python3 -c 'import math,itertools,math
s=math.sqrt(3);b=math.sqrt(2)
def f(j,m):return math.sqrt(1/3)*(math.sin((j+1)*m*math.pi/6)-math.sin(j*m*math.pi/6))
def C(r,p,q,t):return sum(f(j,r)*f(j,p)*f(j,q)*f(j,t) for j in range(6))
def D(r,p,q,t,u,v):return sum(f(j,r)*f(j,p)*f(j,q)*f(j,t)*f(j,u)*f(j,v) for j in range(6))
w=[0,math.sqrt(2-s),1,math.sqrt(2),math.sqrt(3),math.sqrt(2+s)];a=[0,0,2,math.sqrt(3),2,0];M=[2,3,4];R=[1,3,5];nu=2+b+2*s
parts=[((2,2,3),(4,4)),((2,2,4),(3,4)),((2,3,4),(2,4)),((2,4,4),(2,3)),((3,4,4),(2,2))]
for r in R:
 xd=0
 for t in itertools.product(M,repeat=5):
  if abs(sum(w[i] for i in t)-nu)<1e-9:xd=xd+D(r,*t)*math.prod(a[i] for i in t)
 fd=-xd/1920
 xf=0
 for p in range(1,6):
  for qt in itertools.product(M,repeat=2):
   O=nu-w[qt[0]]-w[qt[1]];z=0
   for tr in itertools.product(M,repeat=3):
    if abs(sum(w[i] for i in tr)-O)<1e-9:z=z+C(p,*tr)*math.prod(a[i] for i in tr)
   xf=xf+C(r,p,*qt)*(z/24)*math.prod(a[i] for i in qt)/8/(w[p]**2-O**2)
 xp=0
 det=[]
 for T,P in parts:
  nt=math.factorial(3)/math.prod(math.factorial(T.count(q)) for q in set(T))
  np=math.factorial(2)/math.prod(math.factorial(P.count(q)) for q in set(P))
  O=sum(w[i] for i in T);S=0
  for p in range(1,6):S=S+C(r,p,*P)*C(p,*T)*math.prod(a[i] for i in T)*math.prod(a[i] for i in P)/(w[p]**2-O**2)
  xp=xp+nt*np*S/192;det.append((T,P,nt*np*S/192))
 print(r,fd,xf,xp,det)'