Présentation du problème

On cherche à déterminer le chemin le plus rapide entre un sauveteur sur la plage et un nageur en perdition dans la mer.

La modélisation est représentée par le dessin suivant, la seule variable endogène étant $x$.

Initialisations

import sympy as sp

# Pour éviter des affichages impestifs dûe à un appel obsolète de sympy vers matplotlib
import warnings
warnings.filterwarnings("ignore", category=UserWarning)

sp.init_printing()

Définition des objets fondamentaux

xs, ys, xn, yn, vs, ve, x = sp.symbols("x_s y_s x_n y_n v_s v_e x")

T = sp.sqrt((xs - x) ** 2 + ys ** 2) / vs +  sp.sqrt((xn - x) ** 2 + yn ** 2) / ve
T

$\displaystyle \frac{\sqrt{y_{s}^{2} + \left(- x + x_{s}\right)^{2}}}{v_{s}} + \frac{\sqrt{y_{n}^{2} + \left(- x + x_{n}\right)^{2}}}{v_{e}}$

derive = T.diff(x)
derive

$\displaystyle \frac{x - x_{s}}{v_{s} \sqrt{y_{s}^{2} + \left(- x + x_{s}\right)^{2}}} + \frac{x - x_{n}}{v_{e} \sqrt{y_{n}^{2} + \left(- x + x_{n}\right)^{2}}}$

#sp.solve(sp.Eq(derive, 0), x)

REMARQUE la résolution naïve tourne indéfiniment et a donc été commentée.

Manipulations de l'expression pour guider le calcul

Visualisation de l'expression elle même

Le graphique est généré en utilisant graphviz (commande dot).

sp.srepr(derive)

"Add(Mul(Pow(Symbol('v_s'), Integer(-1)), Add(Symbol('x'), Mul(Integer(-1), Symbol('x_s'))), Pow(Add(Pow(Symbol('y_s'), Integer(2)), Pow(Add(Mul(Integer(-1), Symbol('x')), Symbol('x_s')), Integer(2))), Rational(-1, 2))), Mul(Pow(Symbol('v_e'), Integer(-1)), Add(Symbol('x'), Mul(Integer(-1), Symbol('x_n'))), Pow(Add(Pow(Symbol('y_n'), Integer(2)), Pow(Add(Mul(Integer(-1), Symbol('x')), Symbol('x_n')), Integer(2))), Rational(-1, 2))))"

with open("arbre.dot", "w") as fichier:
    fichier.write(sp.printing.dot.dotprint(derive))

%%sh
dot -Tsvg arbre.dot > arbre.svg

Manipulations

• On va récupérer les premiers termes de l'arbre grâce aux attributs func et args.
• On pourrait itérer sur les sommets en utilisant les fonctions preorder_traversal ou postorder_traversal.

derive.func

sympy.core.add.Add

gauche, droite = derive.args

gauche

$\displaystyle \frac{x - x_{n}}{v_{e} \sqrt{y_{n}^{2} + \left(- x + x_{n}\right)^{2}}}$

droite

$\displaystyle \frac{x - x_{s}}{v_{s} \sqrt{y_{s}^{2} + \left(- x + x_{s}\right)^{2}}}$

Attention contrairement à ce qu'on pourrait croire les termes gauche et droite sont des produits. Pour les traiter comme des fractions et récupérer numérateurs et dénominateurs il faut une méthode spécifique.

num_g, den_g = gauche.as_numer_denom()
num_d, den_d = droite.as_numer_denom()

p = num_g ** 2 * den_d ** 2 -  num_d ** 2 * den_g ** 2
p

$\displaystyle - v_{e}^{2} \left(x - x_{s}\right)^{2} \left(y_{n}^{2} + \left(- x + x_{n}\right)^{2}\right) + v_{s}^{2} \left(x - x_{n}\right)^{2} \left(y_{s}^{2} + \left(- x + x_{s}\right)^{2}\right)$

p = p.as_poly(x)
p

$\displaystyle \operatorname{Poly}{\left( \left(- v_{e}^{2} + v_{s}^{2}\right) x^{4} + \left(2 v_{e}^{2} x_{n} + 2 v_{e}^{2} x_{s} - 2 v_{s}^{2} x_{n} - 2 v_{s}^{2} x_{s}\right) x^{3} + \left(- v_{e}^{2} x_{n}^{2} - 4 v_{e}^{2} x_{n} x_{s} - v_{e}^{2} x_{s}^{2} - v_{e}^{2} y_{n}^{2} + v_{s}^{2} x_{n}^{2} + 4 v_{s}^{2} x_{n} x_{s} + v_{s}^{2} x_{s}^{2} + v_{s}^{2} y_{s}^{2}\right) x^{2} + \left(2 v_{e}^{2} x_{n}^{2} x_{s} + 2 v_{e}^{2} x_{n} x_{s}^{2} + 2 v_{e}^{2} x_{s} y_{n}^{2} - 2 v_{s}^{2} x_{n}^{2} x_{s} - 2 v_{s}^{2} x_{n} x_{s}^{2} - 2 v_{s}^{2} x_{n} y_{s}^{2}\right) x - v_{e}^{2} x_{n}^{2} x_{s}^{2} - v_{e}^{2} x_{s}^{2} y_{n}^{2} + v_{s}^{2} x_{n}^{2} x_{s}^{2} + v_{s}^{2} x_{n}^{2} y_{s}^{2}, x, domain=\mathbb{Z}\left[v_{e}, v_{s}, x_{n}, x_{s}, y_{n}, y_{s}\right] \right)}$

REMARQUE à ce stade on s'est ramené à trouver des racines d'un polynôme de degré 4 en $x$. On notera par contre que les coefficients sont considérés êtres des polynômes à coefficients entiers en les variables exogènes.

p.get_domain()

ZZ[v_e,v_s,x_n,x_s,y_n,y_s]

p = p.to_field()
p

$\displaystyle \operatorname{Poly}{\left( \left(- v_{e}^{2} + v_{s}^{2}\right) x^{4} + \left(2 v_{e}^{2} x_{n} + 2 v_{e}^{2} x_{s} - 2 v_{s}^{2} x_{n} - 2 v_{s}^{2} x_{s}\right) x^{3} + \left(- v_{e}^{2} x_{n}^{2} - 4 v_{e}^{2} x_{n} x_{s} - v_{e}^{2} x_{s}^{2} - v_{e}^{2} y_{n}^{2} + v_{s}^{2} x_{n}^{2} + 4 v_{s}^{2} x_{n} x_{s} + v_{s}^{2} x_{s}^{2} + v_{s}^{2} y_{s}^{2}\right) x^{2} + \left(2 v_{e}^{2} x_{n}^{2} x_{s} + 2 v_{e}^{2} x_{n} x_{s}^{2} + 2 v_{e}^{2} x_{s} y_{n}^{2} - 2 v_{s}^{2} x_{n}^{2} x_{s} - 2 v_{s}^{2} x_{n} x_{s}^{2} - 2 v_{s}^{2} x_{n} y_{s}^{2}\right) x - v_{e}^{2} x_{n}^{2} x_{s}^{2} - v_{e}^{2} x_{s}^{2} y_{n}^{2} + v_{s}^{2} x_{n}^{2} x_{s}^{2} + v_{s}^{2} x_{n}^{2} y_{s}^{2}, x, domain=\mathbb{Z}\left(v_{e}, v_{s}, x_{n}, x_{s}, y_{n}, y_{s}\right) \right)}$

sols = sp.solve(p, x)

type(sols)

list

sols[0]

$\displaystyle \begin{cases} \frac{x_{n}}{2} + \frac{x_{s}}{2} - \frac{\sqrt{\frac{\left(- 2 x_{n} - 2 x_{s}\right)^{2}}{4} - 2 \sqrt[3]{\frac{\left(- \left(- 2 x_{n} - 2 x_{s}\right) \left(\left(- 2 x_{n} - 2 x_{s}\right) \left(\frac{3 \left(- 2 x_{n} - 2 x_{s}\right)^{2}}{256} - \frac{v_{e}^{2} x_{n}^{2} + 4 v_{e}^{2} x_{n} x_{s} + v_{e}^{2} x_{s}^{2} + v_{e}^{2} y_{n}^{2} - v_{s}^{2} x_{n}^{2} - 4 v_{s}^{2} x_{n} x_{s} - v_{s}^{2} x_{s}^{2} - v_{s}^{2} y_{s}^{2}}{16 \left(v_{e}^{2} - v_{s}^{2}\right)}\right) + \frac{- 2 v_{e}^{2} x_{n}^{2} x_{s} - 2 v_{e}^{2} x_{n} x_{s}^{2} - 2 v_{e}^{2} x_{s} y_{n}^{2} + 2 v_{s}^{2} x_{n}^{2} x_{s} + 2 v_{s}^{2} x_{n} x_{s}^{2} + 2 v_{s}^{2} x_{n} y_{s}^{2}}{4 \left(v_{e}^{2} - v_{s}^{2}\right)}\right) + \frac{v_{e}^{2} x_{n}^{2} x_{s}^{2} + v_{e}^{2} x_{s}^{2} y_{n}^{2} - v_{s}^{2} x_{n}^{2} x_{s}^{2} - v_{s}^{2} x_{n}^{2} y_{s}^{2}}{v_{e}^{2} - v_{s}^{2}}\right) \left(- \frac{3 \left(- 2 x_{n} - 2 x_{s}\right)^{2}}{8} + \frac{v_{e}^{2} x_{n}^{2} + 4 v_{e}^{2} x_{n} x_{s} + v_{e}^{2} x_{s}^{2} + v_{e}^{2} y_{n}^{2} - v_{s}^{2} x_{n}^{2} - 4 v_{s}^{2} x_{n} x_{s} - v_{s}^{2} x_{s}^{2} - v_{s}^{2} y_{s}^{2}}{v_{e}^{2} - v_{s}^{2}}\right)}{3} - \frac{\left(\left(- 2 x_{n} - 2 x_{s}\right) \left(\frac{\left(- 2 x_{n} - 2 x_{s}\right)^{2}}{8} - \frac{v_{e}^{2} x_{n}^{2} + 4 v_{e}^{2} x_{n} x_{s} + v_{e}^{2} x_{s}^{2} + v_{e}^{2} y_{n}^{2} - v_{s}^{2} x_{n}^{2} - 4 v_{s}^{2} x_{n} x_{s} - v_{s}^{2} x_{s}^{2} - v_{s}^{2} y_{s}^{2}}{2 \left(v_{e}^{2} - v_{s}^{2}\right)}\right) + \frac{- 2 v_{e}^{2} x_{n}^{2} x_{s} - 2 v_{e}^{2} x_{n} x_{s}^{2} - 2 v_{e}^{2} x_{s} y_{n}^{2} + 2 v_{s}^{2} x_{n}^{2} x_{s} + 2 v_{s}^{2} x_{n} x_{s}^{2} + 2 v_{s}^{2} x_{n} y_{s}^{2}}{v_{e}^{2} - v_{s}^{2}}\right)^{2}}{8} - \frac{\left(- \frac{3 \left(- 2 x_{n} - 2 x_{s}\right)^{2}}{8} + \frac{v_{e}^{2} x_{n}^{2} + 4 v_{e}^{2} x_{n} x_{s} + v_{e}^{2} x_{s}^{2} + v_{e}^{2} y_{n}^{2} - v_{s}^{2} x_{n}^{2} - 4 v_{s}^{2} x_{n} x_{s} - v_{s}^{2} x_{s}^{2} - v_{s}^{2} y_{s}^{2}}{v_{e}^{2} - v_{s}^{2}}\right)^{3}}{108}} - \frac{2 \left(v_{e}^{2} x_{n}^{2} + 4 v_{e}^{2} x_{n} x_{s} + v_{e}^{2} x_{s}^{2} + v_{e}^{2} y_{n}^{2} - v_{s}^{2} x_{n}^{2} - 4 v_{s}^{2} x_{n} x_{s} - v_{s}^{2} x_{s}^{2} - v_{s}^{2} y_{s}^{2}\right)}{3 \left(v_{e}^{2} - v_{s}^{2}\right)}}}{2} - \frac{\sqrt{\frac{\left(- 2 x_{n} - 2 x_{s}\right)^{2}}{2} + \frac{2 \left(- 2 x_{n} - 2 x_{s}\right) \left(\frac{\left(- 2 x_{n} - 2 x_{s}\right)^{2}}{8} - \frac{v_{e}^{2} x_{n}^{2} + 4 v_{e}^{2} x_{n} x_{s} + v_{e}^{2} x_{s}^{2} + v_{e}^{2} y_{n}^{2} - v_{s}^{2} x_{n}^{2} - 4 v_{s}^{2} x_{n} x_{s} - v_{s}^{2} x_{s}^{2} - v_{s}^{2} y_{s}^{2}}{2 \left(v_{e}^{2} - v_{s}^{2}\right)}\right) + \frac{2 \left(- 2 v_{e}^{2} x_{n}^{2} x_{s} - 2 v_{e}^{2} x_{n} x_{s}^{2} - 2 v_{e}^{2} x_{s} y_{n}^{2} + 2 v_{s}^{2} x_{n}^{2} x_{s} + 2 v_{s}^{2} x_{n} x_{s}^{2} + 2 v_{s}^{2} x_{n} y_{s}^{2}\right)}{v_{e}^{2} - v_{s}^{2}}}{\sqrt{\frac{\left(- 2 x_{n} - 2 x_{s}\right)^{2}}{4} - 2 \sqrt[3]{\frac{\left(- \left(- 2 x_{n} - 2 x_{s}\right) \left(\left(- 2 x_{n} - 2 x_{s}\right) \left(\frac{3 \left(- 2 x_{n} - 2 x_{s}\right)^{2}}{256} - \frac{v_{e}^{2} x_{n}^{2} + 4 v_{e}^{2} x_{n} x_{s} + v_{e}^{2} x_{s}^{2} + v_{e}^{2} y_{n}^{2} - v_{s}^{2} x_{n}^{2} - 4 v_{s}^{2} x_{n} x_{s} - v_{s}^{2} x_{s}^{2} - v_{s}^{2} y_{s}^{2}}{16 \left(v_{e}^{2} - v_{s}^{2}\right)}\right) + \frac{- 2 v_{e}^{2} x_{n}^{2} x_{s} - 2 v_{e}^{2} x_{n} x_{s}^{2} - 2 v_{e}^{2} x_{s} y_{n}^{2} + 2 v_{s}^{2} x_{n}^{2} x_{s} + 2 v_{s}^{2} x_{n} x_{s}^{2} + 2 v_{s}^{2} x_{n} y_{s}^{2}}{4 \left(v_{e}^{2} - v_{s}^{2}\right)}\right) + \frac{v_{e}^{2} x_{n}^{2} x_{s}^{2} + v_{e}^{2} x_{s}^{2} y_{n}^{2} - v_{s}^{2} x_{n}^{2} x_{s}^{2} - v_{s}^{2} x_{n}^{2} y_{s}^{2}}{v_{e}^{2} - v_{s}^{2}}\right) \left(- \frac{3 \left(- 2 x_{n} - 2 x_{s}\right)^{2}}{8} + \frac{v_{e}^{2} x_{n}^{2} + 4 v_{e}^{2} x_{n} x_{s} + v_{e}^{2} x_{s}^{2} + v_{e}^{2} y_{n}^{2} - v_{s}^{2} x_{n}^{2} - 4 v_{s}^{2} x_{n} x_{s} - v_{s}^{2} x_{s}^{2} - v_{s}^{2} y_{s}^{2}}{v_{e}^{2} - v_{s}^{2}}\right)}{3} - \frac{\left(\left(- 2 x_{n} - 2 x_{s}\right) \left(\frac{\left(- 2 x_{n} - 2 x_{s}\right)^{2}}{8} - \frac{v_{e}^{2} x_{n}^{2} + 4 v_{e}^{2} x_{n} x_{s} + v_{e}^{2} x_{s}^{2} + v_{e}^{2} y_{n}^{2} - v_{s}^{2} x_{n}^{2} - 4 v_{s}^{2} x_{n} x_{s} - v_{s}^{2} x_{s}^{2} - v_{s}^{2} y_{s}^{2}}{2 \left(v_{e}^{2} - v_{s}^{2}\right)}\right) + \frac{- 2 v_{e}^{2} x_{n}^{2} x_{s} - 2 v_{e}^{2} x_{n} x_{s}^{2} - 2 v_{e}^{2} x_{s} y_{n}^{2} + 2 v_{s}^{2} x_{n}^{2} x_{s} + 2 v_{s}^{2} x_{n} x_{s}^{2} + 2 v_{s}^{2} x_{n} y_{s}^{2}}{v_{e}^{2} - v_{s}^{2}}\right)^{2}}{8} - \frac{\left(- \frac{3 \left(- 2 x_{n} - 2 x_{s}\right)^{2}}{8} + \frac{v_{e}^{2} x_{n}^{2} + 4 v_{e}^{2} x_{n} x_{s} + v_{e}^{2} x_{s}^{2} + v_{e}^{2} y_{n}^{2} - v_{s}^{2} x_{n}^{2} - 4 v_{s}^{2} x_{n} x_{s} - v_{s}^{2} x_{s}^{2} - v_{s}^{2} y_{s}^{2}}{v_{e}^{2} - v_{s}^{2}}\right)^{3}}{108}} - \frac{2 \left(v_{e}^{2} x_{n}^{2} + 4 v_{e}^{2} x_{n} x_{s} + v_{e}^{2} x_{s}^{2} + v_{e}^{2} y_{n}^{2} - v_{s}^{2} x_{n}^{2} - 4 v_{s}^{2} x_{n} x_{s} - v_{s}^{2} x_{s}^{2} - v_{s}^{2} y_{s}^{2}\right)}{3 \left(v_{e}^{2} - v_{s}^{2}\right)}}} + 2 \sqrt[3]{\frac{\left(- \left(- 2 x_{n} - 2 x_{s}\right) \left(\left(- 2 x_{n} - 2 x_{s}\right) \left(\frac{3 \left(- 2 x_{n} - 2 x_{s}\right)^{2}}{256} - \frac{v_{e}^{2} x_{n}^{2} + 4 v_{e}^{2} x_{n} x_{s} + v_{e}^{2} x_{s}^{2} + v_{e}^{2} y_{n}^{2} - v_{s}^{2} x_{n}^{2} - 4 v_{s}^{2} x_{n} x_{s} - v_{s}^{2} x_{s}^{2} - v_{s}^{2} y_{s}^{2}}{16 \left(v_{e}^{2} - v_{s}^{2}\right)}\right) + \frac{- 2 v_{e}^{2} x_{n}^{2} x_{s} - 2 v_{e}^{2} x_{n} x_{s}^{2} - 2 v_{e}^{2} x_{s} y_{n}^{2} + 2 v_{s}^{2} x_{n}^{2} x_{s} + 2 v_{s}^{2} x_{n} x_{s}^{2} + 2 v_{s}^{2} x_{n} y_{s}^{2}}{4 \left(v_{e}^{2} - v_{s}^{2}\right)}\right) + \frac{v_{e}^{2} x_{n}^{2} x_{s}^{2} + v_{e}^{2} x_{s}^{2} y_{n}^{2} - v_{s}^{2} x_{n}^{2} x_{s}^{2} - v_{s}^{2} x_{n}^{2} y_{s}^{2}}{v_{e}^{2} - v_{s}^{2}}\right) \left(- \frac{3 \left(- 2 x_{n} - 2 x_{s}\right)^{2}}{8} + \frac{v_{e}^{2} x_{n}^{2} + 4 v_{e}^{2} x_{n} x_{s} + v_{e}^{2} x_{s}^{2} + v_{e}^{2} y_{n}^{2} - v_{s}^{2} x_{n}^{2} - 4 v_{s}^{2} x_{n} x_{s} - v_{s}^{2} x_{s}^{2} - v_{s}^{2} y_{s}^{2}}{v_{e}^{2} - v_{s}^{2}}\right)}{3} - \frac{\left(\left(- 2 x_{n} - 2 x_{s}\right) \left(\frac{\left(- 2 x_{n} - 2 x_{s}\right)^{2}}{8} - \frac{v_{e}^{2} x_{n}^{2} + 4 v_{e}^{2} x_{n} x_{s} + v_{e}^{2} x_{s}^{2} + v_{e}^{2} y_{n}^{2} - v_{s}^{2} x_{n}^{2} - 4 v_{s}^{2} x_{n} x_{s} - v_{s}^{2} x_{s}^{2} - v_{s}^{2} y_{s}^{2}}{2 \left(v_{e}^{2} - v_{s}^{2}\right)}\right) + \frac{- 2 v_{e}^{2} x_{n}^{2} x_{s} - 2 v_{e}^{2} x_{n} x_{s}^{2} - 2 v_{e}^{2} x_{s} y_{n}^{2} + 2 v_{s}^{2} x_{n}^{2} x_{s} + 2 v_{s}^{2} x_{n} x_{s}^{2} + 2 v_{s}^{2} x_{n} y_{s}^{2}}{v_{e}^{2} - v_{s}^{2}}\right)^{2}}{8} - \frac{\left(- \frac{3 \left(- 2 x_{n} - 2 x_{s}\right)^{2}}{8} + \frac{v_{e}^{2} x_{n}^{2} + 4 v_{e}^{2} x_{n} x_{s} + v_{e}^{2} x_{s}^{2} + v_{e}^{2} y_{n}^{2} - v_{s}^{2} x_{n}^{2} - 4 v_{s}^{2} x_{n} x_{s} - v_{s}^{2} x_{s}^{2} - v_{s}^{2} y_{s}^{2}}{v_{e}^{2} - v_{s}^{2}}\right)^{3}}{108}} - \frac{4 \left(v_{e}^{2} x_{n}^{2} + 4 v_{e}^{2} x_{n} x_{s} + v_{e}^{2} x_{s}^{2} + v_{e}^{2} y_{n}^{2} - v_{s}^{2} x_{n}^{2} - 4 v_{s}^{2} x_{n} x_{s} - v_{s}^{2} x_{s}^{2} - v_{s}^{2} y_{s}^{2}\right)}{3 \left(v_{e}^{2} - v_{s}^{2}\right)}}}{2} & \text{for}\: - \left(- 2 x_{n} - 2 x_{s}\right) \left(\left(- 2 x_{n} - 2 x_{s}\right) \left(\frac{3 \left(- 2 x_{n} - 2 x_{s}\right)^{2}}{256} - \frac{v_{e}^{2} x_{n}^{2} + 4 v_{e}^{2} x_{n} x_{s} + v_{e}^{2} x_{s}^{2} + v_{e}^{2} y_{n}^{2} - v_{s}^{2} x_{n}^{2} - 4 v_{s}^{2} x_{n} x_{s} - v_{s}^{2} x_{s}^{2} - v_{s}^{2} y_{s}^{2}}{16 \left(v_{e}^{2} - v_{s}^{2}\right)}\right) + \frac{- 2 v_{e}^{2} x_{n}^{2} x_{s} - 2 v_{e}^{2} x_{n} x_{s}^{2} - 2 v_{e}^{2} x_{s} y_{n}^{2} + 2 v_{s}^{2} x_{n}^{2} x_{s} + 2 v_{s}^{2} x_{n} x_{s}^{2} + 2 v_{s}^{2} x_{n} y_{s}^{2}}{4 \left(v_{e}^{2} - v_{s}^{2}\right)}\right) + \frac{\left(- \frac{3 \left(- 2 x_{n} - 2 x_{s}\right)^{2}}{8} + \frac{v_{e}^{2} x_{n}^{2} + 4 v_{e}^{2} x_{n} x_{s} + v_{e}^{2} x_{s}^{2} + v_{e}^{2} y_{n}^{2} - v_{s}^{2} x_{n}^{2} - 4 v_{s}^{2} x_{n} x_{s} - v_{s}^{2} x_{s}^{2} - v_{s}^{2} y_{s}^{2}}{v_{e}^{2} - v_{s}^{2}}\right)^{2}}{12} + \frac{v_{e}^{2} x_{n}^{2} x_{s}^{2} + v_{e}^{2} x_{s}^{2} y_{n}^{2} - v_{s}^{2} x_{n}^{2} x_{s}^{2} - v_{s}^{2} x_{n}^{2} y_{s}^{2}}{v_{e}^{2} - v_{s}^{2}} = 0 \\\frac{x_{n}}{2} + \frac{x_{s}}{2} - \frac{\sqrt{\frac{\left(- 2 x_{n} - 2 x_{s}\right)^{2}}{4} - \frac{2 \left(\left(- 2 x_{n} - 2 x_{s}\right) \left(\left(- 2 x_{n} - 2 x_{s}\right) \left(\frac{3 \left(- 2 x_{n} - 2 x_{s}\right)^{2}}{256} - \frac{v_{e}^{2} x_{n}^{2} + 4 v_{e}^{2} x_{n} x_{s} + v_{e}^{2} x_{s}^{2} + v_{e}^{2} y_{n}^{2} - v_{s}^{2} x_{n}^{2} - 4 v_{s}^{2} x_{n} x_{s} - v_{s}^{2} x_{s}^{2} - v_{s}^{2} y_{s}^{2}}{16 \left(v_{e}^{2} - v_{s}^{2}\right)}\right) + \frac{- 2 v_{e}^{2} x_{n}^{2} x_{s} - 2 v_{e}^{2} x_{n} x_{s}^{2} - 2 v_{e}^{2} x_{s} y_{n}^{2} + 2 v_{s}^{2} x_{n}^{2} x_{s} + 2 v_{s}^{2} x_{n} x_{s}^{2} + 2 v_{s}^{2} x_{n} y_{s}^{2}}{4 \left(v_{e}^{2} - v_{s}^{2}\right)}\right) - \frac{\left(- \frac{3 \left(- 2 x_{n} - 2 x_{s}\right)^{2}}{8} + \frac{v_{e}^{2} x_{n}^{2} + 4 v_{e}^{2} x_{n} x_{s} + v_{e}^{2} x_{s}^{2} + v_{e}^{2} y_{n}^{2} - v_{s}^{2} x_{n}^{2} - 4 v_{s}^{2} x_{n} x_{s} - v_{s}^{2} x_{s}^{2} - v_{s}^{2} y_{s}^{2}}{v_{e}^{2} - v_{s}^{2}}\right)^{2}}{12} - \frac{v_{e}^{2} x_{n}^{2} x_{s}^{2} + v_{e}^{2} x_{s}^{2} y_{n}^{2} - v_{s}^{2} x_{n}^{2} x_{s}^{2} - v_{s}^{2} x_{n}^{2} y_{s}^{2}}{v_{e}^{2} - v_{s}^{2}}\right)}{3 \sqrt[3]{- \frac{\left(- \left(- 2 x_{n} - 2 x_{s}\right) \left(\left(- 2 x_{n} - 2 x_{s}\right) \left(\frac{3 \left(- 2 x_{n} - 2 x_{s}\right)^{2}}{256} - \frac{v_{e}^{2} x_{n}^{2} + 4 v_{e}^{2} x_{n} x_{s} + v_{e}^{2} x_{s}^{2} + v_{e}^{2} y_{n}^{2} - v_{s}^{2} x_{n}^{2} - 4 v_{s}^{2} x_{n} x_{s} - v_{s}^{2} x_{s}^{2} - v_{s}^{2} y_{s}^{2}}{16 \left(v_{e}^{2} - v_{s}^{2}\right)}\right) + \frac{- 2 v_{e}^{2} x_{n}^{2} x_{s} - 2 v_{e}^{2} x_{n} x_{s}^{2} - 2 v_{e}^{2} x_{s} y_{n}^{2} + 2 v_{s}^{2} x_{n}^{2} x_{s} + 2 v_{s}^{2} x_{n} x_{s}^{2} + 2 v_{s}^{2} x_{n} y_{s}^{2}}{4 \left(v_{e}^{2} - v_{s}^{2}\right)}\right) + \frac{v_{e}^{2} x_{n}^{2} x_{s}^{2} + v_{e}^{2} x_{s}^{2} y_{n}^{2} - v_{s}^{2} x_{n}^{2} x_{s}^{2} - v_{s}^{2} x_{n}^{2} y_{s}^{2}}{v_{e}^{2} - v_{s}^{2}}\right) \left(- \frac{3 \left(- 2 x_{n} - 2 x_{s}\right)^{2}}{8} + \frac{v_{e}^{2} x_{n}^{2} + 4 v_{e}^{2} x_{n} x_{s} + v_{e}^{2} x_{s}^{2} + v_{e}^{2} y_{n}^{2} - v_{s}^{2} x_{n}^{2} - 4 v_{s}^{2} x_{n} x_{s} - v_{s}^{2} x_{s}^{2} - v_{s}^{2} y_{s}^{2}}{v_{e}^{2} - v_{s}^{2}}\right)}{6} + \frac{\left(\left(- 2 x_{n} - 2 x_{s}\right) \left(\frac{\left(- 2 x_{n} - 2 x_{s}\right)^{2}}{8} - \frac{v_{e}^{2} x_{n}^{2} + 4 v_{e}^{2} x_{n} x_{s} + v_{e}^{2} x_{s}^{2} + v_{e}^{2} y_{n}^{2} - v_{s}^{2} x_{n}^{2} - 4 v_{s}^{2} x_{n} x_{s} - v_{s}^{2} x_{s}^{2} - v_{s}^{2} y_{s}^{2}}{2 \left(v_{e}^{2} - v_{s}^{2}\right)}\right) + \frac{- 2 v_{e}^{2} x_{n}^{2} x_{s} - 2 v_{e}^{2} x_{n} x_{s}^{2} - 2 v_{e}^{2} x_{s} y_{n}^{2} + 2 v_{s}^{2} x_{n}^{2} x_{s} + 2 v_{s}^{2} x_{n} x_{s}^{2} + 2 v_{s}^{2} x_{n} y_{s}^{2}}{v_{e}^{2} - v_{s}^{2}}\right)^{2}}{16} + \frac{\left(- \frac{3 \left(- 2 x_{n} - 2 x_{s}\right)^{2}}{8} + \frac{v_{e}^{2} x_{n}^{2} + 4 v_{e}^{2} x_{n} x_{s} + v_{e}^{2} x_{s}^{2} + v_{e}^{2} y_{n}^{2} - v_{s}^{2} x_{n}^{2} - 4 v_{s}^{2} x_{n} x_{s} - v_{s}^{2} x_{s}^{2} - v_{s}^{2} y_{s}^{2}}{v_{e}^{2} - v_{s}^{2}}\right)^{3}}{216} + \sqrt{\frac{\left(\left(- 2 x_{n} - 2 x_{s}\right) \left(\left(- 2 x_{n} - 2 x_{s}\right) \left(\frac{3 \left(- 2 x_{n} - 2 x_{s}\right)^{2}}{256} - \frac{v_{e}^{2} x_{n}^{2} + 4 v_{e}^{2} x_{n} x_{s} + v_{e}^{2} x_{s}^{2} + v_{e}^{2} y_{n}^{2} - v_{s}^{2} x_{n}^{2} - 4 v_{s}^{2} x_{n} x_{s} - v_{s}^{2} x_{s}^{2} - v_{s}^{2} y_{s}^{2}}{16 \left(v_{e}^{2} - v_{s}^{2}\right)}\right) + \frac{- 2 v_{e}^{2} x_{n}^{2} x_{s} - 2 v_{e}^{2} x_{n} x_{s}^{2} - 2 v_{e}^{2} x_{s} y_{n}^{2} + 2 v_{s}^{2} x_{n}^{2} x_{s} + 2 v_{s}^{2} x_{n} x_{s}^{2} + 2 v_{s}^{2} x_{n} y_{s}^{2}}{4 \left(v_{e}^{2} - v_{s}^{2}\right)}\right) - \frac{\left(- \frac{3 \left(- 2 x_{n} - 2 x_{s}\right)^{2}}{8} + \frac{v_{e}^{2} x_{n}^{2} + 4 v_{e}^{2} x_{n} x_{s} + v_{e}^{2} x_{s}^{2} + v_{e}^{2} y_{n}^{2} - v_{s}^{2} x_{n}^{2} - 4 v_{s}^{2} x_{n} x_{s} - v_{s}^{2} x_{s}^{2} - v_{s}^{2} y_{s}^{2}}{v_{e}^{2} - v_{s}^{2}}\right)^{2}}{12} - \frac{v_{e}^{2} x_{n}^{2} x_{s}^{2} + v_{e}^{2} x_{s}^{2} y_{n}^{2} - v_{s}^{2} x_{n}^{2} x_{s}^{2} - v_{s}^{2} x_{n}^{2} y_{s}^{2}}{v_{e}^{2} - v_{s}^{2}}\right)^{3}}{27} + \frac{\left(\frac{\left(- \left(- 2 x_{n} - 2 x_{s}\right) \left(\left(- 2 x_{n} - 2 x_{s}\right) \left(\frac{3 \left(- 2 x_{n} - 2 x_{s}\right)^{2}}{256} - \frac{v_{e}^{2} x_{n}^{2} + 4 v_{e}^{2} x_{n} x_{s} + v_{e}^{2} x_{s}^{2} + v_{e}^{2} y_{n}^{2} - v_{s}^{2} x_{n}^{2} - 4 v_{s}^{2} x_{n} x_{s} - v_{s}^{2} x_{s}^{2} - v_{s}^{2} y_{s}^{2}}{16 \left(v_{e}^{2} - v_{s}^{2}\right)}\right) + \frac{- 2 v_{e}^{2} x_{n}^{2} x_{s} - 2 v_{e}^{2} x_{n} x_{s}^{2} - 2 v_{e}^{2} x_{s} y_{n}^{2} + 2 v_{s}^{2} x_{n}^{2} x_{s} + 2 v_{s}^{2} x_{n} x_{s}^{2} + 2 v_{s}^{2} x_{n} y_{s}^{2}}{4 \left(v_{e}^{2} - v_{s}^{2}\right)}\right) + \frac{v_{e}^{2} x_{n}^{2} x_{s}^{2} + v_{e}^{2} x_{s}^{2} y_{n}^{2} - v_{s}^{2} x_{n}^{2} x_{s}^{2} - v_{s}^{2} x_{n}^{2} y_{s}^{2}}{v_{e}^{2} - v_{s}^{2}}\right) \left(- \frac{3 \left(- 2 x_{n} - 2 x_{s}\right)^{2}}{8} + \frac{v_{e}^{2} x_{n}^{2} + 4 v_{e}^{2} x_{n} x_{s} + v_{e}^{2} x_{s}^{2} + v_{e}^{2} y_{n}^{2} - v_{s}^{2} x_{n}^{2} - 4 v_{s}^{2} x_{n} x_{s} - v_{s}^{2} x_{s}^{2} - v_{s}^{2} y_{s}^{2}}{v_{e}^{2} - v_{s}^{2}}\right)}{3} - \frac{\left(\left(- 2 x_{n} - 2 x_{s}\right) \left(\frac{\left(- 2 x_{n} - 2 x_{s}\right)^{2}}{8} - \frac{v_{e}^{2} x_{n}^{2} + 4 v_{e}^{2} x_{n} x_{s} + v_{e}^{2} x_{s}^{2} + v_{e}^{2} y_{n}^{2} - v_{s}^{2} x_{n}^{2} - 4 v_{s}^{2} x_{n} x_{s} - v_{s}^{2} x_{s}^{2} - v_{s}^{2} y_{s}^{2}}{2 \left(v_{e}^{2} - v_{s}^{2}\right)}\right) + \frac{- 2 v_{e}^{2} x_{n}^{2} x_{s} - 2 v_{e}^{2} x_{n} x_{s}^{2} - 2 v_{e}^{2} x_{s} y_{n}^{2} + 2 v_{s}^{2} x_{n}^{2} x_{s} + 2 v_{s}^{2} x_{n} x_{s}^{2} + 2 v_{s}^{2} x_{n} y_{s}^{2}}{v_{e}^{2} - v_{s}^{2}}\right)^{2}}{8} - \frac{\left(- \frac{3 \left(- 2 x_{n} - 2 x_{s}\right)^{2}}{8} + \frac{v_{e}^{2} x_{n}^{2} + 4 v_{e}^{2} x_{n} x_{s} + v_{e}^{2} x_{s}^{2} + v_{e}^{2} y_{n}^{2} - v_{s}^{2} x_{n}^{2} - 4 v_{s}^{2} x_{n} x_{s} - v_{s}^{2} x_{s}^{2} - v_{s}^{2} y_{s}^{2}}{v_{e}^{2} - v_{s}^{2}}\right)^{3}}{108}\right)^{2}}{4}}}} + 2 \sqrt[3]{- \frac{\left(- \left(- 2 x_{n} - 2 x_{s}\right) \left(\left(- 2 x_{n} - 2 x_{s}\right) \left(\frac{3 \left(- 2 x_{n} - 2 x_{s}\right)^{2}}{256} - \frac{v_{e}^{2} x_{n}^{2} + 4 v_{e}^{2} x_{n} x_{s} + v_{e}^{2} x_{s}^{2} + v_{e}^{2} y_{n}^{2} - v_{s}^{2} x_{n}^{2} - 4 v_{s}^{2} x_{n} x_{s} - v_{s}^{2} x_{s}^{2} - v_{s}^{2} y_{s}^{2}}{16 \left(v_{e}^{2} - v_{s}^{2}\right)}\right) + \frac{- 2 v_{e}^{2} x_{n}^{2} x_{s} - 2 v_{e}^{2} x_{n} x_{s}^{2} - 2 v_{e}^{2} x_{s} y_{n}^{2} + 2 v_{s}^{2} x_{n}^{2} x_{s} + 2 v_{s}^{2} x_{n} x_{s}^{2} + 2 v_{s}^{2} x_{n} y_{s}^{2}}{4 \left(v_{e}^{2} - v_{s}^{2}\right)}\right) + \frac{v_{e}^{2} x_{n}^{2} x_{s}^{2} + v_{e}^{2} x_{s}^{2} y_{n}^{2} - v_{s}^{2} x_{n}^{2} x_{s}^{2} - v_{s}^{2} x_{n}^{2} y_{s}^{2}}{v_{e}^{2} - v_{s}^{2}}\right) \left(- \frac{3 \left(- 2 x_{n} - 2 x_{s}\right)^{2}}{8} + \frac{v_{e}^{2} x_{n}^{2} + 4 v_{e}^{2} x_{n} x_{s} + v_{e}^{2} x_{s}^{2} + v_{e}^{2} y_{n}^{2} - v_{s}^{2} x_{n}^{2} - 4 v_{s}^{2} x_{n} x_{s} - v_{s}^{2} x_{s}^{2} - v_{s}^{2} y_{s}^{2}}{v_{e}^{2} - v_{s}^{2}}\right)}{6} + \frac{\left(\left(- 2 x_{n} - 2 x_{s}\right) \left(\frac{\left(- 2 x_{n} - 2 x_{s}\right)^{2}}{8} - \frac{v_{e}^{2} x_{n}^{2} + 4 v_{e}^{2} x_{n} x_{s} + v_{e}^{2} x_{s}^{2} + v_{e}^{2} y_{n}^{2} - v_{s}^{2} x_{n}^{2} - 4 v_{s}^{2} x_{n} x_{s} - v_{s}^{2} x_{s}^{2} - v_{s}^{2} y_{s}^{2}}{2 \left(v_{e}^{2} - v_{s}^{2}\right)}\right) + \frac{- 2 v_{e}^{2} x_{n}^{2} x_{s} - 2 v_{e}^{2} x_{n} x_{s}^{2} - 2 v_{e}^{2} x_{s} y_{n}^{2} + 2 v_{s}^{2} x_{n}^{2} x_{s} + 2 v_{s}^{2} x_{n} x_{s}^{2} + 2 v_{s}^{2} x_{n} y_{s}^{2}}{v_{e}^{2} - v_{s}^{2}}\right)^{2}}{16} + \frac{\left(- \frac{3 \left(- 2 x_{n} - 2 x_{s}\right)^{2}}{8} + \frac{v_{e}^{2} x_{n}^{2} + 4 v_{e}^{2} x_{n} x_{s} + v_{e}^{2} x_{s}^{2} + v_{e}^{2} y_{n}^{2} - v_{s}^{2} x_{n}^{2} - 4 v_{s}^{2} x_{n} x_{s} - v_{s}^{2} x_{s}^{2} - v_{s}^{2} y_{s}^{2}}{v_{e}^{2} - v_{s}^{2}}\right)^{3}}{216} + \sqrt{\frac{\left(\left(- 2 x_{n} - 2 x_{s}\right) \left(\left(- 2 x_{n} - 2 x_{s}\right) \left(\frac{3 \left(- 2 x_{n} - 2 x_{s}\right)^{2}}{256} - \frac{v_{e}^{2} x_{n}^{2} + 4 v_{e}^{2} x_{n} x_{s} + v_{e}^{2} x_{s}^{2} + v_{e}^{2} y_{n}^{2} - v_{s}^{2} x_{n}^{2} - 4 v_{s}^{2} x_{n} x_{s} - v_{s}^{2} x_{s}^{2} - v_{s}^{2} y_{s}^{2}}{16 \left(v_{e}^{2} - v_{s}^{2}\right)}\right) + \frac{- 2 v_{e}^{2} x_{n}^{2} x_{s} - 2 v_{e}^{2} x_{n} x_{s}^{2} - 2 v_{e}^{2} x_{s} y_{n}^{2} + 2 v_{s}^{2} x_{n}^{2} x_{s} + 2 v_{s}^{2} x_{n} x_{s}^{2} + 2 v_{s}^{2} x_{n} y_{s}^{2}}{4 \left(v_{e}^{2} - v_{s}^{2}\right)}\right) - \frac{\left(- \frac{3 \left(- 2 x_{n} - 2 x_{s}\right)^{2}}{8} + \frac{v_{e}^{2} x_{n}^{2} + 4 v_{e}^{2} x_{n} x_{s} + v_{e}^{2} x_{s}^{2} + v_{e}^{2} y_{n}^{2} - v_{s}^{2} x_{n}^{2} - 4 v_{s}^{2} x_{n} x_{s} - v_{s}^{2} x_{s}^{2} - v_{s}^{2} y_{s}^{2}}{v_{e}^{2} - v_{s}^{2}}\right)^{2}}{12} - \frac{v_{e}^{2} x_{n}^{2} x_{s}^{2} + v_{e}^{2} x_{s}^{2} y_{n}^{2} - v_{s}^{2} x_{n}^{2} x_{s}^{2} - v_{s}^{2} x_{n}^{2} y_{s}^{2}}{v_{e}^{2} - v_{s}^{2}}\right)^{3}}{27} + \frac{\left(\frac{\left(- \left(- 2 x_{n} - 2 x_{s}\right) \left(\left(- 2 x_{n} - 2 x_{s}\right) \left(\frac{3 \left(- 2 x_{n} - 2 x_{s}\right)^{2}}{256} - \frac{v_{e}^{2} x_{n}^{2} + 4 v_{e}^{2} x_{n} x_{s} + v_{e}^{2} x_{s}^{2} + v_{e}^{2} y_{n}^{2} - v_{s}^{2} x_{n}^{2} - 4 v_{s}^{2} x_{n} x_{s} - v_{s}^{2} x_{s}^{2} - v_{s}^{2} y_{s}^{2}}{16 \left(v_{e}^{2} - v_{s}^{2}\right)}\right) + \frac{- 2 v_{e}^{2} x_{n}^{2} x_{s} - 2 v_{e}^{2} x_{n} x_{s}^{2} - 2 v_{e}^{2} x_{s} y_{n}^{2} + 2 v_{s}^{2} x_{n}^{2} x_{s} + 2 v_{s}^{2} x_{n} x_{s}^{2} + 2 v_{s}^{2} x_{n} y_{s}^{2}}{4 \left(v_{e}^{2} - v_{s}^{2}\right)}\right) + \frac{v_{e}^{2} x_{n}^{2} x_{s}^{2} + v_{e}^{2} x_{s}^{2} y_{n}^{2} - v_{s}^{2} x_{n}^{2} x_{s}^{2} - v_{s}^{2} x_{n}^{2} y_{s}^{2}}{v_{e}^{2} - v_{s}^{2}}\right) \left(- \frac{3 \left(- 2 x_{n} - 2 x_{s}\right)^{2}}{8} + \frac{v_{e}^{2} x_{n}^{2} + 4 v_{e}^{2} x_{n} x_{s} + v_{e}^{2} x_{s}^{2} + v_{e}^{2} y_{n}^{2} - v_{s}^{2} x_{n}^{2} - 4 v_{s}^{2} x_{n} x_{s} - v_{s}^{2} x_{s}^{2} - v_{s}^{2} y_{s}^{2}}{v_{e}^{2} - v_{s}^{2}}\right)}{3} - \frac{\left(\left(- 2 x_{n} - 2 x_{s}\right) \left(\frac{\left(- 2 x_{n} - 2 x_{s}\right)^{2}}{8} - \frac{v_{e}^{2} x_{n}^{2} + 4 v_{e}^{2} x_{n} x_{s} + v_{e}^{2} x_{s}^{2} + v_{e}^{2} y_{n}^{2} - v_{s}^{2} x_{n}^{2} - 4 v_{s}^{2} x_{n} x_{s} - v_{s}^{2} x_{s}^{2} - v_{s}^{2} y_{s}^{2}}{2 \left(v_{e}^{2} - v_{s}^{2}\right)}\right) + \frac{- 2 v_{e}^{2} x_{n}^{2} x_{s} - 2 v_{e}^{2} x_{n} x_{s}^{2} - 2 v_{e}^{2} x_{s} y_{n}^{2} + 2 v_{s}^{2} x_{n}^{2} x_{s} + 2 v_{s}^{2} x_{n} x_{s}^{2} + 2 v_{s}^{2} x_{n} y_{s}^{2}}{v_{e}^{2} - v_{s}^{2}}\right)^{2}}{8} - \frac{\left(- \frac{3 \left(- 2 x_{n} - 2 x_{s}\right)^{2}}{8} + \frac{v_{e}^{2} x_{n}^{2} + 4 v_{e}^{2} x_{n} x_{s} + v_{e}^{2} x_{s}^{2} + v_{e}^{2} y_{n}^{2} - v_{s}^{2} x_{n}^{2} - 4 v_{s}^{2} x_{n} x_{s} - v_{s}^{2} x_{s}^{2} - v_{s}^{2} y_{s}^{2}}{v_{e}^{2} - v_{s}^{2}}\right)^{3}}{108}\right)^{2}}{4}}} - \frac{2 \left(v_{e}^{2} x_{n}^{2} + 4 v_{e}^{2} x_{n} x_{s} + v_{e}^{2} x_{s}^{2} + v_{e}^{2} y_{n}^{2} - v_{s}^{2} x_{n}^{2} - 4 v_{s}^{2} x_{n} x_{s} - v_{s}^{2} x_{s}^{2} - v_{s}^{2} y_{s}^{2}\right)}{3 \left(v_{e}^{2} - v_{s}^{2}\right)}}}{2} - \frac{\sqrt{\frac{\left(- 2 x_{n} - 2 x_{s}\right)^{2}}{2} + \frac{2 \left(- 2 x_{n} - 2 x_{s}\right) \left(\frac{\left(- 2 x_{n} - 2 x_{s}\right)^{2}}{8} - \frac{v_{e}^{2} x_{n}^{2} + 4 v_{e}^{2} x_{n} x_{s} + v_{e}^{2} x_{s}^{2} + v_{e}^{2} y_{n}^{2} - v_{s}^{2} x_{n}^{2} - 4 v_{s}^{2} x_{n} x_{s} - v_{s}^{2} x_{s}^{2} - v_{s}^{2} y_{s}^{2}}{2 \left(v_{e}^{2} - v_{s}^{2}\right)}\right) + \frac{2 \left(- 2 v_{e}^{2} x_{n}^{2} x_{s} - 2 v_{e}^{2} x_{n} x_{s}^{2} - 2 v_{e}^{2} x_{s} y_{n}^{2} + 2 v_{s}^{2} x_{n}^{2} x_{s} + 2 v_{s}^{2} x_{n} x_{s}^{2} + 2 v_{s}^{2} x_{n} y_{s}^{2}\right)}{v_{e}^{2} - v_{s}^{2}}}{\sqrt{\frac{\left(- 2 x_{n} - 2 x_{s}\right)^{2}}{4} - \frac{2 \left(\left(- 2 x_{n} - 2 x_{s}\right) \left(\left(- 2 x_{n} - 2 x_{s}\right) \left(\frac{3 \left(- 2 x_{n} - 2 x_{s}\right)^{2}}{256} - \frac{v_{e}^{2} x_{n}^{2} + 4 v_{e}^{2} x_{n} x_{s} + v_{e}^{2} x_{s}^{2} + v_{e}^{2} y_{n}^{2} - v_{s}^{2} x_{n}^{2} - 4 v_{s}^{2} x_{n} x_{s} - v_{s}^{2} x_{s}^{2} - v_{s}^{2} y_{s}^{2}}{16 \left(v_{e}^{2} - v_{s}^{2}\right)}\right) + \frac{- 2 v_{e}^{2} x_{n}^{2} x_{s} - 2 v_{e}^{2} x_{n} x_{s}^{2} - 2 v_{e}^{2} x_{s} y_{n}^{2} + 2 v_{s}^{2} x_{n}^{2} x_{s} + 2 v_{s}^{2} x_{n} x_{s}^{2} + 2 v_{s}^{2} x_{n} y_{s}^{2}}{4 \left(v_{e}^{2} - v_{s}^{2}\right)}\right) - \frac{\left(- \frac{3 \left(- 2 x_{n} - 2 x_{s}\right)^{2}}{8} + \frac{v_{e}^{2} x_{n}^{2} + 4 v_{e}^{2} x_{n} x_{s} + v_{e}^{2} x_{s}^{2} + v_{e}^{2} y_{n}^{2} - v_{s}^{2} x_{n}^{2} - 4 v_{s}^{2} x_{n} x_{s} - v_{s}^{2} x_{s}^{2} - v_{s}^{2} y_{s}^{2}}{v_{e}^{2} - v_{s}^{2}}\right)^{2}}{12} - \frac{v_{e}^{2} x_{n}^{2} x_{s}^{2} + v_{e}^{2} x_{s}^{2} y_{n}^{2} - v_{s}^{2} x_{n}^{2} x_{s}^{2} - v_{s}^{2} x_{n}^{2} y_{s}^{2}}{v_{e}^{2} - v_{s}^{2}}\right)}{3 \sqrt[3]{- \frac{\left(- \left(- 2 x_{n} - 2 x_{s}\right) \left(\left(- 2 x_{n} - 2 x_{s}\right) \left(\frac{3 \left(- 2 x_{n} - 2 x_{s}\right)^{2}}{256} - \frac{v_{e}^{2} x_{n}^{2} + 4 v_{e}^{2} x_{n} x_{s} + v_{e}^{2} x_{s}^{2} + v_{e}^{2} y_{n}^{2} - v_{s}^{2} x_{n}^{2} - 4 v_{s}^{2} x_{n} x_{s} - v_{s}^{2} x_{s}^{2} - v_{s}^{2} y_{s}^{2}}{16 \left(v_{e}^{2} - v_{s}^{2}\right)}\right) + \frac{- 2 v_{e}^{2} x_{n}^{2} x_{s} - 2 v_{e}^{2} x_{n} x_{s}^{2} - 2 v_{e}^{2} x_{s} y_{n}^{2} + 2 v_{s}^{2} x_{n}^{2} x_{s} + 2 v_{s}^{2} x_{n} x_{s}^{2} + 2 v_{s}^{2} x_{n} y_{s}^{2}}{4 \left(v_{e}^{2} - v_{s}^{2}\right)}\right) + \frac{v_{e}^{2} x_{n}^{2} x_{s}^{2} + v_{e}^{2} x_{s}^{2} y_{n}^{2} - v_{s}^{2} x_{n}^{2} x_{s}^{2} - v_{s}^{2} x_{n}^{2} y_{s}^{2}}{v_{e}^{2} - v_{s}^{2}}\right) \left(- \frac{3 \left(- 2 x_{n} - 2 x_{s}\right)^{2}}{8} + \frac{v_{e}^{2} x_{n}^{2} + 4 v_{e}^{2} x_{n} x_{s} + v_{e}^{2} x_{s}^{2} + v_{e}^{2} y_{n}^{2} - v_{s}^{2} x_{n}^{2} - 4 v_{s}^{2} x_{n} x_{s} - v_{s}^{2} x_{s}^{2} - v_{s}^{2} y_{s}^{2}}{v_{e}^{2} - v_{s}^{2}}\right)}{6} + \frac{\left(\left(- 2 x_{n} - 2 x_{s}\right) \left(\frac{\left(- 2 x_{n} - 2 x_{s}\right)^{2}}{8} - \frac{v_{e}^{2} x_{n}^{2} + 4 v_{e}^{2} x_{n} x_{s} + v_{e}^{2} x_{s}^{2} + v_{e}^{2} y_{n}^{2} - v_{s}^{2} x_{n}^{2} - 4 v_{s}^{2} x_{n} x_{s} - v_{s}^{2} x_{s}^{2} - v_{s}^{2} y_{s}^{2}}{2 \left(v_{e}^{2} - v_{s}^{2}\right)}\right) + \frac{- 2 v_{e}^{2} x_{n}^{2} x_{s} - 2 v_{e}^{2} x_{n} x_{s}^{2} - 2 v_{e}^{2} x_{s} y_{n}^{2} + 2 v_{s}^{2} x_{n}^{2} x_{s} + 2 v_{s}^{2} x_{n} x_{s}^{2} + 2 v_{s}^{2} x_{n} y_{s}^{2}}{v_{e}^{2} - v_{s}^{2}}\right)^{2}}{16} + \frac{\left(- \frac{3 \left(- 2 x_{n} - 2 x_{s}\right)^{2}}{8} + \frac{v_{e}^{2} x_{n}^{2} + 4 v_{e}^{2} x_{n} x_{s} + v_{e}^{2} x_{s}^{2} + v_{e}^{2} y_{n}^{2} - v_{s}^{2} x_{n}^{2} - 4 v_{s}^{2} x_{n} x_{s} - v_{s}^{2} x_{s}^{2} - v_{s}^{2} y_{s}^{2}}{v_{e}^{2} - v_{s}^{2}}\right)^{3}}{216} + \sqrt{\frac{\left(\left(- 2 x_{n} - 2 x_{s}\right) \left(\left(- 2 x_{n} - 2 x_{s}\right) \left(\frac{3 \left(- 2 x_{n} - 2 x_{s}\right)^{2}}{256} - \frac{v_{e}^{2} x_{n}^{2} + 4 v_{e}^{2} x_{n} x_{s} + v_{e}^{2} x_{s}^{2} + v_{e}^{2} y_{n}^{2} - v_{s}^{2} x_{n}^{2} - 4 v_{s}^{2} x_{n} x_{s} - v_{s}^{2} x_{s}^{2} - v_{s}^{2} y_{s}^{2}}{16 \left(v_{e}^{2} - v_{s}^{2}\right)}\right) + \frac{- 2 v_{e}^{2} x_{n}^{2} x_{s} - 2 v_{e}^{2} x_{n} x_{s}^{2} - 2 v_{e}^{2} x_{s} y_{n}^{2} + 2 v_{s}^{2} x_{n}^{2} x_{s} + 2 v_{s}^{2} x_{n} x_{s}^{2} + 2 v_{s}^{2} x_{n} y_{s}^{2}}{4 \left(v_{e}^{2} - v_{s}^{2}\right)}\right) - \frac{\left(- \frac{3 \left(- 2 x_{n} - 2 x_{s}\right)^{2}}{8} + \frac{v_{e}^{2} x_{n}^{2} + 4 v_{e}^{2} x_{n} x_{s} + v_{e}^{2} x_{s}^{2} + v_{e}^{2} y_{n}^{2} - v_{s}^{2} x_{n}^{2} - 4 v_{s}^{2} x_{n} x_{s} - v_{s}^{2} x_{s}^{2} - v_{s}^{2} y_{s}^{2}}{v_{e}^{2} - v_{s}^{2}}\right)^{2}}{12} - \frac{v_{e}^{2} x_{n}^{2} x_{s}^{2} + v_{e}^{2} x_{s}^{2} y_{n}^{2} - v_{s}^{2} x_{n}^{2} x_{s}^{2} - v_{s}^{2} x_{n}^{2} y_{s}^{2}}{v_{e}^{2} - v_{s}^{2}}\right)^{3}}{27} + \frac{\left(\frac{\left(- \left(- 2 x_{n} - 2 x_{s}\right) \left(\left(- 2 x_{n} - 2 x_{s}\right) \left(\frac{3 \left(- 2 x_{n} - 2 x_{s}\right)^{2}}{256} - \frac{v_{e}^{2} x_{n}^{2} + 4 v_{e}^{2} x_{n} x_{s} + v_{e}^{2} x_{s}^{2} + v_{e}^{2} y_{n}^{2} - v_{s}^{2} x_{n}^{2} - 4 v_{s}^{2} x_{n} x_{s} - v_{s}^{2} x_{s}^{2} - v_{s}^{2} y_{s}^{2}}{16 \left(v_{e}^{2} - v_{s}^{2}\right)}\right) + \frac{- 2 v_{e}^{2} x_{n}^{2} x_{s} - 2 v_{e}^{2} x_{n} x_{s}^{2} - 2 v_{e}^{2} x_{s} y_{n}^{2} + 2 v_{s}^{2} x_{n}^{2} x_{s} + 2 v_{s}^{2} x_{n} x_{s}^{2} + 2 v_{s}^{2} x_{n} y_{s}^{2}}{4 \left(v_{e}^{2} - v_{s}^{2}\right)}\right) + \frac{v_{e}^{2} x_{n}^{2} x_{s}^{2} + v_{e}^{2} x_{s}^{2} y_{n}^{2} - v_{s}^{2} x_{n}^{2} x_{s}^{2} - v_{s}^{2} x_{n}^{2} y_{s}^{2}}{v_{e}^{2} - v_{s}^{2}}\right) \left(- \frac{3 \left(- 2 x_{n} - 2 x_{s}\right)^{2}}{8} + \frac{v_{e}^{2} x_{n}^{2} + 4 v_{e}^{2} x_{n} x_{s} + v_{e}^{2} x_{s}^{2} + v_{e}^{2} y_{n}^{2} - v_{s}^{2} x_{n}^{2} - 4 v_{s}^{2} x_{n} x_{s} - v_{s}^{2} x_{s}^{2} - v_{s}^{2} y_{s}^{2}}{v_{e}^{2} - v_{s}^{2}}\right)}{3} - \frac{\left(\left(- 2 x_{n} - 2 x_{s}\right) \left(\frac{\left(- 2 x_{n} - 2 x_{s}\right)^{2}}{8} - \frac{v_{e}^{2} x_{n}^{2} + 4 v_{e}^{2} x_{n} x_{s} + v_{e}^{2} x_{s}^{2} + v_{e}^{2} y_{n}^{2} - v_{s}^{2} x_{n}^{2} - 4 v_{s}^{2} x_{n} x_{s} - v_{s}^{2} x_{s}^{2} - v_{s}^{2} y_{s}^{2}}{2 \left(v_{e}^{2} - v_{s}^{2}\right)}\right) + \frac{- 2 v_{e}^{2} x_{n}^{2} x_{s} - 2 v_{e}^{2} x_{n} x_{s}^{2} - 2 v_{e}^{2} x_{s} y_{n}^{2} + 2 v_{s}^{2} x_{n}^{2} x_{s} + 2 v_{s}^{2} x_{n} x_{s}^{2} + 2 v_{s}^{2} x_{n} y_{s}^{2}}{v_{e}^{2} - v_{s}^{2}}\right)^{2}}{8} - \frac{\left(- \frac{3 \left(- 2 x_{n} - 2 x_{s}\right)^{2}}{8} + \frac{v_{e}^{2} x_{n}^{2} + 4 v_{e}^{2} x_{n} x_{s} + v_{e}^{2} x_{s}^{2} + v_{e}^{2} y_{n}^{2} - v_{s}^{2} x_{n}^{2} - 4 v_{s}^{2} x_{n} x_{s} - v_{s}^{2} x_{s}^{2} - v_{s}^{2} y_{s}^{2}}{v_{e}^{2} - v_{s}^{2}}\right)^{3}}{108}\right)^{2}}{4}}}} + 2 \sqrt[3]{- \frac{\left(- \left(- 2 x_{n} - 2 x_{s}\right) \left(\left(- 2 x_{n} - 2 x_{s}\right) \left(\frac{3 \left(- 2 x_{n} - 2 x_{s}\right)^{2}}{256} - \frac{v_{e}^{2} x_{n}^{2} + 4 v_{e}^{2} x_{n} x_{s} + v_{e}^{2} x_{s}^{2} + v_{e}^{2} y_{n}^{2} - v_{s}^{2} x_{n}^{2} - 4 v_{s}^{2} x_{n} x_{s} - v_{s}^{2} x_{s}^{2} - v_{s}^{2} y_{s}^{2}}{16 \left(v_{e}^{2} - v_{s}^{2}\right)}\right) + \frac{- 2 v_{e}^{2} x_{n}^{2} x_{s} - 2 v_{e}^{2} x_{n} x_{s}^{2} - 2 v_{e}^{2} x_{s} y_{n}^{2} + 2 v_{s}^{2} x_{n}^{2} x_{s} + 2 v_{s}^{2} x_{n} x_{s}^{2} + 2 v_{s}^{2} x_{n} y_{s}^{2}}{4 \left(v_{e}^{2} - v_{s}^{2}\right)}\right) + \frac{v_{e}^{2} x_{n}^{2} x_{s}^{2} + v_{e}^{2} x_{s}^{2} y_{n}^{2} - v_{s}^{2} x_{n}^{2} x_{s}^{2} - v_{s}^{2} x_{n}^{2} y_{s}^{2}}{v_{e}^{2} - v_{s}^{2}}\right) \left(- \frac{3 \left(- 2 x_{n} - 2 x_{s}\right)^{2}}{8} + \frac{v_{e}^{2} x_{n}^{2} + 4 v_{e}^{2} x_{n} x_{s} + v_{e}^{2} x_{s}^{2} + v_{e}^{2} y_{n}^{2} - v_{s}^{2} x_{n}^{2} - 4 v_{s}^{2} x_{n} x_{s} - v_{s}^{2} x_{s}^{2} - v_{s}^{2} y_{s}^{2}}{v_{e}^{2} - v_{s}^{2}}\right)}{6} + \frac{\left(\left(- 2 x_{n} - 2 x_{s}\right) \left(\frac{\left(- 2 x_{n} - 2 x_{s}\right)^{2}}{8} - \frac{v_{e}^{2} x_{n}^{2} + 4 v_{e}^{2} x_{n} x_{s} + v_{e}^{2} x_{s}^{2} + v_{e}^{2} y_{n}^{2} - v_{s}^{2} x_{n}^{2} - 4 v_{s}^{2} x_{n} x_{s} - v_{s}^{2} x_{s}^{2} - v_{s}^{2} y_{s}^{2}}{2 \left(v_{e}^{2} - v_{s}^{2}\right)}\right) + \frac{- 2 v_{e}^{2} x_{n}^{2} x_{s} - 2 v_{e}^{2} x_{n} x_{s}^{2} - 2 v_{e}^{2} x_{s} y_{n}^{2} + 2 v_{s}^{2} x_{n}^{2} x_{s} + 2 v_{s}^{2} x_{n} x_{s}^{2} + 2 v_{s}^{2} x_{n} y_{s}^{2}}{v_{e}^{2} - v_{s}^{2}}\right)^{2}}{16} + \frac{\left(- \frac{3 \left(- 2 x_{n} - 2 x_{s}\right)^{2}}{8} + \frac{v_{e}^{2} x_{n}^{2} + 4 v_{e}^{2} x_{n} x_{s} + v_{e}^{2} x_{s}^{2} + v_{e}^{2} y_{n}^{2} - v_{s}^{2} x_{n}^{2} - 4 v_{s}^{2} x_{n} x_{s} - v_{s}^{2} x_{s}^{2} - v_{s}^{2} y_{s}^{2}}{v_{e}^{2} - v_{s}^{2}}\right)^{3}}{216} + \sqrt{\frac{\left(\left(- 2 x_{n} - 2 x_{s}\right) \left(\left(- 2 x_{n} - 2 x_{s}\right) \left(\frac{3 \left(- 2 x_{n} - 2 x_{s}\right)^{2}}{256} - \frac{v_{e}^{2} x_{n}^{2} + 4 v_{e}^{2} x_{n} x_{s} + v_{e}^{2} x_{s}^{2} + v_{e}^{2} y_{n}^{2} - v_{s}^{2} x_{n}^{2} - 4 v_{s}^{2} x_{n} x_{s} - v_{s}^{2} x_{s}^{2} - v_{s}^{2} y_{s}^{2}}{16 \left(v_{e}^{2} - v_{s}^{2}\right)}\right) + \frac{- 2 v_{e}^{2} x_{n}^{2} x_{s} - 2 v_{e}^{2} x_{n} x_{s}^{2} - 2 v_{e}^{2} x_{s} y_{n}^{2} + 2 v_{s}^{2} x_{n}^{2} x_{s} + 2 v_{s}^{2} x_{n} x_{s}^{2} + 2 v_{s}^{2} x_{n} y_{s}^{2}}{4 \left(v_{e}^{2} - v_{s}^{2}\right)}\right) - \frac{\left(- \frac{3 \left(- 2 x_{n} - 2 x_{s}\right)^{2}}{8} + \frac{v_{e}^{2} x_{n}^{2} + 4 v_{e}^{2} x_{n} x_{s} + v_{e}^{2} x_{s}^{2} + v_{e}^{2} y_{n}^{2} - v_{s}^{2} x_{n}^{2} - 4 v_{s}^{2} x_{n} x_{s} - v_{s}^{2} x_{s}^{2} - v_{s}^{2} y_{s}^{2}}{v_{e}^{2} - v_{s}^{2}}\right)^{2}}{12} - \frac{v_{e}^{2} x_{n}^{2} x_{s}^{2} + v_{e}^{2} x_{s}^{2} y_{n}^{2} - v_{s}^{2} x_{n}^{2} x_{s}^{2} - v_{s}^{2} x_{n}^{2} y_{s}^{2}}{v_{e}^{2} - v_{s}^{2}}\right)^{3}}{27} + \frac{\left(\frac{\left(- \left(- 2 x_{n} - 2 x_{s}\right) \left(\left(- 2 x_{n} - 2 x_{s}\right) \left(\frac{3 \left(- 2 x_{n} - 2 x_{s}\right)^{2}}{256} - \frac{v_{e}^{2} x_{n}^{2} + 4 v_{e}^{2} x_{n} x_{s} + v_{e}^{2} x_{s}^{2} + v_{e}^{2} y_{n}^{2} - v_{s}^{2} x_{n}^{2} - 4 v_{s}^{2} x_{n} x_{s} - v_{s}^{2} x_{s}^{2} - v_{s}^{2} y_{s}^{2}}{16 \left(v_{e}^{2} - v_{s}^{2}\right)}\right) + \frac{- 2 v_{e}^{2} x_{n}^{2} x_{s} - 2 v_{e}^{2} x_{n} x_{s}^{2} - 2 v_{e}^{2} x_{s} y_{n}^{2} + 2 v_{s}^{2} x_{n}^{2} x_{s} + 2 v_{s}^{2} x_{n} x_{s}^{2} + 2 v_{s}^{2} x_{n} y_{s}^{2}}{4 \left(v_{e}^{2} - v_{s}^{2}\right)}\right) + \frac{v_{e}^{2} x_{n}^{2} x_{s}^{2} + v_{e}^{2} x_{s}^{2} y_{n}^{2} - v_{s}^{2} x_{n}^{2} x_{s}^{2} - v_{s}^{2} x_{n}^{2} y_{s}^{2}}{v_{e}^{2} - v_{s}^{2}}\right) \left(- \frac{3 \left(- 2 x_{n} - 2 x_{s}\right)^{2}}{8} + \frac{v_{e}^{2} x_{n}^{2} + 4 v_{e}^{2} x_{n} x_{s} + v_{e}^{2} x_{s}^{2} + v_{e}^{2} y_{n}^{2} - v_{s}^{2} x_{n}^{2} - 4 v_{s}^{2} x_{n} x_{s} - v_{s}^{2} x_{s}^{2} - v_{s}^{2} y_{s}^{2}}{v_{e}^{2} - v_{s}^{2}}\right)}{3} - \frac{\left(\left(- 2 x_{n} - 2 x_{s}\right) \left(\frac{\left(- 2 x_{n} - 2 x_{s}\right)^{2}}{8} - \frac{v_{e}^{2} x_{n}^{2} + 4 v_{e}^{2} x_{n} x_{s} + v_{e}^{2} x_{s}^{2} + v_{e}^{2} y_{n}^{2} - v_{s}^{2} x_{n}^{2} - 4 v_{s}^{2} x_{n} x_{s} - v_{s}^{2} x_{s}^{2} - v_{s}^{2} y_{s}^{2}}{2 \left(v_{e}^{2} - v_{s}^{2}\right)}\right) + \frac{- 2 v_{e}^{2} x_{n}^{2} x_{s} - 2 v_{e}^{2} x_{n} x_{s}^{2} - 2 v_{e}^{2} x_{s} y_{n}^{2} + 2 v_{s}^{2} x_{n}^{2} x_{s} + 2 v_{s}^{2} x_{n} x_{s}^{2} + 2 v_{s}^{2} x_{n} y_{s}^{2}}{v_{e}^{2} - v_{s}^{2}}\right)^{2}}{8} - \frac{\left(- \frac{3 \left(- 2 x_{n} - 2 x_{s}\right)^{2}}{8} + \frac{v_{e}^{2} x_{n}^{2} + 4 v_{e}^{2} x_{n} x_{s} + v_{e}^{2} x_{s}^{2} + v_{e}^{2} y_{n}^{2} - v_{s}^{2} x_{n}^{2} - 4 v_{s}^{2} x_{n} x_{s} - v_{s}^{2} x_{s}^{2} - v_{s}^{2} y_{s}^{2}}{v_{e}^{2} - v_{s}^{2}}\right)^{3}}{108}\right)^{2}}{4}}} - \frac{2 \left(v_{e}^{2} x_{n}^{2} + 4 v_{e}^{2} x_{n} x_{s} + v_{e}^{2} x_{s}^{2} + v_{e}^{2} y_{n}^{2} - v_{s}^{2} x_{n}^{2} - 4 v_{s}^{2} x_{n} x_{s} - v_{s}^{2} x_{s}^{2} - v_{s}^{2} y_{s}^{2}\right)}{3 \left(v_{e}^{2} - v_{s}^{2}\right)}}} + \frac{2 \left(\left(- 2 x_{n} - 2 x_{s}\right) \left(\left(- 2 x_{n} - 2 x_{s}\right) \left(\frac{3 \left(- 2 x_{n} - 2 x_{s}\right)^{2}}{256} - \frac{v_{e}^{2} x_{n}^{2} + 4 v_{e}^{2} x_{n} x_{s} + v_{e}^{2} x_{s}^{2} + v_{e}^{2} y_{n}^{2} - v_{s}^{2} x_{n}^{2} - 4 v_{s}^{2} x_{n} x_{s} - v_{s}^{2} x_{s}^{2} - v_{s}^{2} y_{s}^{2}}{16 \left(v_{e}^{2} - v_{s}^{2}\right)}\right) + \frac{- 2 v_{e}^{2} x_{n}^{2} x_{s} - 2 v_{e}^{2} x_{n} x_{s}^{2} - 2 v_{e}^{2} x_{s} y_{n}^{2} + 2 v_{s}^{2} x_{n}^{2} x_{s} + 2 v_{s}^{2} x_{n} x_{s}^{2} + 2 v_{s}^{2} x_{n} y_{s}^{2}}{4 \left(v_{e}^{2} - v_{s}^{2}\right)}\right) - \frac{\left(- \frac{3 \left(- 2 x_{n} - 2 x_{s}\right)^{2}}{8} + \frac{v_{e}^{2} x_{n}^{2} + 4 v_{e}^{2} x_{n} x_{s} + v_{e}^{2} x_{s}^{2} + v_{e}^{2} y_{n}^{2} - v_{s}^{2} x_{n}^{2} - 4 v_{s}^{2} x_{n} x_{s} - v_{s}^{2} x_{s}^{2} - v_{s}^{2} y_{s}^{2}}{v_{e}^{2} - v_{s}^{2}}\right)^{2}}{12} - \frac{v_{e}^{2} x_{n}^{2} x_{s}^{2} + v_{e}^{2} x_{s}^{2} y_{n}^{2} - v_{s}^{2} x_{n}^{2} x_{s}^{2} - v_{s}^{2} x_{n}^{2} y_{s}^{2}}{v_{e}^{2} - v_{s}^{2}}\right)}{3 \sqrt[3]{- \frac{\left(- \left(- 2 x_{n} - 2 x_{s}\right) \left(\left(- 2 x_{n} - 2 x_{s}\right) \left(\frac{3 \left(- 2 x_{n} - 2 x_{s}\right)^{2}}{256} - \frac{v_{e}^{2} x_{n}^{2} + 4 v_{e}^{2} x_{n} x_{s} + v_{e}^{2} x_{s}^{2} + v_{e}^{2} y_{n}^{2} - v_{s}^{2} x_{n}^{2} - 4 v_{s}^{2} x_{n} x_{s} - v_{s}^{2} x_{s}^{2} - v_{s}^{2} y_{s}^{2}}{16 \left(v_{e}^{2} - v_{s}^{2}\right)}\right) + \frac{- 2 v_{e}^{2} x_{n}^{2} x_{s} - 2 v_{e}^{2} x_{n} x_{s}^{2} - 2 v_{e}^{2} x_{s} y_{n}^{2} + 2 v_{s}^{2} x_{n}^{2} x_{s} + 2 v_{s}^{2} x_{n} x_{s}^{2} + 2 v_{s}^{2} x_{n} y_{s}^{2}}{4 \left(v_{e}^{2} - v_{s}^{2}\right)}\right) + \frac{v_{e}^{2} x_{n}^{2} x_{s}^{2} + v_{e}^{2} x_{s}^{2} y_{n}^{2} - v_{s}^{2} x_{n}^{2} x_{s}^{2} - v_{s}^{2} x_{n}^{2} y_{s}^{2}}{v_{e}^{2} - v_{s}^{2}}\right) \left(- \frac{3 \left(- 2 x_{n} - 2 x_{s}\right)^{2}}{8} + \frac{v_{e}^{2} x_{n}^{2} + 4 v_{e}^{2} x_{n} x_{s} + v_{e}^{2} x_{s}^{2} + v_{e}^{2} y_{n}^{2} - v_{s}^{2} x_{n}^{2} - 4 v_{s}^{2} x_{n} x_{s} - v_{s}^{2} x_{s}^{2} - v_{s}^{2} y_{s}^{2}}{v_{e}^{2} - v_{s}^{2}}\right)}{6} + \frac{\left(\left(- 2 x_{n} - 2 x_{s}\right) \left(\frac{\left(- 2 x_{n} - 2 x_{s}\right)^{2}}{8} - \frac{v_{e}^{2} x_{n}^{2} + 4 v_{e}^{2} x_{n} x_{s} + v_{e}^{2} x_{s}^{2} + v_{e}^{2} y_{n}^{2} - v_{s}^{2} x_{n}^{2} - 4 v_{s}^{2} x_{n} x_{s} - v_{s}^{2} x_{s}^{2} - v_{s}^{2} y_{s}^{2}}{2 \left(v_{e}^{2} - v_{s}^{2}\right)}\right) + \frac{- 2 v_{e}^{2} x_{n}^{2} x_{s} - 2 v_{e}^{2} x_{n} x_{s}^{2} - 2 v_{e}^{2} x_{s} y_{n}^{2} + 2 v_{s}^{2} x_{n}^{2} x_{s} + 2 v_{s}^{2} x_{n} x_{s}^{2} + 2 v_{s}^{2} x_{n} y_{s}^{2}}{v_{e}^{2} - v_{s}^{2}}\right)^{2}}{16} + \frac{\left(- \frac{3 \left(- 2 x_{n} - 2 x_{s}\right)^{2}}{8} + \frac{v_{e}^{2} x_{n}^{2} + 4 v_{e}^{2} x_{n} x_{s} + v_{e}^{2} x_{s}^{2} + v_{e}^{2} y_{n}^{2} - v_{s}^{2} x_{n}^{2} - 4 v_{s}^{2} x_{n} x_{s} - v_{s}^{2} x_{s}^{2} - v_{s}^{2} y_{s}^{2}}{v_{e}^{2} - v_{s}^{2}}\right)^{3}}{216} + \sqrt{\frac{\left(\left(- 2 x_{n} - 2 x_{s}\right) \left(\left(- 2 x_{n} - 2 x_{s}\right) \left(\frac{3 \left(- 2 x_{n} - 2 x_{s}\right)^{2}}{256} - \frac{v_{e}^{2} x_{n}^{2} + 4 v_{e}^{2} x_{n} x_{s} + v_{e}^{2} x_{s}^{2} + v_{e}^{2} y_{n}^{2} - v_{s}^{2} x_{n}^{2} - 4 v_{s}^{2} x_{n} x_{s} - v_{s}^{2} x_{s}^{2} - v_{s}^{2} y_{s}^{2}}{16 \left(v_{e}^{2} - v_{s}^{2}\right)}\right) + \frac{- 2 v_{e}^{2} x_{n}^{2} x_{s} - 2 v_{e}^{2} x_{n} x_{s}^{2} - 2 v_{e}^{2} x_{s} y_{n}^{2} + 2 v_{s}^{2} x_{n}^{2} x_{s} + 2 v_{s}^{2} x_{n} x_{s}^{2} + 2 v_{s}^{2} x_{n} y_{s}^{2}}{4 \left(v_{e}^{2} - v_{s}^{2}\right)}\right) - \frac{\left(- \frac{3 \left(- 2 x_{n} - 2 x_{s}\right)^{2}}{8} + \frac{v_{e}^{2} x_{n}^{2} + 4 v_{e}^{2} x_{n} x_{s} + v_{e}^{2} x_{s}^{2} + v_{e}^{2} y_{n}^{2} - v_{s}^{2} x_{n}^{2} - 4 v_{s}^{2} x_{n} x_{s} - v_{s}^{2} x_{s}^{2} - v_{s}^{2} y_{s}^{2}}{v_{e}^{2} - v_{s}^{2}}\right)^{2}}{12} - \frac{v_{e}^{2} x_{n}^{2} x_{s}^{2} + v_{e}^{2} x_{s}^{2} y_{n}^{2} - v_{s}^{2} x_{n}^{2} x_{s}^{2} - v_{s}^{2} x_{n}^{2} y_{s}^{2}}{v_{e}^{2} - v_{s}^{2}}\right)^{3}}{27} + \frac{\left(\frac{\left(- \left(- 2 x_{n} - 2 x_{s}\right) \left(\left(- 2 x_{n} - 2 x_{s}\right) \left(\frac{3 \left(- 2 x_{n} - 2 x_{s}\right)^{2}}{256} - \frac{v_{e}^{2} x_{n}^{2} + 4 v_{e}^{2} x_{n} x_{s} + v_{e}^{2} x_{s}^{2} + v_{e}^{2} y_{n}^{2} - v_{s}^{2} x_{n}^{2} - 4 v_{s}^{2} x_{n} x_{s} - v_{s}^{2} x_{s}^{2} - v_{s}^{2} y_{s}^{2}}{16 \left(v_{e}^{2} - v_{s}^{2}\right)}\right) + \frac{- 2 v_{e}^{2} x_{n}^{2} x_{s} - 2 v_{e}^{2} x_{n} x_{s}^{2} - 2 v_{e}^{2} x_{s} y_{n}^{2} + 2 v_{s}^{2} x_{n}^{2} x_{s} + 2 v_{s}^{2} x_{n} x_{s}^{2} + 2 v_{s}^{2} x_{n} y_{s}^{2}}{4 \left(v_{e}^{2} - v_{s}^{2}\right)}\right) + \frac{v_{e}^{2} x_{n}^{2} x_{s}^{2} + v_{e}^{2} x_{s}^{2} y_{n}^{2} - v_{s}^{2} x_{n}^{2} x_{s}^{2} - v_{s}^{2} x_{n}^{2} y_{s}^{2}}{v_{e}^{2} - v_{s}^{2}}\right) \left(- \frac{3 \left(- 2 x_{n} - 2 x_{s}\right)^{2}}{8} + \frac{v_{e}^{2} x_{n}^{2} + 4 v_{e}^{2} x_{n} x_{s} + v_{e}^{2} x_{s}^{2} + v_{e}^{2} y_{n}^{2} - v_{s}^{2} x_{n}^{2} - 4 v_{s}^{2} x_{n} x_{s} - v_{s}^{2} x_{s}^{2} - v_{s}^{2} y_{s}^{2}}{v_{e}^{2} - v_{s}^{2}}\right)}{3} - \frac{\left(\left(- 2 x_{n} - 2 x_{s}\right) \left(\frac{\left(- 2 x_{n} - 2 x_{s}\right)^{2}}{8} - \frac{v_{e}^{2} x_{n}^{2} + 4 v_{e}^{2} x_{n} x_{s} + v_{e}^{2} x_{s}^{2} + v_{e}^{2} y_{n}^{2} - v_{s}^{2} x_{n}^{2} - 4 v_{s}^{2} x_{n} x_{s} - v_{s}^{2} x_{s}^{2} - v_{s}^{2} y_{s}^{2}}{2 \left(v_{e}^{2} - v_{s}^{2}\right)}\right) + \frac{- 2 v_{e}^{2} x_{n}^{2} x_{s} - 2 v_{e}^{2} x_{n} x_{s}^{2} - 2 v_{e}^{2} x_{s} y_{n}^{2} + 2 v_{s}^{2} x_{n}^{2} x_{s} + 2 v_{s}^{2} x_{n} x_{s}^{2} + 2 v_{s}^{2} x_{n} y_{s}^{2}}{v_{e}^{2} - v_{s}^{2}}\right)^{2}}{8} - \frac{\left(- \frac{3 \left(- 2 x_{n} - 2 x_{s}\right)^{2}}{8} + \frac{v_{e}^{2} x_{n}^{2} + 4 v_{e}^{2} x_{n} x_{s} + v_{e}^{2} x_{s}^{2} + v_{e}^{2} y_{n}^{2} - v_{s}^{2} x_{n}^{2} - 4 v_{s}^{2} x_{n} x_{s} - v_{s}^{2} x_{s}^{2} - v_{s}^{2} y_{s}^{2}}{v_{e}^{2} - v_{s}^{2}}\right)^{3}}{108}\right)^{2}}{4}}}} - 2 \sqrt[3]{- \frac{\left(- \left(- 2 x_{n} - 2 x_{s}\right) \left(\left(- 2 x_{n} - 2 x_{s}\right) \left(\frac{3 \left(- 2 x_{n} - 2 x_{s}\right)^{2}}{256} - \frac{v_{e}^{2} x_{n}^{2} + 4 v_{e}^{2} x_{n} x_{s} + v_{e}^{2} x_{s}^{2} + v_{e}^{2} y_{n}^{2} - v_{s}^{2} x_{n}^{2} - 4 v_{s}^{2} x_{n} x_{s} - v_{s}^{2} x_{s}^{2} - v_{s}^{2} y_{s}^{2}}{16 \left(v_{e}^{2} - v_{s}^{2}\right)}\right) + \frac{- 2 v_{e}^{2} x_{n}^{2} x_{s} - 2 v_{e}^{2} x_{n} x_{s}^{2} - 2 v_{e}^{2} x_{s} y_{n}^{2} + 2 v_{s}^{2} x_{n}^{2} x_{s} + 2 v_{s}^{2} x_{n} x_{s}^{2} + 2 v_{s}^{2} x_{n} y_{s}^{2}}{4 \left(v_{e}^{2} - v_{s}^{2}\right)}\right) + \frac{v_{e}^{2} x_{n}^{2} x_{s}^{2} + v_{e}^{2} x_{s}^{2} y_{n}^{2} - v_{s}^{2} x_{n}^{2} x_{s}^{2} - v_{s}^{2} x_{n}^{2} y_{s}^{2}}{v_{e}^{2} - v_{s}^{2}}\right) \left(- \frac{3 \left(- 2 x_{n} - 2 x_{s}\right)^{2}}{8} + \frac{v_{e}^{2} x_{n}^{2} + 4 v_{e}^{2} x_{n} x_{s} + v_{e}^{2} x_{s}^{2} + v_{e}^{2} y_{n}^{2} - v_{s}^{2} x_{n}^{2} - 4 v_{s}^{2} x_{n} x_{s} - v_{s}^{2} x_{s}^{2} - v_{s}^{2} y_{s}^{2}}{v_{e}^{2} - v_{s}^{2}}\right)}{6} + \frac{\left(\left(- 2 x_{n} - 2 x_{s}\right) \left(\frac{\left(- 2 x_{n} - 2 x_{s}\right)^{2}}{8} - \frac{v_{e}^{2} x_{n}^{2} + 4 v_{e}^{2} x_{n} x_{s} + v_{e}^{2} x_{s}^{2} + v_{e}^{2} y_{n}^{2} - v_{s}^{2} x_{n}^{2} - 4 v_{s}^{2} x_{n} x_{s} - v_{s}^{2} x_{s}^{2} - v_{s}^{2} y_{s}^{2}}{2 \left(v_{e}^{2} - v_{s}^{2}\right)}\right) + \frac{- 2 v_{e}^{2} x_{n}^{2} x_{s} - 2 v_{e}^{2} x_{n} x_{s}^{2} - 2 v_{e}^{2} x_{s} y_{n}^{2} + 2 v_{s}^{2} x_{n}^{2} x_{s} + 2 v_{s}^{2} x_{n} x_{s}^{2} + 2 v_{s}^{2} x_{n} y_{s}^{2}}{v_{e}^{2} - v_{s}^{2}}\right)^{2}}{16} + \frac{\left(- \frac{3 \left(- 2 x_{n} - 2 x_{s}\right)^{2}}{8} + \frac{v_{e}^{2} x_{n}^{2} + 4 v_{e}^{2} x_{n} x_{s} + v_{e}^{2} x_{s}^{2} + v_{e}^{2} y_{n}^{2} - v_{s}^{2} x_{n}^{2} - 4 v_{s}^{2} x_{n} x_{s} - v_{s}^{2} x_{s}^{2} - v_{s}^{2} y_{s}^{2}}{v_{e}^{2} - v_{s}^{2}}\right)^{3}}{216} + \sqrt{\frac{\left(\left(- 2 x_{n} - 2 x_{s}\right) \left(\left(- 2 x_{n} - 2 x_{s}\right) \left(\frac{3 \left(- 2 x_{n} - 2 x_{s}\right)^{2}}{256} - \frac{v_{e}^{2} x_{n}^{2} + 4 v_{e}^{2} x_{n} x_{s} + v_{e}^{2} x_{s}^{2} + v_{e}^{2} y_{n}^{2} - v_{s}^{2} x_{n}^{2} - 4 v_{s}^{2} x_{n} x_{s} - v_{s}^{2} x_{s}^{2} - v_{s}^{2} y_{s}^{2}}{16 \left(v_{e}^{2} - v_{s}^{2}\right)}\right) + \frac{- 2 v_{e}^{2} x_{n}^{2} x_{s} - 2 v_{e}^{2} x_{n} x_{s}^{2} - 2 v_{e}^{2} x_{s} y_{n}^{2} + 2 v_{s}^{2} x_{n}^{2} x_{s} + 2 v_{s}^{2} x_{n} x_{s}^{2} + 2 v_{s}^{2} x_{n} y_{s}^{2}}{4 \left(v_{e}^{2} - v_{s}^{2}\right)}\right) - \frac{\left(- \frac{3 \left(- 2 x_{n} - 2 x_{s}\right)^{2}}{8} + \frac{v_{e}^{2} x_{n}^{2} + 4 v_{e}^{2} x_{n} x_{s} + v_{e}^{2} x_{s}^{2} + v_{e}^{2} y_{n}^{2} - v_{s}^{2} x_{n}^{2} - 4 v_{s}^{2} x_{n} x_{s} - v_{s}^{2} x_{s}^{2} - v_{s}^{2} y_{s}^{2}}{v_{e}^{2} - v_{s}^{2}}\right)^{2}}{12} - \frac{v_{e}^{2} x_{n}^{2} x_{s}^{2} + v_{e}^{2} x_{s}^{2} y_{n}^{2} - v_{s}^{2} x_{n}^{2} x_{s}^{2} - v_{s}^{2} x_{n}^{2} y_{s}^{2}}{v_{e}^{2} - v_{s}^{2}}\right)^{3}}{27} + \frac{\left(\frac{\left(- \left(- 2 x_{n} - 2 x_{s}\right) \left(\left(- 2 x_{n} - 2 x_{s}\right) \left(\frac{3 \left(- 2 x_{n} - 2 x_{s}\right)^{2}}{256} - \frac{v_{e}^{2} x_{n}^{2} + 4 v_{e}^{2} x_{n} x_{s} + v_{e}^{2} x_{s}^{2} + v_{e}^{2} y_{n}^{2} - v_{s}^{2} x_{n}^{2} - 4 v_{s}^{2} x_{n} x_{s} - v_{s}^{2} x_{s}^{2} - v_{s}^{2} y_{s}^{2}}{16 \left(v_{e}^{2} - v_{s}^{2}\right)}\right) + \frac{- 2 v_{e}^{2} x_{n}^{2} x_{s} - 2 v_{e}^{2} x_{n} x_{s}^{2} - 2 v_{e}^{2} x_{s} y_{n}^{2} + 2 v_{s}^{2} x_{n}^{2} x_{s} + 2 v_{s}^{2} x_{n} x_{s}^{2} + 2 v_{s}^{2} x_{n} y_{s}^{2}}{4 \left(v_{e}^{2} - v_{s}^{2}\right)}\right) + \frac{v_{e}^{2} x_{n}^{2} x_{s}^{2} + v_{e}^{2} x_{s}^{2} y_{n}^{2} - v_{s}^{2} x_{n}^{2} x_{s}^{2} - v_{s}^{2} x_{n}^{2} y_{s}^{2}}{v_{e}^{2} - v_{s}^{2}}\right) \left(- \frac{3 \left(- 2 x_{n} - 2 x_{s}\right)^{2}}{8} + \frac{v_{e}^{2} x_{n}^{2} + 4 v_{e}^{2} x_{n} x_{s} + v_{e}^{2} x_{s}^{2} + v_{e}^{2} y_{n}^{2} - v_{s}^{2} x_{n}^{2} - 4 v_{s}^{2} x_{n} x_{s} - v_{s}^{2} x_{s}^{2} - v_{s}^{2} y_{s}^{2}}{v_{e}^{2} - v_{s}^{2}}\right)}{3} - \frac{\left(\left(- 2 x_{n} - 2 x_{s}\right) \left(\frac{\left(- 2 x_{n} - 2 x_{s}\right)^{2}}{8} - \frac{v_{e}^{2} x_{n}^{2} + 4 v_{e}^{2} x_{n} x_{s} + v_{e}^{2} x_{s}^{2} + v_{e}^{2} y_{n}^{2} - v_{s}^{2} x_{n}^{2} - 4 v_{s}^{2} x_{n} x_{s} - v_{s}^{2} x_{s}^{2} - v_{s}^{2} y_{s}^{2}}{2 \left(v_{e}^{2} - v_{s}^{2}\right)}\right) + \frac{- 2 v_{e}^{2} x_{n}^{2} x_{s} - 2 v_{e}^{2} x_{n} x_{s}^{2} - 2 v_{e}^{2} x_{s} y_{n}^{2} + 2 v_{s}^{2} x_{n}^{2} x_{s} + 2 v_{s}^{2} x_{n} x_{s}^{2} + 2 v_{s}^{2} x_{n} y_{s}^{2}}{v_{e}^{2} - v_{s}^{2}}\right)^{2}}{8} - \frac{\left(- \frac{3 \left(- 2 x_{n} - 2 x_{s}\right)^{2}}{8} + \frac{v_{e}^{2} x_{n}^{2} + 4 v_{e}^{2} x_{n} x_{s} + v_{e}^{2} x_{s}^{2} + v_{e}^{2} y_{n}^{2} - v_{s}^{2} x_{n}^{2} - 4 v_{s}^{2} x_{n} x_{s} - v_{s}^{2} x_{s}^{2} - v_{s}^{2} y_{s}^{2}}{v_{e}^{2} - v_{s}^{2}}\right)^{3}}{108}\right)^{2}}{4}}} - \frac{4 \left(v_{e}^{2} x_{n}^{2} + 4 v_{e}^{2} x_{n} x_{s} + v_{e}^{2} x_{s}^{2} + v_{e}^{2} y_{n}^{2} - v_{s}^{2} x_{n}^{2} - 4 v_{s}^{2} x_{n} x_{s} - v_{s}^{2} x_{s}^{2} - v_{s}^{2} y_{s}^{2}\right)}{3 \left(v_{e}^{2} - v_{s}^{2}\right)}}}{2} & \text{otherwise} \end{cases}$

Conclusion On se convaincra facilement que même si on a effectivement trouvé des formules exactes celles-ci sont inexploitables. On passera donc à des méthodes numériques.