In [23]:

    

from sympy import *
init_printing()

s, p0, p1, p2, p3 = symbols('s p0 p1 p2 p3')
a = p0
b = -(11*p0 - 18*p1 + 9*p2 - 2*p3)/(2*s)
c = 9*(2*p0 - 5*p1 + 4*p2 - p3)/(2*s**2)
d = -9*(p0 - 3*p1 + 3*p2 - p3)/(2*s**3)

h = s/8
s0, s1, s2, s3, s4, s5, s6, s7, s8 = 0*h, 1*h, 2*h, 3*h, 4*h, 5*h, 6*h, 7*h, 8*h
theta0 = a*s0 + b*s0**2/2 + c*s0**3/3 + d*s0**4/4
theta1 = a*s1 + b*s1**2/2 + c*s1**3/3 + d*s1**4/4
theta2 = a*s2 + b*s2**2/2 + c*s2**3/3 + d*s2**4/4
theta3 = a*s3 + b*s3**2/2 + c*s3**3/3 + d*s3**4/4
theta4 = a*s4 + b*s4**2/2 + c*s4**3/3 + d*s4**4/4
theta5 = a*s5 + b*s5**2/2 + c*s5**3/3 + d*s5**4/4
theta6 = a*s6 + b*s6**2/2 + c*s6**3/3 + d*s6**4/4
theta7 = a*s7 + b*s7**2/2 + c*s7**3/3 + d*s7**4/4
theta8 = a*s8 + b*s8**2/2 + c*s8**3/3 + d*s8**4/4

f0, f1, f2, f3, f4, f5, f6, f7, f8 = cos(theta0), cos(theta1), cos(theta2), cos(theta3), cos(theta4), cos(theta5), cos(theta6), cos(theta7), cos(theta8)
g0, g1, g2, g3, g4, g5, g6, g7, g8 = sin(theta0), sin(theta1), sin(theta2), sin(theta3), sin(theta4), sin(theta5), sin(theta6), sin(theta7), sin(theta8)

F = h/3*(f0 + 2*(f2+f4+f6) + 4*(f1+f3+f5+f7) + f8)
G = h/3*(g0 + 2*(g2+g4+g6) + 4*(g1+g3+g5+g7) + g8)


    

In [13]:

    

F.diff(p1)


    

    
Out[13]:





$$\frac{s}{24} \left(- \frac{1851 s}{8192} \sin{\left (\frac{p_{0} s}{8} + \frac{s}{256} \left(- 11 p_{0} + 18 p_{1} - 9 p_{2} + 2 p_{3}\right) + \frac{s}{32768} \left(- 9 p_{0} + 27 p_{1} - 27 p_{2} + 9 p_{3}\right) + \frac{s}{3072} \left(18 p_{0} - 45 p_{1} + 36 p_{2} - 9 p_{3}\right) \right )} - \frac{363 s}{1024} \sin{\left (\frac{p_{0} s}{4} + \frac{s}{64} \left(- 11 p_{0} + 18 p_{1} - 9 p_{2} + 2 p_{3}\right) + \frac{s}{2048} \left(- 9 p_{0} + 27 p_{1} - 27 p_{2} + 9 p_{3}\right) + \frac{s}{384} \left(18 p_{0} - 45 p_{1} + 36 p_{2} - 9 p_{3}\right) \right )} - \frac{9963 s}{8192} \sin{\left (\frac{3 p_{0}}{8} s + \frac{9 s}{256} \left(- 11 p_{0} + 18 p_{1} - 9 p_{2} + 2 p_{3}\right) + \frac{81 s}{32768} \left(- 9 p_{0} + 27 p_{1} - 27 p_{2} + 9 p_{3}\right) + \frac{9 s}{1024} \left(18 p_{0} - 45 p_{1} + 36 p_{2} - 9 p_{3}\right) \right )} - \frac{51 s}{64} \sin{\left (\frac{p_{0} s}{2} + \frac{s}{16} \left(- 11 p_{0} + 18 p_{1} - 9 p_{2} + 2 p_{3}\right) + \frac{s}{128} \left(- 9 p_{0} + 27 p_{1} - 27 p_{2} + 9 p_{3}\right) + \frac{s}{48} \left(18 p_{0} - 45 p_{1} + 36 p_{2} - 9 p_{3}\right) \right )} - \frac{14475 s}{8192} \sin{\left (\frac{5 p_{0}}{8} s + \frac{25 s}{256} \left(- 11 p_{0} + 18 p_{1} - 9 p_{2} + 2 p_{3}\right) + \frac{625 s}{32768} \left(- 9 p_{0} + 27 p_{1} - 27 p_{2} + 9 p_{3}\right) + \frac{125 s}{3072} \left(18 p_{0} - 45 p_{1} + 36 p_{2} - 9 p_{3}\right) \right )} - \frac{891 s}{1024} \sin{\left (\frac{3 p_{0}}{4} s + \frac{9 s}{64} \left(- 11 p_{0} + 18 p_{1} - 9 p_{2} + 2 p_{3}\right) + \frac{81 s}{2048} \left(- 9 p_{0} + 27 p_{1} - 27 p_{2} + 9 p_{3}\right) + \frac{9 s}{128} \left(18 p_{0} - 45 p_{1} + 36 p_{2} - 9 p_{3}\right) \right )} - \frac{13083 s}{8192} \sin{\left (\frac{7 p_{0}}{8} s + \frac{49 s}{256} \left(- 11 p_{0} + 18 p_{1} - 9 p_{2} + 2 p_{3}\right) + \frac{2401 s}{32768} \left(- 9 p_{0} + 27 p_{1} - 27 p_{2} + 9 p_{3}\right) + \frac{343 s}{3072} \left(18 p_{0} - 45 p_{1} + 36 p_{2} - 9 p_{3}\right) \right )} - \frac{3 s}{8} \sin{\left (p_{0} s + \frac{s}{4} \left(- 11 p_{0} + 18 p_{1} - 9 p_{2} + 2 p_{3}\right) + \frac{s}{8} \left(- 9 p_{0} + 27 p_{1} - 27 p_{2} + 9 p_{3}\right) + \frac{s}{6} \left(18 p_{0} - 45 p_{1} + 36 p_{2} - 9 p_{3}\right) \right )}\right)$$

In [14]:

    

F.diff(p2)


    

    
Out[14]:





$$\frac{s}{24} \left(\frac{795 s}{8192} \sin{\left (\frac{p_{0} s}{8} + \frac{s}{256} \left(- 11 p_{0} + 18 p_{1} - 9 p_{2} + 2 p_{3}\right) + \frac{s}{32768} \left(- 9 p_{0} + 27 p_{1} - 27 p_{2} + 9 p_{3}\right) + \frac{s}{3072} \left(18 p_{0} - 45 p_{1} + 36 p_{2} - 9 p_{3}\right) \right )} + \frac{123 s}{1024} \sin{\left (\frac{p_{0} s}{4} + \frac{s}{64} \left(- 11 p_{0} + 18 p_{1} - 9 p_{2} + 2 p_{3}\right) + \frac{s}{2048} \left(- 9 p_{0} + 27 p_{1} - 27 p_{2} + 9 p_{3}\right) + \frac{s}{384} \left(18 p_{0} - 45 p_{1} + 36 p_{2} - 9 p_{3}\right) \right )} + \frac{2187 s}{8192} \sin{\left (\frac{3 p_{0}}{8} s + \frac{9 s}{256} \left(- 11 p_{0} + 18 p_{1} - 9 p_{2} + 2 p_{3}\right) + \frac{81 s}{32768} \left(- 9 p_{0} + 27 p_{1} - 27 p_{2} + 9 p_{3}\right) + \frac{9 s}{1024} \left(18 p_{0} - 45 p_{1} + 36 p_{2} - 9 p_{3}\right) \right )} + \frac{3 s}{64} \sin{\left (\frac{p_{0} s}{2} + \frac{s}{16} \left(- 11 p_{0} + 18 p_{1} - 9 p_{2} + 2 p_{3}\right) + \frac{s}{128} \left(- 9 p_{0} + 27 p_{1} - 27 p_{2} + 9 p_{3}\right) + \frac{s}{48} \left(18 p_{0} - 45 p_{1} + 36 p_{2} - 9 p_{3}\right) \right )} - \frac{2325 s}{8192} \sin{\left (\frac{5 p_{0}}{8} s + \frac{25 s}{256} \left(- 11 p_{0} + 18 p_{1} - 9 p_{2} + 2 p_{3}\right) + \frac{625 s}{32768} \left(- 9 p_{0} + 27 p_{1} - 27 p_{2} + 9 p_{3}\right) + \frac{125 s}{3072} \left(18 p_{0} - 45 p_{1} + 36 p_{2} - 9 p_{3}\right) \right )} - \frac{405 s}{1024} \sin{\left (\frac{3 p_{0}}{4} s + \frac{9 s}{64} \left(- 11 p_{0} + 18 p_{1} - 9 p_{2} + 2 p_{3}\right) + \frac{81 s}{2048} \left(- 9 p_{0} + 27 p_{1} - 27 p_{2} + 9 p_{3}\right) + \frac{9 s}{128} \left(18 p_{0} - 45 p_{1} + 36 p_{2} - 9 p_{3}\right) \right )} - \frac{10437 s}{8192} \sin{\left (\frac{7 p_{0}}{8} s + \frac{49 s}{256} \left(- 11 p_{0} + 18 p_{1} - 9 p_{2} + 2 p_{3}\right) + \frac{2401 s}{32768} \left(- 9 p_{0} + 27 p_{1} - 27 p_{2} + 9 p_{3}\right) + \frac{343 s}{3072} \left(18 p_{0} - 45 p_{1} + 36 p_{2} - 9 p_{3}\right) \right )} - \frac{3 s}{8} \sin{\left (p_{0} s + \frac{s}{4} \left(- 11 p_{0} + 18 p_{1} - 9 p_{2} + 2 p_{3}\right) + \frac{s}{8} \left(- 9 p_{0} + 27 p_{1} - 27 p_{2} + 9 p_{3}\right) + \frac{s}{6} \left(18 p_{0} - 45 p_{1} + 36 p_{2} - 9 p_{3}\right) \right )}\right)$$

In [15]:

    

F.diff(s)


    

    
Out[15]:





$$\frac{s}{24} \left(- 4 \left(\frac{2871 p_{0}}{32768} + \frac{1851 p_{1}}{32768} - \frac{795 p_{2}}{32768} + \frac{169 p_{3}}{32768}\right) \sin{\left (\frac{p_{0} s}{8} + \frac{s}{256} \left(- 11 p_{0} + 18 p_{1} - 9 p_{2} + 2 p_{3}\right) + \frac{s}{32768} \left(- 9 p_{0} + 27 p_{1} - 27 p_{2} + 9 p_{3}\right) + \frac{s}{3072} \left(18 p_{0} - 45 p_{1} + 36 p_{2} - 9 p_{3}\right) \right )} - 4 \left(\frac{3655 p_{0}}{32768} + \frac{14475 p_{1}}{32768} + \frac{2325 p_{2}}{32768} + \frac{25 p_{3}}{32768}\right) \sin{\left (\frac{5 p_{0}}{8} s + \frac{25 s}{256} \left(- 11 p_{0} + 18 p_{1} - 9 p_{2} + 2 p_{3}\right) + \frac{625 s}{32768} \left(- 9 p_{0} + 27 p_{1} - 27 p_{2} + 9 p_{3}\right) + \frac{125 s}{3072} \left(18 p_{0} - 45 p_{1} + 36 p_{2} - 9 p_{3}\right) \right )} - 2 \left(\frac{231 p_{0}}{2048} + \frac{891 p_{1}}{2048} + \frac{405 p_{2}}{2048} + \frac{9 p_{3}}{2048}\right) \sin{\left (\frac{3 p_{0}}{4} s + \frac{9 s}{64} \left(- 11 p_{0} + 18 p_{1} - 9 p_{2} + 2 p_{3}\right) + \frac{81 s}{2048} \left(- 9 p_{0} + 27 p_{1} - 27 p_{2} + 9 p_{3}\right) + \frac{9 s}{128} \left(18 p_{0} - 45 p_{1} + 36 p_{2} - 9 p_{3}\right) \right )} - 2 \left(\frac{15 p_{0}}{128} + \frac{51 p_{1}}{128} - \frac{3 p_{2}}{128} + \frac{p_{3}}{128}\right) \sin{\left (\frac{p_{0} s}{2} + \frac{s}{16} \left(- 11 p_{0} + 18 p_{1} - 9 p_{2} + 2 p_{3}\right) + \frac{s}{128} \left(- 9 p_{0} + 27 p_{1} - 27 p_{2} + 9 p_{3}\right) + \frac{s}{48} \left(18 p_{0} - 45 p_{1} + 36 p_{2} - 9 p_{3}\right) \right )} - 4 \left(\frac{3927 p_{0}}{32768} + \frac{13083 p_{1}}{32768} + \frac{10437 p_{2}}{32768} + \frac{1225 p_{3}}{32768}\right) \sin{\left (\frac{7 p_{0}}{8} s + \frac{49 s}{256} \left(- 11 p_{0} + 18 p_{1} - 9 p_{2} + 2 p_{3}\right) + \frac{2401 s}{32768} \left(- 9 p_{0} + 27 p_{1} - 27 p_{2} + 9 p_{3}\right) + \frac{343 s}{3072} \left(18 p_{0} - 45 p_{1} + 36 p_{2} - 9 p_{3}\right) \right )} - 2 \left(\frac{247 p_{0}}{2048} + \frac{363 p_{1}}{2048} - \frac{123 p_{2}}{2048} + \frac{25 p_{3}}{2048}\right) \sin{\left (\frac{p_{0} s}{4} + \frac{s}{64} \left(- 11 p_{0} + 18 p_{1} - 9 p_{2} + 2 p_{3}\right) + \frac{s}{2048} \left(- 9 p_{0} + 27 p_{1} - 27 p_{2} + 9 p_{3}\right) + \frac{s}{384} \left(18 p_{0} - 45 p_{1} + 36 p_{2} - 9 p_{3}\right) \right )} - 4 \left(\frac{4071 p_{0}}{32768} + \frac{9963 p_{1}}{32768} - \frac{2187 p_{2}}{32768} + \frac{441 p_{3}}{32768}\right) \sin{\left (\frac{3 p_{0}}{8} s + \frac{9 s}{256} \left(- 11 p_{0} + 18 p_{1} - 9 p_{2} + 2 p_{3}\right) + \frac{81 s}{32768} \left(- 9 p_{0} + 27 p_{1} - 27 p_{2} + 9 p_{3}\right) + \frac{9 s}{1024} \left(18 p_{0} - 45 p_{1} + 36 p_{2} - 9 p_{3}\right) \right )} - \left(\frac{p_{0}}{8} + \frac{3 p_{1}}{8} + \frac{3 p_{2}}{8} + \frac{p_{3}}{8}\right) \sin{\left (p_{0} s + \frac{s}{4} \left(- 11 p_{0} + 18 p_{1} - 9 p_{2} + 2 p_{3}\right) + \frac{s}{8} \left(- 9 p_{0} + 27 p_{1} - 27 p_{2} + 9 p_{3}\right) + \frac{s}{6} \left(18 p_{0} - 45 p_{1} + 36 p_{2} - 9 p_{3}\right) \right )}\right) + \frac{1}{6} \cos{\left (\frac{p_{0} s}{8} + \frac{s}{256} \left(- 11 p_{0} + 18 p_{1} - 9 p_{2} + 2 p_{3}\right) + \frac{s}{32768} \left(- 9 p_{0} + 27 p_{1} - 27 p_{2} + 9 p_{3}\right) + \frac{s}{3072} \left(18 p_{0} - 45 p_{1} + 36 p_{2} - 9 p_{3}\right) \right )} + \frac{1}{12} \cos{\left (\frac{p_{0} s}{4} + \frac{s}{64} \left(- 11 p_{0} + 18 p_{1} - 9 p_{2} + 2 p_{3}\right) + \frac{s}{2048} \left(- 9 p_{0} + 27 p_{1} - 27 p_{2} + 9 p_{3}\right) + \frac{s}{384} \left(18 p_{0} - 45 p_{1} + 36 p_{2} - 9 p_{3}\right) \right )} + \frac{1}{6} \cos{\left (\frac{3 p_{0}}{8} s + \frac{9 s}{256} \left(- 11 p_{0} + 18 p_{1} - 9 p_{2} + 2 p_{3}\right) + \frac{81 s}{32768} \left(- 9 p_{0} + 27 p_{1} - 27 p_{2} + 9 p_{3}\right) + \frac{9 s}{1024} \left(18 p_{0} - 45 p_{1} + 36 p_{2} - 9 p_{3}\right) \right )} + \frac{1}{12} \cos{\left (\frac{p_{0} s}{2} + \frac{s}{16} \left(- 11 p_{0} + 18 p_{1} - 9 p_{2} + 2 p_{3}\right) + \frac{s}{128} \left(- 9 p_{0} + 27 p_{1} - 27 p_{2} + 9 p_{3}\right) + \frac{s}{48} \left(18 p_{0} - 45 p_{1} + 36 p_{2} - 9 p_{3}\right) \right )} + \frac{1}{6} \cos{\left (\frac{5 p_{0}}{8} s + \frac{25 s}{256} \left(- 11 p_{0} + 18 p_{1} - 9 p_{2} + 2 p_{3}\right) + \frac{625 s}{32768} \left(- 9 p_{0} + 27 p_{1} - 27 p_{2} + 9 p_{3}\right) + \frac{125 s}{3072} \left(18 p_{0} - 45 p_{1} + 36 p_{2} - 9 p_{3}\right) \right )} + \frac{1}{12} \cos{\left (\frac{3 p_{0}}{4} s + \frac{9 s}{64} \left(- 11 p_{0} + 18 p_{1} - 9 p_{2} + 2 p_{3}\right) + \frac{81 s}{2048} \left(- 9 p_{0} + 27 p_{1} - 27 p_{2} + 9 p_{3}\right) + \frac{9 s}{128} \left(18 p_{0} - 45 p_{1} + 36 p_{2} - 9 p_{3}\right) \right )} + \frac{1}{6} \cos{\left (\frac{7 p_{0}}{8} s + \frac{49 s}{256} \left(- 11 p_{0} + 18 p_{1} - 9 p_{2} + 2 p_{3}\right) + \frac{2401 s}{32768} \left(- 9 p_{0} + 27 p_{1} - 27 p_{2} + 9 p_{3}\right) + \frac{343 s}{3072} \left(18 p_{0} - 45 p_{1} + 36 p_{2} - 9 p_{3}\right) \right )} + \frac{1}{24} \cos{\left (p_{0} s + \frac{s}{4} \left(- 11 p_{0} + 18 p_{1} - 9 p_{2} + 2 p_{3}\right) + \frac{s}{8} \left(- 9 p_{0} + 27 p_{1} - 27 p_{2} + 9 p_{3}\right) + \frac{s}{6} \left(18 p_{0} - 45 p_{1} + 36 p_{2} - 9 p_{3}\right) \right )} + \frac{1}{24}$$

In [16]:

    

G.diff(p1)


    

    
Out[16]:





$$\frac{s}{24} \left(\frac{1851 s}{8192} \cos{\left (\frac{p_{0} s}{8} + \frac{s}{256} \left(- 11 p_{0} + 18 p_{1} - 9 p_{2} + 2 p_{3}\right) + \frac{s}{32768} \left(- 9 p_{0} + 27 p_{1} - 27 p_{2} + 9 p_{3}\right) + \frac{s}{3072} \left(18 p_{0} - 45 p_{1} + 36 p_{2} - 9 p_{3}\right) \right )} + \frac{363 s}{1024} \cos{\left (\frac{p_{0} s}{4} + \frac{s}{64} \left(- 11 p_{0} + 18 p_{1} - 9 p_{2} + 2 p_{3}\right) + \frac{s}{2048} \left(- 9 p_{0} + 27 p_{1} - 27 p_{2} + 9 p_{3}\right) + \frac{s}{384} \left(18 p_{0} - 45 p_{1} + 36 p_{2} - 9 p_{3}\right) \right )} + \frac{9963 s}{8192} \cos{\left (\frac{3 p_{0}}{8} s + \frac{9 s}{256} \left(- 11 p_{0} + 18 p_{1} - 9 p_{2} + 2 p_{3}\right) + \frac{81 s}{32768} \left(- 9 p_{0} + 27 p_{1} - 27 p_{2} + 9 p_{3}\right) + \frac{9 s}{1024} \left(18 p_{0} - 45 p_{1} + 36 p_{2} - 9 p_{3}\right) \right )} + \frac{51 s}{64} \cos{\left (\frac{p_{0} s}{2} + \frac{s}{16} \left(- 11 p_{0} + 18 p_{1} - 9 p_{2} + 2 p_{3}\right) + \frac{s}{128} \left(- 9 p_{0} + 27 p_{1} - 27 p_{2} + 9 p_{3}\right) + \frac{s}{48} \left(18 p_{0} - 45 p_{1} + 36 p_{2} - 9 p_{3}\right) \right )} + \frac{14475 s}{8192} \cos{\left (\frac{5 p_{0}}{8} s + \frac{25 s}{256} \left(- 11 p_{0} + 18 p_{1} - 9 p_{2} + 2 p_{3}\right) + \frac{625 s}{32768} \left(- 9 p_{0} + 27 p_{1} - 27 p_{2} + 9 p_{3}\right) + \frac{125 s}{3072} \left(18 p_{0} - 45 p_{1} + 36 p_{2} - 9 p_{3}\right) \right )} + \frac{891 s}{1024} \cos{\left (\frac{3 p_{0}}{4} s + \frac{9 s}{64} \left(- 11 p_{0} + 18 p_{1} - 9 p_{2} + 2 p_{3}\right) + \frac{81 s}{2048} \left(- 9 p_{0} + 27 p_{1} - 27 p_{2} + 9 p_{3}\right) + \frac{9 s}{128} \left(18 p_{0} - 45 p_{1} + 36 p_{2} - 9 p_{3}\right) \right )} + \frac{13083 s}{8192} \cos{\left (\frac{7 p_{0}}{8} s + \frac{49 s}{256} \left(- 11 p_{0} + 18 p_{1} - 9 p_{2} + 2 p_{3}\right) + \frac{2401 s}{32768} \left(- 9 p_{0} + 27 p_{1} - 27 p_{2} + 9 p_{3}\right) + \frac{343 s}{3072} \left(18 p_{0} - 45 p_{1} + 36 p_{2} - 9 p_{3}\right) \right )} + \frac{3 s}{8} \cos{\left (p_{0} s + \frac{s}{4} \left(- 11 p_{0} + 18 p_{1} - 9 p_{2} + 2 p_{3}\right) + \frac{s}{8} \left(- 9 p_{0} + 27 p_{1} - 27 p_{2} + 9 p_{3}\right) + \frac{s}{6} \left(18 p_{0} - 45 p_{1} + 36 p_{2} - 9 p_{3}\right) \right )}\right)$$

In [17]:

    

G.diff(p2)


    

    
Out[17]:





$$\frac{s}{24} \left(- \frac{795 s}{8192} \cos{\left (\frac{p_{0} s}{8} + \frac{s}{256} \left(- 11 p_{0} + 18 p_{1} - 9 p_{2} + 2 p_{3}\right) + \frac{s}{32768} \left(- 9 p_{0} + 27 p_{1} - 27 p_{2} + 9 p_{3}\right) + \frac{s}{3072} \left(18 p_{0} - 45 p_{1} + 36 p_{2} - 9 p_{3}\right) \right )} - \frac{123 s}{1024} \cos{\left (\frac{p_{0} s}{4} + \frac{s}{64} \left(- 11 p_{0} + 18 p_{1} - 9 p_{2} + 2 p_{3}\right) + \frac{s}{2048} \left(- 9 p_{0} + 27 p_{1} - 27 p_{2} + 9 p_{3}\right) + \frac{s}{384} \left(18 p_{0} - 45 p_{1} + 36 p_{2} - 9 p_{3}\right) \right )} - \frac{2187 s}{8192} \cos{\left (\frac{3 p_{0}}{8} s + \frac{9 s}{256} \left(- 11 p_{0} + 18 p_{1} - 9 p_{2} + 2 p_{3}\right) + \frac{81 s}{32768} \left(- 9 p_{0} + 27 p_{1} - 27 p_{2} + 9 p_{3}\right) + \frac{9 s}{1024} \left(18 p_{0} - 45 p_{1} + 36 p_{2} - 9 p_{3}\right) \right )} - \frac{3 s}{64} \cos{\left (\frac{p_{0} s}{2} + \frac{s}{16} \left(- 11 p_{0} + 18 p_{1} - 9 p_{2} + 2 p_{3}\right) + \frac{s}{128} \left(- 9 p_{0} + 27 p_{1} - 27 p_{2} + 9 p_{3}\right) + \frac{s}{48} \left(18 p_{0} - 45 p_{1} + 36 p_{2} - 9 p_{3}\right) \right )} + \frac{2325 s}{8192} \cos{\left (\frac{5 p_{0}}{8} s + \frac{25 s}{256} \left(- 11 p_{0} + 18 p_{1} - 9 p_{2} + 2 p_{3}\right) + \frac{625 s}{32768} \left(- 9 p_{0} + 27 p_{1} - 27 p_{2} + 9 p_{3}\right) + \frac{125 s}{3072} \left(18 p_{0} - 45 p_{1} + 36 p_{2} - 9 p_{3}\right) \right )} + \frac{405 s}{1024} \cos{\left (\frac{3 p_{0}}{4} s + \frac{9 s}{64} \left(- 11 p_{0} + 18 p_{1} - 9 p_{2} + 2 p_{3}\right) + \frac{81 s}{2048} \left(- 9 p_{0} + 27 p_{1} - 27 p_{2} + 9 p_{3}\right) + \frac{9 s}{128} \left(18 p_{0} - 45 p_{1} + 36 p_{2} - 9 p_{3}\right) \right )} + \frac{10437 s}{8192} \cos{\left (\frac{7 p_{0}}{8} s + \frac{49 s}{256} \left(- 11 p_{0} + 18 p_{1} - 9 p_{2} + 2 p_{3}\right) + \frac{2401 s}{32768} \left(- 9 p_{0} + 27 p_{1} - 27 p_{2} + 9 p_{3}\right) + \frac{343 s}{3072} \left(18 p_{0} - 45 p_{1} + 36 p_{2} - 9 p_{3}\right) \right )} + \frac{3 s}{8} \cos{\left (p_{0} s + \frac{s}{4} \left(- 11 p_{0} + 18 p_{1} - 9 p_{2} + 2 p_{3}\right) + \frac{s}{8} \left(- 9 p_{0} + 27 p_{1} - 27 p_{2} + 9 p_{3}\right) + \frac{s}{6} \left(18 p_{0} - 45 p_{1} + 36 p_{2} - 9 p_{3}\right) \right )}\right)$$

In [18]:

    

G.diff(s)


    

    
Out[18]:





$$\frac{s}{24} \left(4 \left(\frac{2871 p_{0}}{32768} + \frac{1851 p_{1}}{32768} - \frac{795 p_{2}}{32768} + \frac{169 p_{3}}{32768}\right) \cos{\left (\frac{p_{0} s}{8} + \frac{s}{256} \left(- 11 p_{0} + 18 p_{1} - 9 p_{2} + 2 p_{3}\right) + \frac{s}{32768} \left(- 9 p_{0} + 27 p_{1} - 27 p_{2} + 9 p_{3}\right) + \frac{s}{3072} \left(18 p_{0} - 45 p_{1} + 36 p_{2} - 9 p_{3}\right) \right )} + 4 \left(\frac{3655 p_{0}}{32768} + \frac{14475 p_{1}}{32768} + \frac{2325 p_{2}}{32768} + \frac{25 p_{3}}{32768}\right) \cos{\left (\frac{5 p_{0}}{8} s + \frac{25 s}{256} \left(- 11 p_{0} + 18 p_{1} - 9 p_{2} + 2 p_{3}\right) + \frac{625 s}{32768} \left(- 9 p_{0} + 27 p_{1} - 27 p_{2} + 9 p_{3}\right) + \frac{125 s}{3072} \left(18 p_{0} - 45 p_{1} + 36 p_{2} - 9 p_{3}\right) \right )} + 2 \left(\frac{231 p_{0}}{2048} + \frac{891 p_{1}}{2048} + \frac{405 p_{2}}{2048} + \frac{9 p_{3}}{2048}\right) \cos{\left (\frac{3 p_{0}}{4} s + \frac{9 s}{64} \left(- 11 p_{0} + 18 p_{1} - 9 p_{2} + 2 p_{3}\right) + \frac{81 s}{2048} \left(- 9 p_{0} + 27 p_{1} - 27 p_{2} + 9 p_{3}\right) + \frac{9 s}{128} \left(18 p_{0} - 45 p_{1} + 36 p_{2} - 9 p_{3}\right) \right )} + 2 \left(\frac{15 p_{0}}{128} + \frac{51 p_{1}}{128} - \frac{3 p_{2}}{128} + \frac{p_{3}}{128}\right) \cos{\left (\frac{p_{0} s}{2} + \frac{s}{16} \left(- 11 p_{0} + 18 p_{1} - 9 p_{2} + 2 p_{3}\right) + \frac{s}{128} \left(- 9 p_{0} + 27 p_{1} - 27 p_{2} + 9 p_{3}\right) + \frac{s}{48} \left(18 p_{0} - 45 p_{1} + 36 p_{2} - 9 p_{3}\right) \right )} + 4 \left(\frac{3927 p_{0}}{32768} + \frac{13083 p_{1}}{32768} + \frac{10437 p_{2}}{32768} + \frac{1225 p_{3}}{32768}\right) \cos{\left (\frac{7 p_{0}}{8} s + \frac{49 s}{256} \left(- 11 p_{0} + 18 p_{1} - 9 p_{2} + 2 p_{3}\right) + \frac{2401 s}{32768} \left(- 9 p_{0} + 27 p_{1} - 27 p_{2} + 9 p_{3}\right) + \frac{343 s}{3072} \left(18 p_{0} - 45 p_{1} + 36 p_{2} - 9 p_{3}\right) \right )} + 2 \left(\frac{247 p_{0}}{2048} + \frac{363 p_{1}}{2048} - \frac{123 p_{2}}{2048} + \frac{25 p_{3}}{2048}\right) \cos{\left (\frac{p_{0} s}{4} + \frac{s}{64} \left(- 11 p_{0} + 18 p_{1} - 9 p_{2} + 2 p_{3}\right) + \frac{s}{2048} \left(- 9 p_{0} + 27 p_{1} - 27 p_{2} + 9 p_{3}\right) + \frac{s}{384} \left(18 p_{0} - 45 p_{1} + 36 p_{2} - 9 p_{3}\right) \right )} + 4 \left(\frac{4071 p_{0}}{32768} + \frac{9963 p_{1}}{32768} - \frac{2187 p_{2}}{32768} + \frac{441 p_{3}}{32768}\right) \cos{\left (\frac{3 p_{0}}{8} s + \frac{9 s}{256} \left(- 11 p_{0} + 18 p_{1} - 9 p_{2} + 2 p_{3}\right) + \frac{81 s}{32768} \left(- 9 p_{0} + 27 p_{1} - 27 p_{2} + 9 p_{3}\right) + \frac{9 s}{1024} \left(18 p_{0} - 45 p_{1} + 36 p_{2} - 9 p_{3}\right) \right )} + \left(\frac{p_{0}}{8} + \frac{3 p_{1}}{8} + \frac{3 p_{2}}{8} + \frac{p_{3}}{8}\right) \cos{\left (p_{0} s + \frac{s}{4} \left(- 11 p_{0} + 18 p_{1} - 9 p_{2} + 2 p_{3}\right) + \frac{s}{8} \left(- 9 p_{0} + 27 p_{1} - 27 p_{2} + 9 p_{3}\right) + \frac{s}{6} \left(18 p_{0} - 45 p_{1} + 36 p_{2} - 9 p_{3}\right) \right )}\right) + \frac{1}{6} \sin{\left (\frac{p_{0} s}{8} + \frac{s}{256} \left(- 11 p_{0} + 18 p_{1} - 9 p_{2} + 2 p_{3}\right) + \frac{s}{32768} \left(- 9 p_{0} + 27 p_{1} - 27 p_{2} + 9 p_{3}\right) + \frac{s}{3072} \left(18 p_{0} - 45 p_{1} + 36 p_{2} - 9 p_{3}\right) \right )} + \frac{1}{12} \sin{\left (\frac{p_{0} s}{4} + \frac{s}{64} \left(- 11 p_{0} + 18 p_{1} - 9 p_{2} + 2 p_{3}\right) + \frac{s}{2048} \left(- 9 p_{0} + 27 p_{1} - 27 p_{2} + 9 p_{3}\right) + \frac{s}{384} \left(18 p_{0} - 45 p_{1} + 36 p_{2} - 9 p_{3}\right) \right )} + \frac{1}{6} \sin{\left (\frac{3 p_{0}}{8} s + \frac{9 s}{256} \left(- 11 p_{0} + 18 p_{1} - 9 p_{2} + 2 p_{3}\right) + \frac{81 s}{32768} \left(- 9 p_{0} + 27 p_{1} - 27 p_{2} + 9 p_{3}\right) + \frac{9 s}{1024} \left(18 p_{0} - 45 p_{1} + 36 p_{2} - 9 p_{3}\right) \right )} + \frac{1}{12} \sin{\left (\frac{p_{0} s}{2} + \frac{s}{16} \left(- 11 p_{0} + 18 p_{1} - 9 p_{2} + 2 p_{3}\right) + \frac{s}{128} \left(- 9 p_{0} + 27 p_{1} - 27 p_{2} + 9 p_{3}\right) + \frac{s}{48} \left(18 p_{0} - 45 p_{1} + 36 p_{2} - 9 p_{3}\right) \right )} + \frac{1}{6} \sin{\left (\frac{5 p_{0}}{8} s + \frac{25 s}{256} \left(- 11 p_{0} + 18 p_{1} - 9 p_{2} + 2 p_{3}\right) + \frac{625 s}{32768} \left(- 9 p_{0} + 27 p_{1} - 27 p_{2} + 9 p_{3}\right) + \frac{125 s}{3072} \left(18 p_{0} - 45 p_{1} + 36 p_{2} - 9 p_{3}\right) \right )} + \frac{1}{12} \sin{\left (\frac{3 p_{0}}{4} s + \frac{9 s}{64} \left(- 11 p_{0} + 18 p_{1} - 9 p_{2} + 2 p_{3}\right) + \frac{81 s}{2048} \left(- 9 p_{0} + 27 p_{1} - 27 p_{2} + 9 p_{3}\right) + \frac{9 s}{128} \left(18 p_{0} - 45 p_{1} + 36 p_{2} - 9 p_{3}\right) \right )} + \frac{1}{6} \sin{\left (\frac{7 p_{0}}{8} s + \frac{49 s}{256} \left(- 11 p_{0} + 18 p_{1} - 9 p_{2} + 2 p_{3}\right) + \frac{2401 s}{32768} \left(- 9 p_{0} + 27 p_{1} - 27 p_{2} + 9 p_{3}\right) + \frac{343 s}{3072} \left(18 p_{0} - 45 p_{1} + 36 p_{2} - 9 p_{3}\right) \right )} + \frac{1}{24} \sin{\left (p_{0} s + \frac{s}{4} \left(- 11 p_{0} + 18 p_{1} - 9 p_{2} + 2 p_{3}\right) + \frac{s}{8} \left(- 9 p_{0} + 27 p_{1} - 27 p_{2} + 9 p_{3}\right) + \frac{s}{6} \left(18 p_{0} - 45 p_{1} + 36 p_{2} - 9 p_{3}\right) \right )}$$

In [19]:

    

theta8.diff(p1)


    

    
Out[19]:





$$\frac{3 s}{8}$$

In [20]:

    

theta8.diff(p2)


    

    
Out[20]:





$$\frac{3 s}{8}$$

In [21]:

    

theta8.diff(s)


    

    
Out[21]:





$$\frac{p_{0}}{8} + \frac{3 p_{1}}{8} + \frac{3 p_{2}}{8} + \frac{p_{3}}{8}$$

In [ ]: