Simplify outputs huge result for diff symbolic expression with hyperbolic functions. Part 2.

This question relates to the previous one.

After the 2D-case was successfully solved I finaly end up with the 3D-case task:

syms x y z real;
syms f(x,y,z) g(x,y,z);
syms Nc(x,y,z) r(x,y,z);
syms xd_dot(x,y,z) yd_dot(x,y,z);
syms chi_d(x,y,z);
syms fx fy fz d1 real;
syms gx gy gz d2 real;
syms k A B real;
syms F G R NC;

f(x,y,z) = fx*x + fy*y + fz*z + d1;
g(x,y,z) = gx*x + gy*y + gz*z + d2;

Nc(x,y,z) = sqrt((f(x,y,z)*fx + g(x,y,z)*gx)^2 + (f(x,y,z)*fy + g(x,y,z)*gy)^2 + (f(x,y,z)*fz + g(x,y,z)*gz)^2);
r(x,y,z) = sqrt(f(x,y,z)^2 + g(x,y,z)^2);

xd_dot(x,y,z) = -(f(x,y,z)*fx + g(x,y,z)*gx)*tanh(k*r(x,y,z))/Nc(x,y,z) + A*sech(k*r(x,y,z));
yd_dot(x,y,z) = -(f(x,y,z)*fy + g(x,y,z)*gy)*tanh(k*r(x,y,z))/Nc(x,y,z) + B*sech(k*r(x,y,z));

chi_d(x,y,z) = atan2(yd_dot(x,y,z), xd_dot(x,y,z));
d_chi_d_dx = diff(chi_d(x,y,z), x);

% substitute back original functions to shorten output
d_chi_d_dx = subs(d_chi_d_dx, fx*x + fy*y + fz*z + d1, F);
d_chi_d_dx = subs(d_chi_d_dx, gx*x + gy*y + gz*z + d2, G);
d_chi_d_dx = subs(d_chi_d_dx, sqrt(F^2 + G^2), R);
d_chi_d_dx = subs(d_chi_d_dx, sqrt((F*fx + G*gx)^2 + (F*fy + G*gy)^2 + (F*fz + G*gz)^2), NC);

d_chi_d_dx = simplify(d_chi_d_dx, 'Steps', 200);

Output is sustainable but huge anyway:

((B/cosh(R*k) - (tanh(R*k)*(F*fy + G*gy))/NC)*((tanh(R*k)*(fx^2 + gx^2))/NC - (tanh(R*k)*(F*fx + G*gx)*(2*(F*fy + G*gy)*(fx*fy + gx*gy) + 2*(F*fz + G*gz)*(fx*fz + gx*gz) + 2*(F*fx + G*gx)*(fx^2 + gx^2)))/(2*NC^3) - (k*(tanh(R*k)^2 - 1)*(F*fx + G*gx)*(2*F*fx + 2*G*gx))/(2*NC*R) + (A*k*sinh(R*k)*(2*F*fx + 2*G*gx))/(2*R*cosh(R*k)^2)))/((A/cosh(R*k) - (tanh(R*k)*(F*fx + G*gx))/NC)^2 + (B/cosh(R*k) - (tanh(R*k)*(F*fy + G*gy))/NC)^2) - ((A/cosh(R*k) - (tanh(R*k)*(F*fx + G*gx))/NC)*((tanh(R*k)*(fx*fy + gx*gy))/NC - (tanh(R*k)*(F*fy + G*gy)*(2*(F*fy + G*gy)*(fx*fy + gx*gy) + 2*(F*fz + G*gz)*(fx*fz + gx*gz) + 2*(F*fx + G*gx)*(fx^2 + gx^2)))/(2*NC^3) - (k*(tanh(R*k)^2 - 1)*(2*F*fx + 2*G*gx)*(F*fy + G*gy))/(2*NC*R) + (B*k*sinh(R*k)*(2*F*fx + 2*G*gx))/(2*R*cosh(R*k)^2)))/((A/cosh(R*k) - (tanh(R*k)*(F*fx + G*gx))/NC)^2 + (B/cosh(R*k) - (tanh(R*k)*(F*fy + G*gy))/NC)^2)

There are certian parts of code that can be simplified futher:

1.

simplifyFraction((tanh(R*k)*(F*fx + G*gx)*(2*(F*fy + G*gy)*(fx*fy + gx*gy) + 2*(F*fz + G*gz)*(fx*fz + gx*gz) + 2*(F*fx + G*gx)*(fx^2 + gx^2)))/(2*NC^3));

2.

simplifyFraction((k*(tanh(R*k)^2 - 1)*(F*fx + G*gx)*(2*F*fx + 2*G*gx))/(2*NC*R));

3.

simplifyFraction((A*k*sinh(R*k)*(2*F*fx + 2*G*gx))/(2*R*cosh(R*k)^2));

4.

simplifyFraction((tanh(R*k)*(F*fy + G*gy)*(2*(F*fy + G*gy)*(fx*fy + gx*gy) + 2*(F*fz + G*gz)*(fx*fz + gx*gz) + 2*(F*fx + G*gx)*(fx^2 + gx^2)))/(2*NC^3));

5.

simplifyFraction((k*(tanh(R*k)^2 - 1)*(2*F*fx + 2*G*gx)*(F*fy + G*gy))/(2*NC*R));

But the issue is that these simpifications do not shorten output much and also should be identified manually.

Is there any way to simplify the original output more automatically?

Answer 1

[Andrey Mironenko]

The task was solved numerically instead of symbolically.