + c) d)^3 (-2 (1 + c)^2 (17 + c) \\&+ (1 + c)^2 d + (-3 + c) d^2 \\&+ a (1 + c) (2 (1 + c) (5 + c) - (5 + c) d \\&- d^2)) X Y \mu )/( 2 d^4 (-(1 + c) (-1 + a + (-5 + a) c)\\&+ (-9 + a + (-5 + a) c) d) ((-1 + a) (-1 + c^2) \\&- (3 + a + (-1 + a) c) d)^2 ((-1 + a) (-1 + c^2) \\&+ (-9 + a + c - a c) d)) - ( a^3 (-1 + c) (1 + c)^3\\&\times (-2 (2 + c (3 + c - d)) + a (1 + c) (1 + c - d)) (3 + 3 c - 2 d) (-1 \\&+ c^2 - (3 + c) d)^5 X^2 Y \mu )/((-3 + a) d^6 (-(1 + c) \\&\times (-1 + a + (-5 + a) c) + (-9 + a\\&+ (-5 + a) c) d) (-(-1 + a) (-1 + c^2) \\&+ (3 + a + (-1 + a) c) d)^3 ((-1 + a) (-1 + c^2) \\&+ (-9 + a + c - a c) d))\\&- ((-3 + a) a^2 (-1 + c) (1 + c)^3 (1 \\&+ c - d)^2 (1 - c^2 + (3 + c) d)^2 ((1 + c)^2 (-17 - c + a (5 + c))\\&- (1 + c) (-1 - 3 c + a (3 + c)) d - 2 (3 + c) d^2) Y^2 \mu )/\\&\times ( 4 d^4 (-(1 + c) (-1 + a + (-5 + a) c) + (-9 + a\\&+ (-5 + a) c) d) ((-1 + a) (-1 + c^2) \\&- (3 + a + (-1 + a) c) d)^2 ((-1 + a) (-1 + c^2)\\&+ (-9 + a + c - a c) d)) - ( a^3 (-1 + c) (1 + c)^4\\&\times (3 + 3 c - 2 d) (1 + c - d) (1 \\&- c^2 + (3 + c) d)^4 ((1 + c) (-7 + a + c + a c)\\&- (9 + a + c + a c) d) X Y^2 \mu )/( 4 d^6 (-(1 + c) (-1 + a + (-5 + a) c) \\&+ (-9 + a\\&+ (-5 + a) c) d) (-(-1 + a) (-1 + c^2) \\&+ (3 + a + (-1 + a) c) d)^3 ((-1 + a) (-1 + c^2) \\&+ (-9 + a + c - a c) d))\\&+ ( 2 (-3 + a) a (1 + c)^2 (1 + c - d) (1 - c^2\\&+ (3 + c) d)^2 \mu ^2)/( d (-(1 + c) (-1 + a + (-5 + a) c)\\&+ (-9 + a + (-5 + a) c) d) ((-1 + a) (-1 + c^2) \\&+ (-9 + a + c - a c) d)^2)\\&- ( a^2 (-1 + c) (1 + c)^2 (1 + c - d) (1 - c^2\\&+ (3 + c) d)^4 (6 (-3 + a) (1 + c)^2\\&+ (-1 + a) (-1 + c^2) d\\&- (3 + a + (-1 + a) c) d^2) X \mu ^2)/( d^4 (-(1 + c) (-1 + a + (-5 + a) c)\\&+ (-9 + a + (-5 + a) c) d) ((-1 + a) (-1 + c^2)\\&- (3 + a + (-1 + a) c) d)^2 ((-1 + a) (-1+ c^2)\\&+ (-9 + a + c - a c) d)^2) - ( a^3 (-1 + c) (1 + c)^3 (1 - c^2\\&+ (3 + c) d)^6 (6 (1 + c)^2 (-7 \\&- 2 c + a (2 + c)) - 2 (1 + c) (-17 - 7 c + 4 a (2 + c)) d \\&+ (1 + a) (3 + c) d^2\\&+ (-3 + a) d^3) X^2 \mu ^2)/((-3 + a) d^6 (-(1 + c) (-1 + a + (-5 + a) c) \\&+ (-9 + a+ (-5 + a) c) d) (-(-1 + a) (-1 + c^2) \\&+ (3 + a + (-1 + a) c) d)^3 ((-1 + a) (-1 + c^2) \\&+ (-9 + a + c - a c) d)^2)\\&- ((-3 + a) a^2 (-1 + c) (1 + c)^3 (1 + c - d)^2 (-1 + c^2\\&- (3 + c) d)^3 (6 (-3 + a) (1 + c)^2 \\&+ (-1 + a) (-1 + c^2) d\\&- (3 + a + (-1 + a) c) d^2) Y \mu ^2)/( 2 d^4 (-(1 + c) (-1 + a + (-5 + a) c) \\&+ (-9 + a+ (-5 + a) c) d) ((-1 + a) (-1 + c^2) \\&- (3 + a + (-1 + a) c) d)^2 ((-1 + a) (-1 + c^2) \\&+ (-9 + a + c - a c) d)^2) - ( a^3 (-1 + c) (1 + c)^3 (1 + c - d) (-1\\&+ c^2 - (3 + c) d)^5 (3 (1 + c)^3 (-3 (9 + c) \\&+ a (7 + 3 c)) \\&- (1 + c)^2 (-41 + 29 a + 13 (-1 + a) c) d\\&+ (1 + c) (5 + 7 a + 3 (-1 + a) c) d^2 \\&+ (3 + a + (-1 + a) c) d^3) X Y \mu ^2)/( 2 d^6 (-(1 + c) (-1 + a + (-5 + a) c) \\&+ (-9 + a + (-5 + a) c) d) (-(-1 + a) (-1 + c^2) + (3 + a\\&+ (-1 + a) c) d)^3 ((-1 + a) (-1 + c^2) \\&+ (-9 + a + c - a c) d)^2)\\&- ( 3 a^4 (-1 + c) (1 + c)^4 (-2 (2 + c (3 + c - d))\\&+ a (1 + c) (1 + c - d)) (1 + c - d) (2 + 2 c - d) (-1+ c^2\\&- (3 + c) d)^7 X^2 Y \mu ^2)/((-3 + a) d^8 (-(1 + c) \\&\times (-1 + a + (-5 + a) c) + (-9 + a\\&+ (-5 + a) c) d) ((-1 + a) (-1 + c^2) - (3 \\&+ a + (-1 + a) c) d)^4 ((-1 + a) (-1 + c^2) + (-9 + a + c\\&- a c) d)^2) - ((-3 + a) a^3 (-1 + c) (1 + c)^4 (1 + c \\&- d)^2 (1 - c^2 + (3 + c) d)^4 (3 (1 + c)^3 (-13 + c + a (3 + c)) - (1 \\&+ c)^2 (-7 + c + a (13 + 5 c)) d + 2 (1 + c) (1 - 2 c + a (2 + c)) d^2\\&+ 2 (3 + c) d^3) Y^2 \mu ^2)/( 4 d^6 (-(1 + c) (-1 + a + (-5 + a) c) \\&+ (-9 + a+ (-5 + a) c) d) (-(-1 + a) (-1 + c^2) \\&+ (3 + a + (-1 + a) c) d)^3 ((-1 + a) (-1 + c^2)\\&+ (-9 + a + c - a c) d)^2) - ( 3 a^4 (-1 + c) (1 + c)^5 \\&\times (1 + c - d)^2 (2 + 2 c - d) (1 - c^2\\&+ (3 + c) d)^6 ((1 + c) (-7 + a + c + a c) - (9 + a + c\\&+ a c) d) X Y^2 \mu ^2)/( 4 d^8 (-(1 + c) (-1 + a + (-5 + a) c) + (-9 \\&+ a+ (-5 + a) c) d) ((-1 + a) (-1 + c^2) - (3 \\&+ a + (-1 + a) c) d)^4 ((-1 + a) (-1 + c^2)\\&+ (-9 + a + c - a c) d)^2) + {{\varvec{O}}}((|X|+|\mu |+|Y|)^{3}). \end{aligned}$$
$$\begin{aligned} F_2 (X, \mu ,Y)= & {} -((2 a (1- c^2 + (3 + c) d)^2 (-(-2 + a - c) (-1 + c) (1 + c)^2 + (1\\&+ c) (3 + a + (-7 + a) c - 2 c^2) d + (3 + c)^2 d^2) X^2)/((-3\\&+ a) (1 + c - d) d^2 ((1 + c) (-1 + a + (-5 + a) c) - (-9\\&+ a + (-5 + a) c) d) ((-1 + a) (-1 + c^2) - (3\\&+ a + (-1 + a) c) d))) - ( a (1 + c) (-1\\&+ c^2 - (3 + c) d) ((7 + a (-3 + c) - c) (1 + c)^2 (2 - a + c)\\&- 2 (7 + a (-3 + c) - c) (1 + c) (3 + c) d + (a + c) (3\\&+ a + (-1 + a) c) d^2) X Y)/((-3\\&+ a) d^2 (-(1 + c) (-1 + a + (-5 + a) c) + (-9\\&+ a + (-5 + a) c) d) (-(-1 + a) (-1 + c^2) + (3\\&+ a + (-1 + a) c) d)) + ( 2 a^2 (1 + c)^2 (-2 (2 + c (3 + c - d))\\&+ a (1 + c) (1 + c - d)) (-1 + c^2 - (3 + c) d)^3 ((-2 + a - c) (1 + c) + (6 - a\\&+ c) d) X^2 Y)/((-3 + a)^2 (1 + c\\&- d) d^4 ((1 + c) (-1 + a + (-5 + a) c) - (-9\\&+ a + (-5 + a) c) d) ((-1 + a) (-1 + c^2) - (3\\&+ a + (-1 + a) c) d)^2) + (a (1\\&+ c)^2 (-2 (1 + c)^3 (2 - a + c)^2 + (1 + c)^2 (41 + a^2 (3 + c)\\&+ c (37 + 6 c) - 2 a (13 + 7 c)) d\\&- 2 (1 + c) (30 + a^2 (1 + c) - 4 a (3 + 2 c)\\&+ c (25 + 3 c)) d^2 + (-6 a (1 + c)\\&+ a^2 (1 + c) + (9 + c) (3 + 2 c)) d^3) Y^2)/(2 d^2 (-(1\\&+ c) (-1 + a + (-5 + a) c) + (-9 + a + (-5 + a) c) d) (-(-1 + a) (-1 + c^2) \\&+ (3 + a + (-1 + a) c) d)) - ( a^2 (1 + c)^3 (1\\&- c^2 + (3 + c) d)^2 ((-2 + a - c) (1 + c) + (6 - a + c) d) ((1\\&+ c) (-7 + a + c + a c)\\&- (9 + a + c + a c) d) X Y^2)/( 2 (-3 + a) d^4 (-(1 + c) (-1 + a + (-5 + a) c)\\&+ (-9 + a + (-5 + a) c) d) ((-1 + a) (-1 + c^2) - (3 + a + (-1 + a) c) d)^2)\\&- ( 2 a (1 + c) (1 - c^2 + (3 + c) d)^2 ((-2 + a - c) (1 + c) + (a + c) d) X \mu )/((-3\\&+ a) d^2 (-(1 + c) (-1 + a + (-5 + a) c) + (-9 + a \\&+ (-5 + a) c) d) ((-1 + a) (-1 + c^2)\\&+ (-9 + a + c - a c) d)) + ( 2 a^2 (1 + c)^2 (1\\&- c^2 + (3 + c) d)^4 ((-2 + a - c) (1 + c) \\&+ (6 - a + c) d) ((1 + c) (-17 - c + a (5 + c))\\&- 2 (a + c) d - (-3 + a) d^2) X^2 \mu )/((-3 + a)^2 (1 + c - d) d^4 \\&\times ((1 + c) (-1 + a + (-5 + a) c)\\&- (-9 + a + (-5 + a) c) d) ((-1 + a) (-1 + c^2) - (3 + a\\&+ (-1 + a) c) d)^2 ((-1 + a) (-1 + c^2) + (-9 + a + c - a c) d)) \\&- ( a (1 + c)^2 (1 + c - d) (-1\\&+ c^2 - (3 + c) d) ((-2 + a - c) (1 + c) \\&+ (a + c) d) Y \mu )/( d^2 (-(1 + c) (-1 + a + (-5 + a) c)\\&+ (-9 + a + (-5 + a) c) d) ((-1 + a) (-1 + c^2)\\&+ (-9 + a + c - a c) d)) - ( a^2 (1 + c)^2 (-1\\&+ c^2 - (3 + c) d)^3 ((-2 + a - c) (1 + c) + (6 - a + c) d) (-2 (1 + c)^2 (17 + c)\\&+ (1 + c)^2 d + (-3 + c) d^2 + a (1 + c) (2 (1 + c) (5 + c) \\&- (5 + c) d - d^2)) X Y \mu )/((-3\\&+ a) d^4 (-(1 + c) (-1 + a + (-5 + a) c) + (-9\\&+ a + (-5 + a) c) d) ((-1 + a) (-1 + c^2) - (3\\&+ a + (-1 + a) c) d)^2 ((-1 + a) (-1 + c^2) + (-9 + a + c\\&- a c) d)) - ( 2 a^3 (1 + c)^3 (-2 (2 + c (3 + c - d))\\&+ a (1 + c) (1 + c - d)) (3 + 3 c - 2 d) (-1\\&+ c^2 - (3 + c) d)^5 ((-2 + a - c) (1 + c) + (6 - a\\&+ c) d) X^2 Y \mu )/((-3 + a)^2 (1 + c - d) d^6 ((1 + c) (-1 + a \\&+ (-5 + a) c) - (-9+ a + (-5 + a) c) d) ((-1 + a) (-1 + c^2) - (3\\&+ a + (-1 + a) c) d)^3 ((-1 + a) (-1 + c^2) + (-9 + a + c - a c) d))\\&- ( a^2 (1 + c)^3 (1 + c - d) (1 - c^2 + (3 + c) d)^2 ((-2 + a - c) (1 + c)\\&+ (6 - a + c) d) ((1 + c)^2 (-17 - c + a (5 + c)) - (1 + c) (-1 - 3 c\\&+ a (3 + c)) d - 2 (3 + c) d^2) Y^2 \mu )/( 2 d^4 (-(1 + c) (-1 + a + (-5 + a) c)\\&+ (-9 + a + (-5 + a) c) d) ((-1 + a) (-1 + c^2) - (3\\&+ a + (-1 + a) c) d)^2 ((-1 + a) (-1 + c^2) + (-9 + a + c - a c) d))\\&- ( a^3 (1 + c)^4 (3 + 3 c - 2 d) (1 - c^2\\&+ (3 + c) d)^4 ((-2 + a - c) (1 + c) + (6 - a + c) d) ((1 + c) (-7 + a + c + a c)\\&- (9 + a + c + a c) d) X Y^2 \mu )/( 2 (-3 + a) d^6 (-(1 + c) (-1 + a \\&+ (-5 + a) c) + (-9 + a\\&+ (-5 + a) c) d) (-(-1 + a) (-1 + c^2) + (3 + a\\&+ (-1 + a) c) d)^3 ((-1 + a) (-1 + c^2) + (-9 + a + c - a c) d))\\&- ( 2 (-3 + a) a (1 + c)^2 (1 + c - d) (1 - c^2\\&+ (3 + c) d)^2 \mu ^2)/( d (-(1 + c) (-1 + a + (-5 + a) c) \\&+ (-9 + a + (-5 + a) c) d) ((-1\\&+ a) (-1 + c^2) + (-9 + a + c - a c) d)^2) - ( 2 a^2 (1 + c)^2 (1\\&- c^2 + (3 + c) d)^4 ((-2 + a - c) (1 + c) + (6 - a + c) d) (6 (-3\\&+ a) (1 + c)^2 + (-1 + a) (-1 + c^2) d - (3 + a\\&+ (-1 + a) c) d^2) X \mu ^2)/((-3 + a) d^4 (-(1 + c) (-1 + a + (-5 + a) c) + (-9\\&+ a + (-5 + a) c) d) ((-1 + a) (-1 + c^2) - (3 + a\\&+ (-1 + a) c) d)^2 ((-1 + a) (-1 + c^2) \\&+ (-9 + a + c - a c) d)^2) - ( 2 a^3 (1 + c)^3 (1\\&- c^2 + (3 + c) d)^6 ((-2 + a - c) (1 + c) \\&+ (6 - a + c) d) (6 (1 + c)^2 (-7 - 2 c + a (2 + c))\\&- 2 (1 + c) (-17 - 7 c + 4 a (2 + c)) d + (1 + a) (3 + c) d^2\\&+ (-3 + a) d^3) X^2 \mu ^2)/((-3 + a)^2 (1 + c - d) d^6 ((1 + c) (-1 + a + (-5 + a) c)\\&- (-9 + a + (-5 + a) c) d) ((-1 + a) (-1 + c^2) - (3\\&+ a + (-1 + a) c) d)^3 ((-1 + a) (-1 + c^2) \\&+ (-9 + a + c - a c) d)^2) - ( a^2 (1 + c)^3 (1 + c - d) (-1\\&+ c^2 - (3 + c) d)^3 ((-2 + a - c) (1 + c)\\&+ (6 - a + c) d) (6 (-3 + a) (1 + c)^2 + (-1 + a) (-1 + c^2) d\\&- (3 + a + (-1 + a) c) d^2) Y \mu ^2)/( d^4 (-(1 + c) (-1 + a + (-5 + a) c)\\&+ (-9 + a + (-5 + a) c) d) ((-1 + a) (-1 + c^2)\\&- (3 + a + (-1 + a) c) d)^2 ((-1 + a) (-1 + c^2)\\&+ (-9 + a + c - a c) d)^2) - (a^3 (1 + c)^3 (-1 + c^2\\&- (3 + c) d)^5 ((-2 + a - c) (1 + c) \\&+ (6 - a + c) d) (3 (1 + c)^3 (-3 (9 + c) + a (7 + 3 c))\\&- (1 + c)^2 (-41 + 29 a + 13 (-1 + a) c) d + (1 + c) (5 + 7 a + 3 (-1 + a) c) d^2\\&+ (3 + a + (-1 + a) c) d^3) X Y \mu ^2)/((-3 + a) d^6 (-(1 + c)\\&\times (-1 + a + (-5 + a) c) + (-9\\&+ a + (-5 + a) c) d) (-(-1 + a) (-1 + c^2) \\&+ (3 + a + (-1 + a) c) d)^3 ((-1 + a) (-1 + c^2)\\&+ (-9 + a + c - a c) d)^2) - ( 6 a^4 (1 + c)^4 (-2 (2 + c (3 + c - d))\\&+ a (1 + c) (1 + c - d)) (2 + 2 c - d) (-1 + c^2 - (3 + c) d)^7 ((-2 + a - c) (1 + c)\\&+ (6 - a + c) d) X^2 Y \mu ^2)/((-3 + a)^2 d^8 (-(1 + c) (-1 + a + (-5 + a) c)\\&+ (-9 + a + (-5 + a) c) d) ((-1 + a) (-1 + c^2)\\&- (3 + a + (-1 + a) c) d)^4 ((-1 + a) (-1 + c^2) + (-9 + a\\&+ c - a c) d)^2) - (a^3 (1 + c)^4 (1 + c - d) (1 - c^2\\&+ (3 + c) d)^4 ((-2 + a - c) (1 + c)\\&+ (6 - a + c) d) (3 (1 + c)^3 (-13 + c + a (3 + c)) - (1 + c)^2 (-7\\&+ c + a (13 + 5 c)) d + 2 (1 + c) (1 - 2 c\\&+ a (2 + c)) d^2 + 2 (3 + c) d^3) Y^2 \mu ^2)/(2 d^6 (-(1 + c) (-1\\&+ a + (-5 + a) c) + (-9 + a + (-5 + a) c) d) (-(-1 + a) (-1\\&+ c^2) + (3 + a + (-1 + a) c) d)^3 ((-1 + a) (-1 + c^2) + (-9 + a\\&+ c - a c) d)^2) - ( 3 a^4 (1 + c)^5 (1 + c - d) (2 + 2 c - d) (1 - c^2\\&+ (3 + c) d)^6 ((-2 + a - c) (1 + c) + (6 - a + c) d) ((1 + c) (-7 + a + c + a c)\\&- (9 + a + c + a c) d) X Y^2 \mu ^2)/( 2 (-3 + a) d^8 (-(1 + c) (-1 + a + (-5 + a) c)\\&+ (-9 + a + (-5 + a) c) d) ((-1 + a) (-1 + c^2)\\&- (3 + a + (-1 + a) c) d)^4 ((-1 + a) (-1 + c^2)\\&+ (-9 + a + c - a c) d)^2)+ {{\varvec{O}}}((|X|+|\mu |+|Y|)^{3}). \end{aligned}$$

Expressions \(\tau _1,\, \tau _2\) and \(\tau _3\)

$$\begin{aligned} \tau _1= & {} (-3 + a)^2 (-1 + a) (-1 + c) (1 + c)^5 (-1 + a - 5 c + a c) d^2\\&- 2 (-3 + a)^2 (1 + c)^4 (-7 - 9 c + 10 c^2 + a (8 - 12 c^2)\\&+a^2 (-1 + c + 2 c^2)) d^3\\&+ 2 (-3 + a)^2 (1 + c)^3 (-26 - 15 c + 15 c^2 + 3 a^2 c (1 + c)\\&- 2 a (-5 + 6 c + 9 c^2)) d^4\\&- 2 (-3 + a)^2 (1 + c)^2 (-33 - 11 c + 10 c^2 - 4 a c (4 + 3 c)\\&+ a^2 (1 + 3 c + 2 c^2)) d^5 + (-3 + a)^2 (1 + c) (-27 - 6 c\\&+ 5 c^2 - 6 a (1 + c)^2 + a^2 (1 + c)^2) d^6.\\ \tau _2= & {} (3 - 4 a + a^2) (-1 + c) (1 + c)^3 (-1 + a - 5 c + a c)^2 d^2 - (-3\\&+ a) (1 + c)^2 (3 a^3 (-1 + c) (1 + c)^2\\&+ a^2 (33 + 49 c - 17 c^2 - 33 c^3) \\&+ 3 (9 + 57 c + 55 c^2 - 25 c^3)\\&+ a (-57 - 217 c - 23 c^2 + 105 c^3)) d^3 + (-3 + a) (1 + c) (243\\&+ 891 c + 285 c^2 - 75 c^3 + 3 a^3 (-1 + c) (1 + c)^2\\&+ a^2 (57 + 65 c - 25 c^2 - 33 c^3)\\&+ a (-297 - 441 c - 7 c^2 + 105 c^3)) d^4 - (-3 + a) (9\\&+ a (-1 + c) - c) (-9 + a - 5 c + a c)^2 d^5.\\ \tau _3= & {} -(-1 + a)^2 (-1 + c)^2 (1 + c)^3 (-1 + a - 5 c + a c) d + (-1\\&+ a) (-1 + c) (1 + c)^2 (a (30 + 20 c - 18 c^2)\\&+ 3 a^2 (-1 + c^2) + 3 (-9 - 28 c + 5 c^2)) d^2 - (1\\&+ c) (3 a^3 (-1 + c)^2 (1 + c)\\&+ a^2 (-57 + 29 c + 49 c^2 - 21 c^3)\\&- 3 (81 + 99 c - 57 c^2 + 5 c^3)\\&+ a (297 + 15 c - 217 c^2 + 33 c^3)) d^3 + (9 + a (-1 + c)\\&- c)^2 (-9 + a - 5 c + a c) d^4. \end{aligned}$$

Expressions \({h_1,h_2,h_3,h_4,h_5,h_6,h_7,h_8,h_9, h_{10},h_{11},h_{12} }\) and \({h_{13}}\)

$$\begin{aligned} \mathbf{h} _1= & {} -((a (1 - c^2 + 3 d + c d)^2 (-3\\&- 3 c^4 - 2 d + 6 c^3 d + 9 d^2 - 2 c d (3 + d) + a (1 + c)^2 (1 - c + d)^2\\&+ c^2 (6 + 2 d - 3 d^2)))/((-3 + a) (1 + c\\&- d) d^2 (1 - c^2 + a (1 + c) (-1 + c - d) - 3 d + c d) (-1\\&- 5 c^2 + a (1 + c) (1 + c - d) + 9 d + c (-6 + 5 d)))). \end{aligned}$$
$$\begin{aligned} \mathbf{h} _2= & {} (a^2 (1 - c^2 + 3 d + c d)^4 (-14 + c^8\\&+ 33 d + 63 d^2 - 81 d^3 - 81 d^4 - c^7 (3 + 4 d)\\&+ c^6 (-25 + 3 d + 6 d^2) + c^5 (-27 + 84 d + 9 d^2 - 4 d^3)\\&+ c^2 (5 - 135 d + 24 d^2 + 144 d^3)\\&+ c (-33 + 4 d + 153 d^2 + 18 d^3 - 54 d^4)\\&+ c^4 (33 + 99 d - 93 d^2 - 15 d^3 + d^4)\\&+ c^3 (63 - 84 d - 162 d^2 + 34 d^3 + 6 d^4)\\&+ a^2 (1 + c)^3 (-3 + c^4 - 5 d - 5 d^2 - 3 d^3 - c^3 (4 + 3 d)\\&+ c^2 (2 + 5 d + 3 d^2) + c (4 + 3 d + 2 d^2 - d^3))\\&- a (1 + c) (-13 + c^7 + c^6 (1 - 4 d) + 4 d + 42 d^2 + 36 d^3\\&+ 27 d^4 + 3 c^5 (-5 - 4 d + 2 d^2) + c^4 (-15 + 36 d + 30 d^2 - 4 d^3)\\&+ c^3 (27 + 48 d - 28 d^3 + d^4) + 3 c^2 (9 - 12 d - 24 d^2 - 16 d^3 + 3 d^4)\\&+ c (-13 - 36 d - 6 d^2 + 12 d^3 + 27 d^4))))/((-3 + a)^2 (1\\&+ c - d)^2 d^4 (1 - c^2 + a (1 + c) (-1 + c - d) - 3 d\\&+ c d)^2 (-1 - 5 c^2 + a (1 + c) (1 + c - d) + 9 d\\&+ c (-6 + 5 d))^2). \end{aligned}$$
$$\begin{aligned} \mathbf{h} _3= & {} -((a (1 + c) (-1 + c^2 - 3 d - c d)^3)/((1 + 5 c^2\\&+ c (6 - 5 d) - a (1 + c) (1 + c - d) - 9 d) d^2 (-1 + c^2\\&- a (-1 + c) (1 + c - d) + 9 d - c d))). \end{aligned}$$
$$\begin{aligned} \mathbf{h} _4= & {} (a^2 (1 + c) (1 - c^2 + 3 d + c d)^4 (33 - 6 c^10 - 412 d\\&+ 1752 d^2 - 1014 d^3 - 513 d^4 - 486 d^5 + 15 c^9 (-3 + 2 d)\\&+ c^8 (-519 + 142 d - 60 d^2)\\&+ 2 c^7 (-954 + 733 d - 59 d^2 + 30 d^3)\\&- 2 c^6 (1380 - 2861 d + 528 d^2 + 24 d^3 + 15 d^4)\\&+ c^2 (1302 - 7858 d + 7224 d^2 + 468 d^3 + 1080 d^4 - 420 d^5)\\&+ c^4 (1950 + 2406 d - 7860 d^2 + 210 d^3 + 775 d^4 - 38 d^5)\\&+ c^5 (-918 + 8502 d - 5082 d^2 - 438 d^3 + 107 d^4 + 6 d^5)\\&- 2 c^3 (-1278 + 3417 d + 693 d^2 - 816 d^3 - 739 d^4 + 114 d^5)\\&- c (-315 + 3164 d - 6586 d^2 + 1638 d^3 + 273 d^4 + 594 d^5)\\&+ a^3 (1 + c)^4 (-9 + 3 c^5 + c^4 (3 - 10 d) + 8 d - 24 d^2\\&+ 14 d^3 + 9 d^4 + 2 d^5 + 2 c^3 (3 - 9 d + 5 d^2)\\&+ 2 c^2 (3 + d + 18 d^2) - c (9 - 18 d + 22 d^2 + 30 d^3 + 5 d^4))\\&- a^2 (1 + c)^2 (-47 + 2 c^8 + c^7 (23 - 10 d) + 168 d - 184 d^2\\&+ 258 d^3 - d^4 - 66 d^5 + c^6 (89 - 90 d + 20 d^2)\\&+ c^5 (195 - 328 d + 130 d^2 - 20 d^3)\\&+ c^4 (201 - 492 d + 472 d^2 - 80 d^3 + 10 d^4)\\&+ c (-191 + 492 d - 582 d^2 + 262 d^3 + 185 d^4 - 22 d^5)\\&- c^3 (27 + 154 d - 452 d^2 + 338 d^3 - 15 d^4 + 2 d^5)\\&+ c^2 (-245 + 414 d - 308 d^2 - 274 d^3 + 127 d^4 + 2 d^5))\\&+ a (1 + c) (-71 + 8 c^9 + c^8 (61 - 40 d) + 556 d - 1048 d^2\\&+ 554 d^3 - 489 d^4 - 270 d^5 + c^7 (388 - 250 d + 80 d^2)\\&- 2 c^6 (-556 + 630 d - 195 d^2 + 40 d^3) + 2 c^5 (654 - 1585 d + 767 d^2 \\&- 140 d^3 + 20 d^4)+c^2 (-1216 + 3740 d - 2282 d^2 + 512 d^3 + 644 d^4 - 82 d^5)\\&+ c^4 (114 - 2996 d + 2940 d^2 - 922 d^3 + 85 d^4 - 8 d^5)\\&- 2 c^3 (610 - 449 d - 730 d^2 + 586 d^3 - 171 d^4 + 3 d^5)\\&- 2 c (242 - 1261 d + 1537 d^2 - 758 d^3 + 7 d^4\\&+ 177 d^5))))/((-3 + a) (1 + c - d) d^4 (1 - c^2\\&+ a (-1 + c) (1 + c - d) - 9 d + c d) (1 - c^2\\&+ a (1 + c) (-1 + c - d) - 3 d + c d)^2 (-1 - 5 c^2\\&+ a (1 + c) (1 + c - d) + 9 d + c (-6 + 5 d))^3). \end{aligned}$$
$$\begin{aligned} \mathbf{h} _5= & {} (a^3 (1 + c) (1 - c^2 + 3 d + c d)^6 (116 - 4 c^13 - 1744 d\\&+ 7995 d^2 - 3078 d^3 - 24948 d^4 + 486 d^5 - 2187 d^6\\&+ 6 c^12 (5 + 4 d) + c^11 (1198 - 105 d - 60 d^2)\\&+ c^10 (6574 - 5034 d + 75 d^2 + 80 d^3)\\&+ c^9 (13870 - 30505 d + 8024 d^2 + 150 d^3 - 60 d^4)\\&+ c^8 (8080 - 70120 d + 53691 d^2 - 5716 d^3 - 300 d^4 + 24 d^5)\\&+ c (1342 - 14797 d + 32276 d^2 + 37116 d^3 - 73170 d^4\\&+ 6237 d^5 - 972 d^6) + c^6 (-29540 + 56308 d + 136050 d^2 - 129670 d^3 + 16264 d^4\\&+ 286 d^5 - 45 d^6) + c^7 (-15540 - 51170 d + 139852 d^2 - 43990 d^3 + 1374 d^4\\&+ 195 d^5 - 4 d^6) - c^5 (13960 - 124182 d + 51804 d^2 + 161338 d^3 - 52534 d^4\\&+ 2241 d^5 + 132 d^6) + c^4 (8510 + 60720 d - 206022 d^2 - 12894 d^3 + 72272 d^4\\&- 8086 d^5 + 207 d^6) + c^3 (13094 - 27605 d - 128288 d^2 + 168062 d^3 + 20474 d^4\\&- 5919 d^5 + 1620 d^6) + c^2 (6230 - 40154 d + 8211 d^2 + 151278 d^3 - 62136 d^4\\&+ 5562 d^5 + 2025 d^6) + a^4 (1 + c)^5 (16 + 4 c^6 + c^5 (12 - 13 d) - 5 d + 6 d^2\\&+ 36 d^3 + 10 d^4 + d^5 + c^4 (8 - 45 d + 12 d^2)\\&+ 2 c^3 (-12 + d + 31 d^2 + d^3) - 2 c^2 (14 - 25 d + 9 d^2 + 18 d^3 + 4 d^4)\\&+ c (12 + 11 d - 62 d^2 - 2 d^3 + 6 d^4 + 3 d^5))\\&- a^3 (1 + c)^3 (118 + 4 c^9 + c^8 (58 - 19 d) - 317 d - 15 d^2\\&+ 330 d^3 - 380 d^4 - 93 d^5 - 27 d^6 + c^7 (292 - 234 d + 35 d^2)\\&+ c^6 (444 - 1042 d + 347 d^2 - 30 d^3)\\&+ c^5 (-164 - 1418 d + 1427 d^2 - 208 d^3 + 10 d^4)\\&+ c^4 (-944 + 416 d + 1659 d^2 - 922 d^3 + 12 d^4 + d^5)\\&+ c^2 (324 + 962 d - 1991 d^2 + 622 d^3 + 432 d^4 + 28 d^5\\&- 9 d^6) - c^3 (564 - 2130 d + 591 d^2 + 992 d^3 - 244 d^4 - 34 d^5\\&+ d^6) - c (-432 + 478 d + 871 d^2 - 1200 d^3 + 190 d^4\\&+ 98 d^5 + 27 d^6)) + a^2 (1 + c)^2 (300 + 26 c^10 + c^9 (378 - 127 d) - 1811 d\\&+ 1415 d^2 + 2610 d^3 - 2934 d^4 + 945 d^5 + 243 d^6 + c^8 (1948 - 1611 d + 245 d^2)\\&+ c^7 (3408 - 7644 d + 2646 d^2 - 230 d^3) + 2 c^6 (18 - 6566 d + 5833 d^2 \\&- 1017 d^3 + 50 d^4)-c^5 (6156 + 1842 d - 19462 d^2 + 8614 d^3 - 666 d^4 + 11 d^5)\\&- 2 c^2 (-1705 + 606 d + 8921 d^2 - 6637 d^3 + 148 d^4 + 507 d^5)\\&+ c^4 (-5720 + 17766 d + 4516 d^2 - 13850 d^3 + 3066 d^4 - 27 d^5 - 3 d^6)\\&- 2 c^3 (-288 - 7802 d + 9847 d^2 + 1369 d^3 - 2482 d^4 + 211 d^5 + 9 d^6)\\&+ c (1794 - 5991 d - 2414 d^2 + 11582 d^3 - 5694 d^4 + 81 d^5 + 162 d^6))\\&+ a (1 + c) (-314 + 4 c^12 + 3227 d - 7559 d^2 - 5670 d^3 + 12420 d^4 \\&- 4293 d^5 - 243 d^6- 8 c^11 (7 + 3 d) + c^10 (-1122 + 221 d + 60 d^2)\\&- c^9 (5816 - 4848 d + 265 d^2 + 80 d^3)\\&+ 3 c^8 (-3758 + 8405 d - 2621 d^2 - 10 d^3 + 20 d^4)\\&+ c^7 (-3680 + 50232 d - 42580 d^2 + 5412 d^3 + 310 d^4\\&- 24 d^5) + c^6 (16636 + 23194 d - 89068 d^2 + 33944 d^3 - 708 d^4\\&- 239 d^5 + 4 d^6) + c^5 (22304 - 55416 d - 55038 d^2 + 76908 d^3 - 11262 d^4\\&- 876 d^5 + 59 d^6) + c^4 (4704 - 78218 d + 64902 d^2 + 56132 d^3 - 30792 d^4\\&- 167 d^5 + 309 d^6) + 2 c^3 (-5012 - 8832 d + 54926 d^2 - 20066 d^3 - 11555 d^4\\&+ 1808 d^5 + 333 d^6) + c^2 (-8634 + 26361 d + 39528 d^2 - 84376 d^3 + 19788 d^4 \\&+2907 d^5 + 378 d^6)- c (2728 - 18024 d + 11969 d^2 + 42108 d^3 - 34830 d^4\\&+ 3996 d^5 + 405 d^6))))/((-3 + a)^2 (1 + c - d) d^6 (1\\&- c^2 + a (-1 + c) (1 + c - d) - 9 d + c d) (1 - c^2\\&+ a (1 + c) (-1 + c - d) - 3 d + c d)^3 (-1 - 5 c^2\\&+ a (1 + c) (1 + c - d) + 9 d + c (-6 + 5 d))^4). \end{aligned}$$
$$\begin{aligned} \mathbf{h} _6= & {} (2 (-3 + a) a (1 + c)^2 (1 + c - d) (1 - c^2 \\&+ 3 d + c d)^2)/((1 + 5 c^2 + c (6 - 5 d)\\&- a (1 + c) (1 + c - d) - 9 d) d (1 - c^2\\&+ a (-1 + c) (1 + c - d) - 9 d + c d)^2). \end{aligned}$$
$$\begin{aligned} \mathbf{h} _7= & {} -((a^2 (1 + c)^2 (1 - c^2 + 3 d + c d)^4 (20 - c^10\\&+ 5 c^9 (-1 + d) - 344 d + 1702 d^2 - 1290 d^3 + 270 d^4\\&- 486 d^5 - 2 c^8 (228 + 6 d + 5 d^2)\\&+ 2 c^7 (-986 + 660 d + 49 d^2 + 5 d^3)\\&- c^6 (2966 - 6068 d + 1228 d^2 + 172 d^3 + 5 d^4)\\&+ c^4 (2140 + 2244 d - 8772 d^2 + 2638 d^3 + 36 d^4 - 32 d^5)\\&+ c^5 (-966 + 9042 d - 6330 d^2 + 324 d^3 + 123 d^4 + d^5)\\&+ c^3 (2684 - 7376 d - 594 d^2 + 3002 d^3 - 698 d^4 + 4 d^5)\\&+ c (259 - 2991 d + 6826 d^2 - 3144 d^3 + 543 d^4 + 27 d^5)\\&+ c^2 (1263 - 7956 d + 8308 d^2 - 984 d^3 - 333 d^4 + 294 d^5)\\&+ a^3 (1 + c)^3 (-7 + c^6 - c^5 (-6 + d) + 11 d - 28 d^2\\&+ 8 d^3 + 11 d^4 + 5 d^5 + c^4 (15 - 25 d - 6 d^2)\\&+ 2 c^3 (6 - 17 d + 17 d^2 + 7 d^3)\\&+ c^2 (-9 + 14 d + 34 d^2 - 12 d^3 - 11 d^4)\\&+ c (-18 + 35 d - 34 d^2 - 26 d^3 - 8 d^4 + 3 d^5))\\&- a^2 (1 + c)^2 (-34 + c^8 - 5 c^7 (-1 + d) + 166 d - 254 d^2\\&+ 234 d^3 + 40 d^4 - 24 d^5 + c^6 (79 - 4 d + 10 d^2)\\&- c^5 (-249 + 259 d + 34 d^2 + 10 d^3)\\&+ c^4 (237 - 666 d + 278 d^2 + 76 d^3 + 5 d^4)\\&- c^3 (81 + 307 d - 652 d^2 + 70 d^3 + 59 d^4 + d^5)\\&+ c^2 (-283 + 504 d - 34 d^2 - 326 d^3 - 53 d^4 + 16 d^5)\\&+ c (-173 + 571 d - 618 d^2 + 64 d^3 + 115 d^4 + 25 d^5))\\&+ a (1 + c) (-47 + 2 c^9 + c^8 (9 - 10 d) + 499 d - 1144 d^2\\&+ 892 d^3 - 321 d^4 - 135 d^5 + c^7 (336 + 9 d + 20 d^2)\\&- c^6 (-1244 + 997 d + 126 d^2 + 20 d^3)\\&+ c^5 (1500 - 3651 d + 902 d^2 + 234 d^3 + 10 d^4)\\&+ c (-414 + 2559 d - 3618 d^2 + 1426 d^3 + 236 d^4\\&- 141 d^5) - c^4 (-30 + 3669 d - 3668 d^2 + 84 d^3 + 171 d^4 + 2 d^5)\\&+ c^3 (-1424 + 1083 d + 2696 d^2 - 1500 d^3 - 230 d^4 + 45 d^5)\\&+ c^2 (-1236 + 4177 d - 2398 d^2 - 628 d^3 + 380 d^4\\&+ 73 d^5))))/((1 + 5 c^2 + c (6 - 5 d)\\&- a (1 + c) (1 + c - d) - 9 d)^3 d^4 (1 - c^2\\&+ a (-1 + c) (1 + c - d) - 9 d + c d)^2 (-1 + c^2 + 3 d - c d\\&+ a (1 - c^2 + d + c d))^2)). \end{aligned}$$
$$\begin{aligned} \mathbf{h} _8= & {} (a^4 (-1 + c) (1 + c)^2 (1 - c^2 + 3 d + c d)^8 (-374 - 9 c^15\\&+ 7927 d - 62959 d^2 + 204153 d^3 - 80685 d^4 - 528903 d^5\\&+ 248751 d^6 - 12393 d^7 + 19683 d^8 + c^14 (1148 + 83 d)\\&+ c^13 (22187 - 7262 d - 329 d^2)\\&+ c^12 (151358 - 135971 d + 19163 d^2 + 735 d^3)\\&+ c^11 (539191 - 940312 d + 349778 d^2 - 26764 d^3 - 1015 d^4)\\&+ c^10 (1128604 - 3425021 d + 2431052 d^2 - 484448 d^3\\&+ 20185 d^4 + 889 d^5) + c^9 (1380259 - 7363326 d + 9065745 d^2 - 3359408 d^3\\&+ 381465 d^4 - 6478 d^5 - 483 d^6)\\&+ c^8 (737982 - 9445043 d + 20224883 d^2 - 12775821 d^3\\&+ 2634571 d^4 - 160427 d^5 - 1123 d^6 + 149 d^7)\\&+ c^5 (-1213831 + 5321446 d + 5796793 d^2 - 39178080 d^3 \\&+ 37174526 d^4 - 10466936 d^5 + 882206 d^6 - 6156 d^7\\&- 1422 d^8) + c^6 (-1295388 + 742417 d + 21684224 d^2 - 43317256 d^3\\&+ 24195606 d^4 - 4362198 d^5 + 220518 d^6 + 4382 d^7 - 301 d^8)\\&+ c^7 (-485523 - 6136576 d + 27678396 d^2 - 29625112 d^3\\&+ 10134086 d^4 - 1130372 d^5 + 24456 d^6 + 1432 d^7 - 20 d^8)\\&- c^4 (671494 - 5046431 d + 5808575 d^2 + 19450283 d^3\\ \end{aligned}$$
$$\begin{aligned}&- 36677794 d^4 + 16220478 d^5 - 2036268 d^6 + 67908 d^7\\&+ 1251 d^8) + c^3 (-235675 + 2539224 d - 6867918 d^2 - 2298236 d^3\\&+ 22267617 d^4 - 16210700 d^5 + 2884920 d^6 - 163440 d^7 + 9720 d^8)\\&+ c^2 (-51836 + 739545 d - 3181292 d^2 + 2721608 d^3\\&+ 7363665 d^4 - 10064691 d^5 + 2486754 d^6 - 177606 d^7\\&+ 33453 d^8) + c (-6599 + 117174 d - 715969 d^2 + 1386288 d^3 + 854457 d^4\\&- 3521322 d^5 + 1200069 d^6 - 85212 d^7 + 42282 d^8)\\&- 2 a^5 (1 + c)^6 (1 + c - d)^2 (-19 + 6 c^5 + c^4 (23 - 22 d)\\&+ 2 d + 5 d^2 - 13 d^3 + 6 d^4 + 3 d^5 + c^3 (30 - 70 d + 27 d^2) - c^2 (4 + 64 d \\&- 77 d^2 + 9 d^3)-c (36 + 14 d - 47 d^2 + 36 d^3 + 5 d^4))\\&+ a^4 (1 + c)^4 (-347 + 9 c^10 + c^9 (296 - 68 d) + 1512 d\\&- 1744 d^2 - 422 d^3 + 2440 d^4 - 1932 d^5 + 556 d^6 - 54 d^7\\&- 9 d^8 + c^8 (1909 - 1728 d + 218 d^2)\\&- 4 c^7 (-1395 + 2539 d - 1051 d^2 + 96 d^3)\\&+ c^6 (8358 - 27056 d + 22182 d^2 - 5454 d^3 + 400 d^4)\\&+ c^5 (4972 - 37212 d + 53464 d^2 - 25372 d^3 + 3970 d^4\\&- 244 d^5) + 2 c^4 (-1717 - 11612 d + 33467 d^2 - 27421 d^3 + 7982 d^4\\&- 758 d^5 + 39 d^6) - 4 c^3 (2119 - 627 d - 10403 d^2 + 15358 d^3 - 7664 d^4\\&+ 1305 d^5 - 52 d^6 + 2 d^7) + c^2 (-6495 + 13248 d + 7322 d^2 - 35282 d^3 + 29676 d^4\\&- 8896 d^5 + 690 d^6 + 26 d^7 - d^8)\\&- 2 c (1186 - 3840 d + 2184 d^2 + 4406 d^3 - 6927 d^4\\&+ 3440 d^5 - 574 d^6 - 6 d^7 + 3 d^8))\\&+ a^3 (1 + c)^3 (1157 + 3 c^12 - 8905 d + 17933 d^2 - 6325 d^3\\&- 22477 d^4 + 28373 d^5 - 11313 d^6 + 1449 d^7 + 108 d^8\\&- 7 c^11 (22 + 3 d) + c^10 (-2864 + 1045 d + 63 d^2)\\&- c^9 (17458 - 17405 d + 3024 d^2 + 105 d^3)\\&+ c^8 (-52761 + 98859 d - 44579 d^2 + 4823 d^3 + 105 d^4)\\&- c^7 (86452 - 280758 d + 233980 d^2 - 61976 d^3 + 4550 d^4\\&+ 63 d^5) + c^6 (-66552 + 438090 d - 621534 d^2 + 297968 d^3\\&- 49890 d^4 + 2499 d^5 + 21 d^6)\\&+ c^5 (13068 + 347970 d - 919532 d^2 + 736334 d^3\\&- 218042 d^4 + 22589 d^5 - 700 d^6 - 3 d^7)\\&+ c^4 (80497 + 41550 d - 738826 d^2 + 1023398 d^3\\&- 498324 d^4 + 89671 d^5 - 4815 d^6 + 49 d^7)\\&+ c^3 (80334 - 175281 d - 236212 d^2 + 800128 d^3\\&- 638410 d^4 + 188811 d^5 - 17956 d^6 + 94 d^7 + 12 d^8)\\&+ c^2 (40520 - 160527 d + 74943 d^2 + 316488 d^3 - 455078 d^4\\&+ 214465 d^5 - 35429 d^6 + 758 d^7 + 84 d^8)\\&+ c (10662 - 60719 d + 80748 d^2 + 38019 d^3 - 164662 d^4\\&+ 123671 d^5 - 33008 d^6 + 2037 d^7 + 180 d^8))\\&- a^2 (1 + c)^2 (1799 + 15 c^13 - 21185 d + 75337 d^2\\&- 61517 d^3 - 70701 d^4 + 165837 d^5 - 91341 d^6 + 9153 d^7\\&+ 810 d^8 - c^12 (865 + 109 d) + c^11 (-13198 + 5624 d + 343 d^2)\\&- c^10 (78770 - 82602 d + 15473 d^2 + 609 d^3)\\&+ c^9 (-247947 + 470488 d - 217903 d^2 + 23174 d^3 + 665 d^4)\\&- c^8 (442855 - 1423097 d + 1184931 d^2 - 311719 d^3\\&+ 20065 d^4 + 455 d^5) + c^7 (-412724 + 2480400 d - 3442458 d^2 + 1620024 d^3\\&- 257260 d^4 + 9580 d^5 + 189 d^6) - c^6 (48988 - 2384796 d + 5843962 d^2 \\&- 4525118 d^3+1281760 d^4 - 117928 d^5 + 1879 d^6 + 43 d^7)\\&+ c^5 (357769 + 761872 d - 5752702 d^2 + 7455124 d^3\\&- 3452018 d^4 + 569460 d^5 - 23967 d^6 - 202 d^7 + 4 d^8)\\&+ c^4 (462113 - 920427 d - 2734102 d^2 + 7340870 d^3\\&- 5481450 d^4 + 1488306 d^5 - 118783 d^6 - 617 d^7\\&+ 106 d^8) + c^3 (294690 - 1285768 d + 231763 d^2 + 4056472 d^3\\&- 5203164 d^4 + 2249620 d^5 - 309641 d^6 + 2436 d^7\\&+ 696 d^8) + c^2 (107566 - 709286 d + 989403 d^2 + 952707 d^3 \\&- 2829736 d^4 + 1949536 d^5 - 437181 d^6 + 13491 d^7\\&+ 1836 d^8) + c (21395 - 193128 d + 467229 d^2 - 86506 d^3 - 771935 d^4\\&+ 892492 d^5 - 315765 d^6 + 19494 d^7 + 2052 d^8))\\&+ a (1 + c) (1325 + 21 c^14 - 21639 d + 121207 d^2 - 218933 d^3\\&- 106161 d^4 + 400671 d^5 - 293463 d^6 + 32157 d^7 + 2916 d^8\\&- c^13 (1868 + 171 d) + 3 c^12 (-9465 + 3975 d + 203 d^2)\\&- c^11 (174770 - 179274 d + 31878 d^2 + 1239 d^3) + c^10 (-579809 + 1080186 d \\&- 475914 d^2 + 45451 d^3+ 1575 d^4) - c^9 (1122914 - 3531999 d + 2814256 d^2 - 681417 d^3\\&+ 35700 d^4 + 1281 d^5) + c^8 (-1215653 + 6831327 d - 9083691 d^2 + 3971487 d^3\\&- 555315 d^4 + 13083 d^5 + 651 d^6)\\&+ c^7 (-425332 + 7675548 d - 17583684 d^2 + 12691378 d^3\\&- 3221446 d^4 + 240900 d^5 + 490 d^6 - 189 d^7)\\&+ c^6 (720763 + 3794556 d - 20569636 d^2 + 24570070 d^3\\&- 10247784 d^4 + 1436644 d^5 - 35964 d^6 - 1947 d^7\\&+ 24 d^8) + c^5 (1256336 - 1849773 d - 12972792 d^2 + 29555738 d^3\\&- 19785774 d^4 + 4614606 d^5 - 272632 d^6 - 8871 d^7 \\&+ 444 d^8) + c^4 (976835 - 4387821 d - 1365501 d^2 + 21321622 d^3\\&- 23867652 d^4 + 8864214 d^5 - 952856 d^6 - 13925 d^7\\&+ 2868 d^8) + c^3 (449030 - 3225702 d + 4297114 d^2 + 7755077 d^3 \\&- 17755938 d^4 + 10387660 d^5 - 1850486 d^6 + 12765 d^7 \\&+ 8712 d^8) + c (19518 - 256263 d + 1028952 d^2 - 915971 d^3\\&- 1636982 d^4 + 2674851 d^5 - 1206684 d^6 + 78759 d^7\\&+ 10044 d^8) + c^2 (124913 - 1253622 d + 3296382 d^2 + 76703 d^3 \\&- 7660503 d^4 + 7202124 d^5 - 2047680 d^6 + 67203 d^7 \\&+ 13392 d^8))))/((-3 + a)^2 (1 + c - d) d^8 (1 - c^2 \\&+ a (-1 + c) (1 + c - d) - 9 d + c d)^2 (1 - c^2 + a (1 + c) (-1 + c - d) - 3 d \\&+ c d)^4 (-1 - 5 c^2+a (1 + c) (1 + c - d) + 9 d + c (-6 + 5 d))^5) \end{aligned}$$
$$\begin{aligned} \mathbf{h} _9= & {} -((2 a^3 (1 + c)^2 (1 - c^2 + 3 d + c d)^6 (81 - c^16 - 2177 d \\&+ 23219 d^2 - 119048 d^3 + 285792 d^4 - 231057 d^5 + 38799 d^6 \\&+ 18954 d^7 - 19683 d^8 + 4 c^15 (3 + 2 d) \\&- c^14 (2037 + 62 d + 28 d^2) \\&+ c^13 (-28003 + 13098 d + 98 d^2 + 56 d^3) \\&+ c^12 (-148560 + 171823 d - 34920 d^2 + 42 d^3 - 70 d^4) \\&+ c^11 (-422094 + 900216 d - 437321 d^2 + 48534 d^3 - 350 d^4 \\&+ 56 d^5) + c^10 (-701107 + 2540280 d - 2239895 d^2 + 582446 d^3 \\&- 34710 d^4 + 518 d^5 - 28 d^6) \\&+ c^9 (-636033 + 4151018 d - 6192449 d^2 + 2899882 d^3 \\&- 406010 d^4 + 7542 d^5 - 378 d^6 + 8 d^7) \\&+ c (1625 - 36978 d + 315391 d^2 - 1218974 d^3 + 1977206 d^4 \\&- 1139466 d^5 + 105219 d^6 + 107730 d^7 - 35721 d^8) \\&+ c^2 (14499 - 269300 d + 1785685 d^2 - 4963450 d^3 \\&+ 5495330 d^4 - 2107730 d^5 + 9459 d^6 + 157680 d^7 \\&- 12717 d^8) + c^7 (488744 + 212808 d - 7711770 d^2 + 11614572 d^3 \\&- 5128964 d^4 + 558320 d^5 + 46127 d^6 - 4470 d^7 - 22 d^8) \\&+ c^8 (-110076 + 3565153 d - 9788385 d^2 + 7757068 d^3 \\&- 1986860 d^4 + 100363 d^5 + 6072 d^6 + 142 d^7 - d^8) \\&+ c^6 (703077 - 3270182 d + 800110 d^2 + 7942700 d^3 \\&- 7121900 d^4 + 1511972 d^5 + 100949 d^6 - 36344 d^7 \\&+ 891 d^8) + c^5 (521067 - 4140450 d + 8596784 d^2 - 2936292 d^3 \\&- 3813532 d^4 + 2010132 d^5 + 26675 d^6 - 102490 d^7 \\&+ 6919 d^8) + c^3 (74682 - 1099720 d + 5429267 d^2 - 10407778 d^3 \\&+ 7369346 d^4 - 1426600 d^5 - 204395 d^6 + 30390 d^7 \\&+ 16776 d^8) + c^4 (244124 - 2735535 d + 9454214 d^2 - 11199758 d^3 \\&+ 3360114 d^4 + 735918 d^5 - 182003 d^6 - 109264 d^7 \\&+ 18438 d^8) + a^5 (1 + c)^5 (1 + c - d)^2 (-15 + 2 c^7 + c^6 (9 - 12 d) \\&+ 25 d - 39 d^2 + 13 d^3 - 30 d^4 - 2 d^5 \\&+ 3 c^5 (8 - 13 d + 9 d^2) \\&+ c^4 (39 - 71 d + 69 d^2 - 28 d^3) \\ \end{aligned}$$
$$\begin{aligned}&+ c^3 (18 - 42 d + 50 d^2 - 63 d^3 + 12 d^4) \\&+ c^2 (-33 + 58 d - 30 d^2 + 15 d^3 + 30 d^4) \\&- c (44 - 81 d + 77 d^2 - 63 d^3 + 16 d^4 + 6 d^5 + d^6)) \\&+ a^4 (1 + c)^4 (119 + c^11 - 813 d + 2043 d^2 - 2860 d^3 \\&+ 2452 d^4 - 1509 d^5 + 631 d^6 - 18 d^7 - 45 d^8 \\&- 6 c^10 (7 + d) + 2 c^9 (-160 + 155 d + 7 d^2) \\&- c^8 (1169 - 2095 d + 968 d^2 + 14 d^3) \\&+ 3 c^7 (-870 + 2144 d - 1919 d^2 + 550 d^3) \\&+ c^6 (-3368 + 11148 d - 14415 d^2 + 8566 d^3 - 1640 d^4 \\&+ 14 d^5) + c^5 (-1588 + 9444 d - 17881 d^2 + 16722 d^3 - 7400 d^4 \\&+ 922 d^5 - 14 d^6) + c^4 (1890 - 1338 d - 5807 d^2 + 12130 d^3 - 10508 d^4 \\&+ 3675 d^5 - 240 d^6 + 6 d^7) + c (876 - 4794 d + 10231 d^2 - 12058 d^3 + 8864 d^4 \\&- 4330 d^5 + 779 d^6 + 182 d^7 - 6 d^8) \\&- c^3 (-3641 + 11392 d - 13393 d^2 + 6314 d^3 + 1416 d^4 \\&- 3456 d^5 + 965 d^6 + 2 d^7 + d^8) \\&+ c^2 (2570 - 11086 d + 19147 d^2 - 17822 d^3 + 9744 d^4 \\&- 2132 d^5 - 655 d^6 + 112 d^7 + 10 d^8)) \\&- a^3 (1 + c)^3 (347 + 9 c^12 - 3864 d + 14350 d^2 - 25422 d^3 \\&+ 26146 d^4 - 14804 d^5 + 5006 d^6 - 294 d^7 - 441 d^8 \\&- 2 c^11 (170 + 27 d) + 2 c^10 (-1526 + 1199 d + 63 d^2) \\&- 2 c^9 (6088 - 9886 d + 3593 d^2 + 63 d^3) \\&+ c^8 (-28255 + 69944 d - 53778 d^2 + 11798 d^3) \\&+ 2 c^7 (-19076 + 68626 d - 83012 d^2 + 39534 d^3 \\&- 5665 d^4 + 63 d^5) - 2 c^6 (11176 - 71758 d + 132702 d^2 - 104086 d^3 \\&+ 33525 d^4 - 3085 d^5 + 63 d^6) \\&+ 2 c^5 (7040 + 16084 d - 92438 d^2 + 126816 d^3 \\&- 72785 d^4 + 15986 d^5 - 779 d^6 + 27 d^7) \\&+ c^4 (38299 - 114032 d + 77124 d^2 + 66840 d^3 \\&- 116826 d^4 + 54856 d^5 - 7382 d^6 - 14 d^7 - 9 d^8) \\&+ 2 c^3 (16510 - 79311 d + 133852 d^2 - 99982 d^3 \\&+ 23981 d^4 + 8871 d^5 - 5202 d^6 + 194 d^7 + 31 d^8) \\&+ 2 c^2 (7502 - 48981 d + 113791 d^2 - 130694 d^3 \\&+ 78769 d^4 - 22951 d^5 + 659 d^6 + 802 d^7 + 31 d^8) \\&- 2 c (-1784 + 15258 d - 45191 d^2 + 66305 d^3 - 54373 d^4 \\&+ 24760 d^5 - 5453 d^6 - 491 d^7 + 201 d^8)) \\&- a^2 (1 + c)^2 (c^14 - c^13 (29 + 8 d) \\&+ 2 c^12 (683 + 80 d + 14 d^2) \\&- 4 c^11 (-3545 + 2324 d + 77 d^2 + 14 d^3) \\&+ c^10 (61865 - 90586 d + 26864 d^2 + 112 d^3 + 70 d^4) \\&+ c^9 (151703 - 366092 d + 242474 d^2 - 42464 d^3 + 490 d^4 \\&- 56 d^5) + c^8 (217326 - 811290 d + 898314 d^2 - 348872 d^3 \\&+ 39100 d^4 - 896 d^5 + 28 d^6) \\&- 4 c^7 (-37704 + 250364 d - 440478 d^2 + 291876 d^3 \\&- 71290 d^4 + 5044 d^5 - 175 d^6 + 2 d^7) \\&+ c^4 (-202598 + 1015228 d - 1498256 d^2 + 530112 d^3 \\&+ 454208 d^4 - 327294 d^5 + 43798 d^6 + 2796 d^7 \\&- 226 d^8) + c^6 (-41037 - 474644 d + 1764944 d^2 - 1968592 d^3 \\&+ 843396 d^4 - 124970 d^5 + 4576 d^6 - 272 d^7 + d^8) \\&+ c^5 (-197663 + 482672 d + 279236 d^2 - 1431072 d^3 \\&+ 1166988 d^4 - 316044 d^5 + 20826 d^6 + 256 d^7 \\&+ 43 d^8) - c^2 (36445 - 353454 d + 1148496 d^2 - 1682784 d^3\\&+ 1207706 d^4 - 368410 d^5 + 31308 d^6 + 2376 d^7 \\&+ 45 d^8) - 4 c^3 (28145 - 203192 d + 483749 d^2 - 494274 d^3 \\&+ 201022 d^4 - 7186 d^5 - 6650 d^6 - 578 d^7 + 252 d^8) \\&+ 2 (-239 + 3839 d - 21699 d^2 + 52228 d^3 - 64214 d^4 \\&+ 42183 d^5 - 9027 d^6 - 378 d^7 + 891 d^8) +c (-6427 + 81412 d - 348318 d^2 \\&+ 664000 d^3 - 647910 d^4 + 307148 d^5 - 49086 d^6 - 3168 d^7 + 2349 d^8)) \\&+ a (1 + c) (-316 + 2 c^15 + 6749 d - 54190 d^2 + 194554 d^3 \\&- 305103 d^4 + 219753 d^5 - 54324 d^6 - 22032 d^7 + 3645 d^8 \\&- c^14 (33 + 16 d) + 2 c^13 (1359 + 87 d + 28 d^2) \\&- 2 c^12 (-16027 + 8993 d + 147 d^2 + 56 d^3) \\&+ c^11 (153378 - 201553 d + 50021 d^2 - 42 d^3 + 140 d^4) \\&+ c^10 (402717 - 924227 d + 529186 d^2 - 74628 d^3 \\&+ 840 d^4 - 112 d^5) + c^9 (618878 - 2308893 d + 2304729 d^2 - 738900 d^3 \\&+ 62075 d^4 - 1302 d^5 + 56 d^6) \\&+ c^8 (495048 - 3297203 d + 5393910 d^2 - 3030286 d^3 \\&+ 567285 d^4 - 25138 d^5 + 966 d^6 - 16 d^7) \\&+ c^3 (-156730 + 1652043 d - 5636751 d^2 + 7739262 d^3 \\&- 4288316 d^4 + 582485 d^5 + 84373 d^6 + 50190 d^7 \\&- 11772 d^8) + c^4 (-392034 + 3010568 d - 6932626 d^2 + 5358612 d^3 \\&- 309214 d^4 - 917795 d^5 + 78210 d^6 + 61008 d^7 \\&- 5625 d^8) + c^5 (-603126 + 2996932 d - 3093614 d^2 - 1985016 d^3 \\&+ 3860246 d^4 - 1333173 d^5 + 15289 d^6 + 25894 d^7 \\&- 817 d^8) + c^7 (-9834 - 2231114 d + 6948074 d^2 - 6532644 d^3 \\&+ 2178392 d^4 - 204529 d^5 + 531 d^6 - 366 d^7 + 2 d^8) \\&+ c^6 (-499063 + 692498 d + 3535516 d^2 - 7279320 d^3 \\&+ 4258296 d^4 - 746511 d^5 - 3812 d^6 + 3224 d^7 \\&+ 57 d^8) + c (-5286 + 92411 d - 572515 d^2 + 1517340 d^3 \\&- 1815225 d^4 + 964071 d^5 - 118233 d^6 - 52326 d^7 \\&+ 1539 d^8) - c^2 (38373 - 529617 d + 2471502 d^2 - 4831180 d^3 \\&+ 4214792 d^4 - 1477355 d^5 + 39024 d^6 + 18792 d^7 \\&+ 8613 d^8))))/((-3 + a) (1 + c - d) d^6 (1 - c^2 \\&+ a (-1 + c) (1 + c - d) - 9 d + c d)^2 (1 - c^2 \\&+ a (1 + c) (-1 + c - d) - 3 d + c d)^3 (-1 - 5 c^2 \\&+ a (1 + c) (1 + c - d) + 9 d + c (-6 + 5 d))^5)). \end{aligned}$$
$$\begin{aligned} \mathbf{h} _{10}= & {} ((-3 + a) a^2 (1 + c)^3 (1 + c \\&- d) (-1 + c^2 - 3 d \\&- c d)^5)/(d^3 (-1 + c^2 \\&- a (-1 + c) (1 + c - d) + 9 d \\&- c d)^3 (-1 - 5 c^2 + a (1 + c) (1 + c - d) \\&+ 9 d + c (-6 + 5 d))^2). \end{aligned}$$
$$\begin{aligned} \mathbf{h} _{11}= & {} -((a^4 (-1 + c) (1 + c)^3 (1 - c^2 + 3 d + c d)^8 (c^13 \\&- 4 c^12 (49 + 2 d) + c^11 (-4098 + 1097 d + 27 d^2) \\&- 2 c^10 (15305 - 9709 d + 1308 d^2 + 25 d^3) \\&+ c^9 (-122993 + 129337 d - 37005 d^2 + 3485 d^3 + 55 d^4) \\&- 2 c^8 (154251 - 237712 d + 106744 d^2 - 18518 d^3 + 1430 d^4 \\&+ 18 d^5) + c^7 (-518076 + 1107218 d - 689938 d^2 + 173100 d^3 \\&- 23024 d^4 + 1491 d^5 + 13 d^6) \\&- 2 c^6 (301122 - 869566 d + 706252 d^2 - 231610 d^3 \\&+ 39533 d^4 - 5847 d^5 + 236 d^6 + d^7) \\&+ c^5 (-489825 + 1890682 d - 1946250 d^2 + 761214 d^3 \\&- 157190 d^4 + 36233 d^5 - 5257 d^6 + 71 d^7) \\&+ 2 c^4 (-138304 + 712016 d - 925860 d^2 + 392568 d^3 \\&- 93415 d^4 + 30548 d^5 - 10990 d^6 + 618 d^7) \\&+ 2 (-115 + 1692 d - 7892 d^2 + 3690 d^3 + 9213 d^4 + 5652 d^5 \\&- 486 d^6 + 4374 d^7) + c^3 (-105538 + 726149 d - 1211913 d^2 + 509580 d^3 \\&- 105824 d^4 + 66917 d^5 - 45405 d^6 + 6474 d^7) \\&+ 2 c^2 (-12885 + 118493 d - 262184 d^2 + 103783 d^3 \\&+ 8077 d^4 + 28755 d^5 - 23112 d^6 + 7893 d^7) \\&+ c (-3631 + 43885 d - 135401 d^2 + 52909 d^3 + 51807 d^4 \\&+ 36927 d^5 - 18999 d^6 + 18711 d^7) \\&- a^4 (1 + c)^4 (1 + c - d)^2 (26 + 8 c^5 + c^4 (50 - 11 d) \\&+ d - 18 d^2 + 32 d^3 - 16 d^4 - d^5 \\&- 2 c^3 (-64 + 30 d + 7 d^2) \\&+ 2 c^2 (82 - 43 d - 19 d^2 + 14 d^3) \\&- 2 c (-52 + 18 d + 21 d^2 - 32 d^3 + 5 d^4)) \\&+ a^3 (1 + c)^4 (207 + 7 c^8 + c^7 (168 - 45 d) - 741 d \\&+ 607 d^2 + 649 d^3 - 1467 d^4 + 885 d^5 - 131 d^6 - 9 d^7 \\&+ c^6 (1108 - 659 d + 119 d^2) \\&+ c^5 (3472 - 3625 d + 854 d^2 - 165 d^3) \\&+ c^4 (6130 - 9423 d + 3393 d^2 - 195 d^3 + 125 d^4) \\&+ c^3 (6472 - 13175 d + 6788 d^2 + 622 d^3 - 460 d^4 \\&- 47 d^5) + c^2 (4068 - 10297 d + 6825 d^2 + 1690 d^3 - 2594 d^4 \\&+ 367 d^5 + 5 d^6) + c (1408 - 4275 d + 3302 d^2 + 1687 d^3 - 3412 d^4 \\&+ 1227 d^5 - 66 d^6 + d^7)) \\&+ a^2 (1 + c)^2 (-554 + c^11 + 3555 d - 4936 d^2 - 1333 d^3 \\&+ 6654 d^4 - 3991 d^5 - 60 d^6 + 153 d^7 - 2 c^10 (40 + 3 d) \\&+ c^9 (-1355 + 471 d + 15 d^2) - c^8 (9090 - 6075 d + 1152 d^2 + 20 d^3) \\&+ c^7 (-32982 + 34212 d - 10580 d^2 + 1495 d^3 + 15 d^4) \\&- c^6 (73104 - 107296 d + 45400 d^2 - 8729 d^3 + 1080 d^4 \\&+ 6 d^5) + c^4 (-99780 + 253130 d - 177368 d^2 + 27301 d^3 \\&+ 3222 d^4 + 329 d^5 - 56 d^6)\\&+ c^5 (-104934 + 206010 d - 114094 d^2 + 21075 d^3 \\&- 3191 d^4 + 405 d^5 + d^6) \\&- c^3 (62475 - 200836 d + 172388 d^2 - 18565 d^3 \\&- 23929 d^4 + 4406 d^5 + 66 d^6 + 3 d^7) \\&+ c^2 (-24752 + 99966 d - 102280 d^2 + 3659 d^3 + 37444 d^4 \\&- 11820 d^5 + 204 d^6 + 59 d^7) \\&+ c (-5615 + 28487 d - 34089 d^2 - 2799 d^3 + 25487 d^4\\&- 11487 d^5 + 217 d^6 + 183 d^7)) \\&- a (1 + c) (-603 + 2 c^12 + 6201 d - 16095 d^2 - 181 d^3\\&+ 13327 d^4 - 3417 d^5 - 4005 d^6 + 1701 d^7\\&- c^11 (269 + 14 d) + c^10 (-4195 + 1525 d + 42 d^2) \\&- c^9 (27931 - 20012 d + 3589 d^2 + 70 d^3) \\&+ c^8 (-104627 + 114509 d - 38315 d^2 + 4485 d^3 + 70 d^4) \\&- c^7 (245890 - 377700 d + 180040 d^2 - 37374 d^3\\&+ 3135 d^4 + 42 d^5) + c^6 (-383054 + 793986 d - 493644 d^2 + 132950 d^3 \\&- 19761 d^4 + 1159 d^5 + 14 d^6) \\&- c^4 (293156 - 1054202 d + 1010670 d^2 - 323056 d^3\\&+ 44529 d^4 - 10609 d^5 + 1553 d^6 + d^7) \\&- c^5 (405718 - 1112696 d + 866682 d^2 - 266666 d^3\\&+ 46331 d^4 - 6104 d^5 + 175 d^6 + 2 d^7) \\&+ c^3 (-141857 + 666874 d - 786464 d^2 + 234026 d^3\\&+ 8355 d^4 + 5670 d^5 - 5362 d^6 + 334 d^7) \\&+ c^2 (-43807 + 269065 d - 396646 d^2 + 92906 d^3 \\&+ 57805 d^4 - 4767 d^5 - 10008 d^6 + 1596 d^7) \\&+ c (-7775 + 62220 d - 118553 d^2 + 15220 d^3 + 48023 d^4 \\&- 8148 d^5 - 9759 d^6 + 2772 d^7))))/((-3 + a) (1 + 5 c^2 \\&+ c (6 - 5 d) - a (1 + c) (1 + c - d) - 9 d)^5 d^8 (-1 + c^2 \\&- a (-1 + c) (1 + c - d) + 9 d - c d)^3 (-1 + c^2 + 3 d - c d \\&+ a (1 - c^2 + d + c d))^3)). \end{aligned}$$
$$\begin{aligned} \mathbf{h} _{12}= & {} (a^3 (1 + c)^3 (1 + c - d) (1 - c^2 + 3 d + c d)^6 (-36 + 312 d \\&+ 343 d^2 - 2793 d^3 + 81 d^4 - 243 d^5 + c^9 (90 + d) \\&+ c^8 (648 - 222 d - 4 d^2) \\&+ c^7 (1836 - 646 d + 153 d^2 + 6 d^3) \\&- c^6 (-2412 + 942 d + 799 d^2 + 27 d^3 + 4 d^4) \\&+ c^3 (-2412 + 662 d + 1395 d^2 + 5082 d^3 - 474 d^4 - 2 d^5) \\&+ c^5 (864 - 1524 d - 4887 d^2 + 942 d^3 + 33 d^4 + d^5) \\&- c^4 (1620 + 1458 d + 5967 d^2 - 4245 d^3 + 143 d^4 + 27 d^5) \\&+ c (-378 + 1507 d + 3339 d^2 - 6030 d^3 + 345 d^4 + 81 d^5) \\&+ c^2 (-1404 + 2310 d + 6427 d^2 - 1425 d^3 - 30 d^4 + 222 d^5)\\&- a^3 (1 + c)^3 (-6 - 7 d + 23 d^2 - 35 d^3 - 5 d^4 - 2 d^5 \\&+ 3 c^5 (2 + d) + c^4 (18 + 5 d - 7 d^2) \\&+ c^3 (12 + 6 d - 38 d^2 + 3 d^3) \\&+ c^2 (-12 + 2 d - 16 d^2 + 35 d^3 + 3 d^4) \\&- c (18 + 9 d - 38 d^2 + 3 d^3 + 6 d^4 + 2 d^5)) \\&+ a^2 (1 + c)^2 (-36 + 26 d + 277 d^2 - 375 d^3 + 107 d^4 \\&+ 33 d^5 + c^7 (6 + d) + c^6 (84 + 6 d - 4 d^2) \\&+ c^5 (246 + 41 d - 45 d^2 + 6 d^3) \\&+ c^4 (228 + 38 d - 407 d^2 + 39 d^3 - 4 d^4) \\&+ c^3 (-78 - 61 d - 602 d^2 + 354 d^3 + 3 d^4 + d^5)\\&- c (174 - 19 d - 647 d^2 + 360 d^3 + 51 d^4 + d^5)\\&- c^2 (276 + 70 d - 134 d^2 - 336 d^3 + 71 d^4 + 9 d^5)) \\&- a (1 + c) (-66 + 249 d + 773 d^2 - 1645 d^3 + 465 d^4\\&+ 2 c^8 (24 + d) + c^7 (402 - 75 d - 8 d^2) \\&+ c^6 (1146 - 99 d - 29 d^2 + 12 d^3) \\&+ c^5 (1314 - 123 d - 1260 d^2 + 57 d^3 - 8 d^4)\\&+ c^2 (-1218 + 415 d + 2673 d^2 + 482 d^3 - 354 d^4 - 58 d^5)\\&+ c^4 (90 - 567 d - 3417 d^2 + 1151 d^3 + 33 d^4 + 2 d^5) \\&- c^3 (1242 + 537 d + 1704 d^2 - 2706 d^3 + 136 d^4 + 34 d^5) \\&+ c (-474 + 735 d + 2972 d^2 - 2763 d^3 + 160 d^4 \\&+ 42 d^5))))/(d^6 (-1 + c^2 - a (-1 + c) (1 + c - d) + 9 d \\&- c d)^3 (-1 - 5 c^2 + a (1 + c) (1 + c - d) + 9 d \\&+ c (-6 + 5 d))^4 (-1 + c^2 + 3 d - c d + a (1 - c^2 + d + c d))^2). \end{aligned}$$
$$\begin{aligned} \mathbf{h} _{13}= & {} -((a^5 (-1 + c) (1 + c)^4 (1 - c^2 + 3 d + c d)^10 (33 c^13 \\&+ c^12 (561 - 180 d) + c^11 (4635 - 3610 d + 367 d^2) \\&+ c^10 (23799 - 29934 d + 9511 d^2 - 270 d^3) \\&+ c^9 (82626 - 141344 d + 77206 d^2 - 12824 d^3 - 165 d^4) \\&+ c^8 (201942 - 433816 d + 338946 d^2 - 99652 d^3 + 8535 d^4 \\&+ 472 d^5) + c^7 (355014 - 926028 d + 930410 d^2 - 399676 d^3 + 62139 d^4 \\&- 1186 d^5 - 375 d^6) + c^6 (453150 - 1422068 d + 1702458 d^2 - 999704 d^3 \\&+ 205955 d^4 - 8026 d^5 - 1975 d^6 + 138 d^7) \\&+ 2 (246 - 2948 d + 8835 d^2 - 2996 d^3 + 2490 d^4 - 21510 d^5 \\&- 8451 d^6 + 14094 d^7) + c^2 (38043 - 270638 d + 522855 d^2 - 411578 d^3 \\&+ 447481 d^4-91130 d^5 - 199431 d^6 + 45270 d^7 - 1512 d^8) \\&+ c^3 (127119 - 722986 d + 1211471 d^2 - 1087540 d^3 \\&+ 765521 d^4 + 45054 d^5 - 157279 d^6 + 20388 d^7 - 864 d^8)\\&+ c^4 (277341 - 1279644 d + 1922096 d^2 - 1680756 d^3 \\&+ 737721 d^4 + 72808 d^5 - 59180 d^6 + 5988 d^7 - 216 d^8) \\&+ c^5 (419301 - 1588280 d + 2153816 d^2 - 1617660 d^3 \\&+ 460657 d^4 + 12624 d^5 - 11492 d^6 + 1204 d^7 - 20 d^8) \\&- 2 c (-3300 + 29964 d - 70133 d^2 + 40126 d^3 - 58260 d^4 \\&+ 62694 d^5 + 54171 d^6 - 27972 d^7 + 486 d^8) \\&+ a^4 (1 + c)^4 (1 + c - d)^3 (33 + 3 c^4 + 23 d + d^2 + 3 d^3\\&- 8 d^4 - 3 c^3 (-6 + 7 d) + 5 c^2 (12 - 7 d + 3 d^2) \\&+ c (78 + 9 d + 24 d^2 + 11 d^3)) \\&+ 2 a^3 (1 + c)^2 (1 + c - d)^2 (3 c^8 + 3 c^7 (-13 + d) \\&+ c^6 (-303 + 194 d - 36 d^2) + c^5 (-1023 + 1051 d - 338 d^2 + 60 d^3) \\&+ c^4 (-2097 + 2418 d - 1110 d^2 + 256 d^3 - 39 d^4)\\&+ c^2 (-2061 + 2206 d - 1144 d^2 + 116 d^3 - 9 d^4 + 6 d^5)\\&+ c (-861 + 921 d - 274 d^2 - 152 d^3 + 71 d^4 + 7 d^5)\\&+ c^3 (-2685 + 3017 d - 1660 d^2 + 364 d^3 - 79 d^4\\&+ 9 d^5) + 2 (-75 + 87 d + 9 d^2 - 50 d^3 + 20 d^4 + 9 d^5)) \\&+ 2 a (-570 + 12 c^13 + 4602 d - 8508 d^2 + 2256 d^3 \\&- 6536 d^4 + 5394 d^5 + 3636 d^6 - 1836 d^7 - 486 d^8 \\&- 3 c^12 (25 + 28 d) + c^11 (-1749 + 483 d + 254 d^2)\\&- c^10 (11439 - 10926 d + 1313 d^2 + 432 d^3) \\&+ c^9 (-43659 + 65901 d - 27809 d^2 + 1935 d^3 + 450 d^4) \\&- c^8 (113664 - 223530 d + 152696 d^2 - 36954 d^3 \\&+ 1635 d^4 + 292 d^5) + c^7 (-213090 + 502398 d - 458296 d^2 + 179418 d^3\\&- 27067 d^4 + 745 d^5 + 114 d^6) \\&- c^6 (292350 - 799164 d + 877734 d^2 - 469938 d^3 + 108571 d^4 - 10442 d^5 \\&+ 123 d^6 + 24 d^7)+c (-6711 + 42921 d - 69833 d^2 + 41241 d^3 - 50303 d^4\\&+ 17467 d^5 + 14685 d^6 - 3357 d^7 - 702 d^8) \\&+ c^2 (-34755 + 180822 d - 269761 d^2 + 210390 d^3 - 160737 d^4 + 24806 d^5 \\&+ 20721 d^6 - 1626 d^7-324 d^8) \\&+ c^3 (-105561 + 456495 d - 641326 d^2 + 533862 d^3\\&- 287433 d^4 + 29499 d^5 + 10716 d^6 + 40 d^7 - 36 d^8) \\&+ c^5 (-292794 + 922458 d - 1139246 d^2 + 764952 d^3 \\&- 235455 d^4 + 28193 d^5 - 1803 d^6 - 27 d^7 + 2 d^8)\\&+ c^4 (-210699 + 771696 d - 1026244 d^2 + 802302 d^3\\&- 322329 d^4 + 35554 d^5 - 522 d^6 + 142 d^7 + 10 d^8))\\&- a^2 (1 + c) (-915 + 9 c^12 + c^11 (24 - 60 d) + 4732 d\\&- 5579 d^2 + 3016 d^3 - 717 d^4 - 804 d^5 - 201 d^6\\&+ 576 d^7 - 108 d^8 + c^10 (-762 - 22 d + 171 d^2)\\&- 2 c^9 (3228 - 2521 d + 166 d^2 + 135 d^3) \\&+ c^8 (-26085 + 35796 d - 13867 d^2 + 1154 d^3 + 255 d^4)\\&- 2 c^7 (33816 - 62104 d + 39662 d^2 - 10297 d^3 + 850 d^4 \\&+ 72 d^5) + c^6 (-122556 + 268268 d - 229302 d^2 + 90218 d^3\\&- 17912 d^4 + 1326 d^5 + 45 d^6) - 2 c^5 (79176 - 196350 d + 201394 d^2 \\&- 102595 d^3 +28238 d^4 - 4625 d^5 + 274 d^6 + 3 d^7) \\&+ c^4 (-144561 + 401152 d - 457802 d^2 + 277966 d^3 \\&- 90714 d^4 + 20104 d^5 - 2759 d^6 + 102 d^7) \\&+ c^2 (-36906 + 133338 d - 162229 d^2 + 113790 d^3\\&- 41440 d^4 + 7310 d^5 - 3789 d^6 + 970 d^7 - 36 d^8) \\&- 2 c^3 (45300 - 141998 d + 170586 d^2 - 115419 d^3\\&+ 41134 d^4 - 9730 d^5 + 2362 d^6 - 225 d^7 + 2 d^8)\\&- 2 c (4380 - 18689 d + 22496 d^2 - 14896 d^3 + 5042 d^4 \\&+ 315 d^5 + 716 d^6 - 570 d^7 + 54 d^8))))/((-3 + a)^2 (1\\&+ c - d) d^10 (1 - c^2 + a (-1 + c) (1 + c - d) - 9 d \\&+ c d)^3 (1 - c^2 + a (1 + c) (-1 + c - d) - 3 d + c d)^4 (-1\\&- 5 c^2 + a (1 + c) (1 + c - d) + 9 d + c (-6 + 5 d))^5)). \end{aligned}$$

Sets \(Q_1,Q_2,Q_3,Q_4,Q5,Q_6\)   and  \(Q_7\)

$$\begin{aligned} Q_1= & {} \left\{ (a,c,d)|0<c<1, \, 3<a\le \frac{4}{c+1}+5\,\,\text{ and } \,\, c+1<d<\frac{(c+1) ((a-5) c+a-1)}{(a-5) c+a-9}\right\} .\\ Q_2= & {} \left\{ (a,c,d)|0<c<1,\,\, \frac{4}{c+1}+5<a\le \frac{c-9}{c-1}\,\, \text{ and } \,\, d>\frac{(c+1) ((a-5) c+a-1)}{(a-5) c+a-9} \right\} .\\ Q_3= & {} \left\{ (a,c,d)|0<c<1,\,\, a>\frac{c-9}{c-1} \,\, \text{ and } \,\, \frac{(c+1) ((a-5) c+a-1)}{(a-5) c+a-9}<d<\frac{(a-1) \left( c^2-1\right) }{a (c-1)-c+9} \right\} .\\ Q_4= & {} \left\{ (a,c,d)|0<c<1,\,\, a>\frac{c-9}{c-1} \,\,\text{ and } \,\, d>\frac{(a-1) \left( c^2-1\right) }{a (c-1)-c+9} \right\} .\\ Q_5= & {} \left\{ (a,c,d)|c\ge 1, \,\, \wedge 3<a\le \frac{4}{c+1}+5 \,\, \text{ and } \,\, d>c+1 \right\} .\\ Q_6= & {} \left\{ (a,c,d)|c\ge 1, \,\, a>\frac{4}{c+1}+5 \,\, \text{ and }\,\, c+1<d<\frac{(c+1) ((a-5) c+a-1)}{(a-5) c+a-9} \right\} .\\ Q_7= & {} \left\{ (a,c,d)|c\ge 1,\,\, a>\frac{4}{c+1}+5\,\, \text{ and } \,\, d>\frac{(c+1) ((a-5) c+a-1)}{(a-5) c+a-9} \right\} . \end{aligned}$$

Expressions \({\gamma _i}\) and \({\beta _i}\) for \({i\in \{1,2,3,4,5\}}\)

$$\begin{aligned} \gamma _{1}= & {} \frac{a \sqrt{c+1} \left( c^2+c d-2 d^2+d-1\right) \sqrt{(a-1) \left( a (c+1) (c-d+1)-5 c^2+c (5 d-6)+9 d-1\right) }}{4 (a-1) d^2 (c-d) \sqrt{-c+d-1}}.\\ \gamma _{2}= & {} \frac{a \left( a (c+1) \left( c^2+c d-2 d^2+d-1\right) -c^4+2 c^3 (d-1)-c^2 (d-1) d+c \left( d^2-4 d+2\right) +4 d^2-3 d+1\right) }{2 (a-1) d^2 (c-d)}.\\ \gamma _{3}= & {} \frac{a^2 (c-1) \sqrt{c+1} \sqrt{-c+d-1} \left( c^2-c d+c-2 d\right) ^2 \sqrt{(a-1) \left( a (c+1) (c-d+1)-5 c^2+c (5 d-6)+9 d-1\right) }}{4 (a-1)^3 d^4 (c-d)^2}.\\ \gamma _{4}= & {} \frac{a \sqrt{c+1} \sqrt{-c+d-1} \left( a (c+1) \left( c^2+c d-2 d^2+d-1\right) -2 c^4+c^3 (4 d-3)+c^2 \left( -2 d^2+3 d+1\right) +c (3-6 d)+6 d^2-5 d+1\right) }{4 d^2 (c-d) \sqrt{(a-1) \left( a (c+1) (c-d+1)-5 c^2+c (5 d-6)+9 d-1\right) }}.\\ \gamma _{5}= & {} \frac{a^2 \left( c^2-1\right) (c-d+1) \left( c^2-c d+c-2 d\right) ^2}{2 (a-1)^2 d^4 (c-d)^2}.\\ \beta _{1}= & {} \frac{a (c+1) \left( a (c+1) (c-d+1)-5 c^2+c (5 d-6)+9 d-1\right) }{4 (a-1) d^2 (c-d)}.\\ \beta _{2}= & {} \frac{a \sqrt{c+1} \sqrt{-c+d-1} \left( a (c+1)-c^2+c (d-2)+2 d-1\right) \sqrt{(a-1) \left( a (c+1) (c-d+1)-5 c^2+c (5 d-6)+9 d-1\right) }}{2 (a-1)^2 d^2 (c-d)}.\\ \beta _{3}= & {} \frac{a^2 (c+1) \left( c^2-c d+c-2 d\right) ^2 \left( a (c+1) (c-d+1)-5 c^2+c (5 d-6)+9 d-1\right) }{4 (a-1)^3 d^4 (c-d)^2}.\\ \beta _{4}= & {} \frac{a (c+1) (c-d+1) \left( a c+a-2 c^2+2 c d-3 c+4 d-1\right) }{4 (a-1) d^2 (c-d)}.\\ \beta _{5}= & {} \frac{a^2 (c+1)^{3/2} \sqrt{-c+d-1} \left( c^2-c d+c-2 d\right) ^2 \sqrt{(a-1) \left( a (c+1) (c-d+1)-5 c^2+c (5 d-6)+9 d-1\right) }}{2 (a-1)^3 d^4 (c-d)^2}. \end{aligned}$$

Parameter Sets

$$\begin{aligned} \varGamma _1= & {} \left\{ (a,c,d,e)|a>\frac{c^2+c-d^2-d}{c d-d^2+d},\,\, c>0,\,\, d>c+1\,\, \text{ and } \,\, e=\frac{d-c-1}{c+d+1} \right\} .\\ \varGamma _2= & {} \left\{ (a,c,d,e)|\frac{(c-d) (c-d+1)}{c^2 (e+1)+c (-d (e+2)+e+1)+d (d-2 e-1)}\right. \\< & {} \left. a< a_1,\,\, 0<c \le 3, \,\, d>c+1 \,\,\text{ and } \,\, 0<e<\frac{d-c-1}{c+d+1} \right\} .\\ \varGamma _3= & {} \left\{ (a,c,d,e)|\frac{(c-d) (c-d+1)}{c^2 (e+1)+c (-d (e+2)+e+1)+d (d-2 e-1)}\right. \\< & {} \left. a< a_1,\,\, 0<c \le 3, \,\, d>c+1 \,\,\text{ and } \,\, \frac{d-c-1}{c+d+1}<e\right. \\< & {} \left. -\frac{4 (c-d) (c-d+1)}{c (5 c-5 d+6)-9 d+1} \right\} .\\ \varGamma _4= & {} \left\{ (a,c,d,e)|\frac{(c-d) (c-d+1)}{c^2 (e+1)+c (-d (e+2)+e+1)+d (d-2 e-1)}\right. \\< & {} \left. a< a_1,\,\, c>3, \,\, d>\frac{c^2-1}{c-3} \,\,\text{ and } 0<e<\frac{d-c-1}{c+d+1} \right\} .\\ \varGamma _5= & {} \left\{ (a,c,d,e)|\frac{(c-d) (c-d+1)}{c^2 (e+1)+c (-d (e+2)+e+1)+d (d-2 e-1)}\right. \\< & {} \left. a< a_1,\,\, c>3, c+1<d\le \frac{c^2-1}{c-3}\,\, \text{ and } \,\, \frac{d-c-1}{c+d+1}<e\right. \\< & {} \left. -\frac{4 (c-d) (c-d+1)}{c (5 c-5 d+6)-9 d+1} \right\} .\\ \varGamma _6= & {} \{(a,c,d,e)|\frac{(c-d) (c-d+1)}{c^2 (e+1)+c (-d (e+2)+e+1)+d (d-2 e-1)}\\< & {} a< a_1,\,\, c>3, d>c+\frac{8}{c-3}+3\,\, \text{ and } \,\, \frac{d-c-1}{c+d+1}<e\\\le & {} -\frac{4 (c-d) (c-d+1)}{c (5 c-5 d+6)-9 d+1} \}.\\ \varGamma _7= & {} \{(a,c,d,e)|\frac{(c-d) (c-d+1)}{c^2 (e+1)+c (-d (e+2)+e+1)+d (d-2 e-1)}\\< & {} a< a_1,\,\, c>3, \,\, d>c+1\,\, \text{ and } 0<e<\frac{d-c-1}{c+d+1} \}.\\ \varGamma _8= & {} \left\{ (a,c,d,e)|\frac{(c-d) (c-d+1)}{c^2 (e+1)+c (-d (e+2)+e+1)+d (d-2 e-1)}\right. \\< & {} a< a_1,\, c>3, d>c+\frac{8}{c-3}+3\, \text{ and }\\&-\frac{4 (c-d) (c-d+1)}{c (5 c-5 d+6)-9 d+1}<e\\< & {} \left. \frac{1}{2} \left( -\sqrt{-\frac{(c-d+1)^2 \left( 4 (c+1) d-c-4 d^2-1\right) }{(c+1) (c (c-d+3)+2)^2}}\right. \right. \\&\left. \left. +\frac{2 c d+c+d (5-2 d)+1}{c (c-d+3)+2}-2\right) \right\} . \end{aligned}$$

Where

$$\begin{aligned} a_1= & {} -\displaystyle \frac{N_1}{d^2 (c-d+1)^2} \\&+ \displaystyle \frac{2 \sqrt{d^2 (c-d+1)^3 \left( (c+1) e^2 (c (c-d+3)+2)+(c+1) e (2 c-2 d+3) (c-d+1)+(c-d+1)^3\right) }}{(c+1) (e (c+d+1)+c-d+1)^2} \end{aligned}$$

and

$$\begin{aligned} N_1= & {} 2 a (c+1)^2 e (c-d+1) \left( d (a+2 c+1)+(a-1) (c+1)-2 d^2\right) \\&+a^2 (c+1)^2 e^2 (c+d+1)^2 +(a-1) (c+1) (c-d+1)^2 \\&\times \left( (a-1) (c+1)+4 (c+1) d-4 d^2\right) . \end{aligned}$$

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Arias, C.F., Blé, G. & Falconi, M. Dynamics of a Discrete-Time Predator–Prey System with Holling II Functional Response. Qual. Theory Dyn. Syst. 21, 31 (2022). https://doi.org/10.1007/s12346-022-00562-5

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  • DOI: https://doi.org/10.1007/s12346-022-00562-5

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