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SandBoxExpOfEnd

Edit detail for SandBoxExpOfEnd revision 11 of 18

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Editor: Bill Page Time: 2014/09/20 03:05:52 GMT+0
Note:

changed:
-eq2_2:= exp(X)=g[0]*id+g[1]*X
eq2_2:= exp(X)=g[0]+g[1]*X

changed:
-eq3_2:= exp(X)=g[0]*id+g[1]*X+g[2]*X^2
eq3_2:= exp(X)=g[0]+g[1]*X+g[2]*X^2

changed:
-eq3_8:= exp(X) = id+X'+X'^2/(1+γ)
eq3_8:= exp(X) = 1+X'+X'^2/(1+γ)

changed:
-eq5_2:= exp(X)=g[0]*id+g[1]*X+g[2]*X^2+g[3]*X^3
eq5_2:= exp(X)=g[0]+g[1]*X+g[2]*X^2+g[3]*X^3

changed:
-X^4 = X^4-eval(rhs eq5_1,x=X)
-test(_rule(X^2*(lhs %),X^2*(rhs %))(lhs eq5_6b)=rhs eq5_6b)
eq5_6c:= X^4 = X^4-eval(rhs eq5_1,x=X)
eq5_6d:= X^2*(lhs %)=X^2*(rhs %)
test(_rule(lhs eq5_6d,rhs eq5_6d)(lhs eq5_6b)=rhs eq5_6b)

changed:
-eq5_9:= exp(X) = exp(Y[1])*exp(Y[2])
-eval(eq5_9,[eq5_5a,eq5_5b])
eq5_9a:= exp(X) = exp(Y[1])*exp(Y[2])
eval(eq5_9a,[eq5_5a,eq5_5b])

added:
eq5_9b:= exp(X) = 1 + Y'[1] +Y'[2] + Y'[1]^2/(1+γ[1]) + Y'[2]^2/(1+γ[2])
normalize eval(eval(rhs eq5_9b,concat [eq5_8a,eq5_8b]),[eq5_5a,eq5_5b])
_rule(lhs eq5_6d,rhs eq5_6d)(%)
_rule(lhs eq5_6c,rhs eq5_6c)(%)
test(normalize(% - rhs eq5_4) = 0)

Choose n items from a list. Returns list of size binomial(#a,n) of lists.

fricas

(1) -> choose(a,n) ==
  j:=[i for i in 1..n]
  r:=[[a(j(i)) for i in 1..n]]
  k:=n
  while k>0 and j(k)+n-k<#a repeat
    j(k):=j(k)+1
    for i in k..n-1 repeat j(i+1):=j(i)+1
    r:=concat(r,[a(j(i)) for i in 1..n])
    k:=n; while j(k)+n-k>=#a and k>1 repeat k:=k-1
  if binomial(#a,n)~=#r then error "error in choose"
  return r

Type: Void

The Main Result

fricas

sum(x)==reduce(+,x,0)

Type: Void

fricas

product(x)==reduce(*,x,1)

Type: Void

fricas

-- specify n
f(i,j) == sum [ product x for x in choose([r[q]::Expression Integer for q in 1..n|q~=j],n-i-1)]

Type: Void

fricas

groupPolyCoeff(i) == (-1)^(i+n+1)*reduce(+,[exp(r[j])/reduce(*,[r[j]-r[m] for m in 1..n | j~=m])*f(i,j) for j in 1..n])

Type: Void

Polynomial of degree 2

fricas

n:=2

$\label{eq1}2$

(1)

Type: PositiveInteger?

fricas

eq2_1:= m[X]=(x-r[1])*(x-r[2])

$\label{eq2}{m_{X}}={{{x}^{2}}+{{\left(-{r_{2}}-{r_{1}}\right)}\ x}+{{r_{1}}\ {r_{2}}}}$

(2)

Type: Equation(Polynomial(Integer))

fricas

eq2_2:= exp(X)=g[0]+g[1]*X

$\label{eq3}{{e}^{X}}={{{g_{1}}\ X}+{g_{0}}}$

(3)

Type: Equation(Expression(Integer))

fricas

eq2_3a:= g[0]=groupPolyCoeff(0)

fricas

Compiling function choose with type (List(Expression(Integer)), 
      Integer) -> List(List(Expression(Integer)))

fricas

Compiling function product with type List(Expression(Integer)) -> 
      Expression(Integer)

fricas

Compiling function sum with type List(Expression(Integer)) -> 
      Expression(Integer)

fricas

Compiling function f with type (NonNegativeInteger, PositiveInteger)
       -> Expression(Integer)

fricas

Compiling function groupPolyCoeff with type NonNegativeInteger -> 
      Expression(Integer)

$\label{eq4}{g_{0}}={\frac{-{{r_{1}}\ {{e}^{r_{2}}}}+{{r_{2}}\ {{e}^{r_{1}}}}}{{r_{2}}-{r_{1}}}}$

(4)

Type: Equation(Expression(Integer))

fricas

eq2_3b:= g[1]=groupPolyCoeff(1)

fricas

Compiling function f with type (PositiveInteger, PositiveInteger)
       -> Expression(Integer)

fricas

Compiling function groupPolyCoeff with type PositiveInteger -> 
      Expression(Integer)

$\label{eq5}{g_{1}}={\frac{{{e}^{r_{2}}}-{{e}^{r_{1}}}}{{r_{2}}-{r_{1}}}}$

(5)

Type: Equation(Expression(Integer))

Example 2.1

fricas

eval(eq2_1,[r[2]=-r[1]])

$\label{eq6}{m_{X}}={{{x}^{2}}-{{r_{1}}^{2}}}$

(6)

Type: Equation(Polynomial(Integer))

fricas

eq2_4:= eval(eval(eq2_2,[eq2_3a,eq2_3b]),r[2]=-r[1])

$\label{eq7}{{e}^{X}}={\frac{{{\left(X +{r_{1}}\right)}\ {{e}^{r_{1}}}}+{{\left(- X +{r_{1}}\right)}\ {{e}^{-{r_{1}}}}}}{2 \ {r_{1}}}}$

(7)

Type: Equation(Expression(Integer))

fricas

htrigs rhs %

$\label{eq8}\frac{{X \ {\sinh \left({r_{1}}\right)}}+{{r_{1}}\ {\cosh \left({r_{1}}\right)}}}{r_{1}}$

(8)

Type: Expression(Integer)

Polynomial of degree 3
fricas
```
n:=3
```
$\label{eq9}3$ (9)

Type: PositiveInteger?

fricas
```
eq3_1:= m[X]=(x-r[1])*(x-r[2])*(x-r[3])
```
$\label{eq10}\begin{array}{@{}l} \displaystyle {m_{X}}={ \begin{array}{@{}l} \displaystyle {{x}^{3}}+{{\left(-{r_{3}}-{r_{2}}-{r_{1}}\right)}\ {{x}^{2}}}+{{\left({{\left({r_{2}}+{r_{1}}\right)}\ {r_{3}}}+{{r_{1}}\ {r_{2}}}\right)}\ x}- \ \ \displaystyle {{r_{1}}\ {r_{2}}\ {r_{3}}}$ (10)

Type: Equation(Polynomial(Integer))

fricas
```
eq3_2:= exp(X)=g[0]+g[1]*X+g[2]*X^2
```
$\label{eq11}{{e}^{X}}={{{g_{2}}\ {{X}^{2}}}+{{g_{1}}\ X}+{g_{0}}}$ (11)

Type: Equation(Expression(Integer))

fricas
```
eq3_3a:= g[0]=groupPolyCoeff(0)
```
$\label{eq12}\begin{array}{@{}l} \displaystyle {g_{0}}= \ \ \displaystyle {\frac{{{\left({{r_{1}}\ {{r_{2}}^{2}}}-{{{r_{1}}^{2}}\ {r_{2}}}\right)}\ {{e}^{r_{3}}}}+{{\left(-{{r_{1}}\ {{r_{3}}^{2}}}+{{{r_{1}}^{2}}\ {r_{3}}}\right)}\ {{e}^{r_{2}}}}+{{\left({{r_{2}}\ {{r_{3}}^{2}}}-{{{r_{2}}^{2}}\ {r_{3}}}\right)}\ {{e}^{r_{1}}}}}{{{\left({r_{2}}-{r_{1}}\right)}\ {{r_{3}}^{2}}}+{{\left(-{{r_{2}}^{2}}+{{r_{1}}^{2}}\right)}\ {r_{3}}}+{{r_{1}}\ {{r_{2}}^{2}}}-{{{r_{1}}^{2}}\ {r_{2}}}}}$ (12)

Type: Equation(Expression(Integer))

fricas
```
eq3_3b:= g[1]=groupPolyCoeff(1)
```
$\label{eq13}\begin{array}{@{}l} \displaystyle {g_{1}}= \ \ \displaystyle {\frac{{{\left(-{{r_{2}}^{2}}+{{r_{1}}^{2}}\right)}\ {{e}^{r_{3}}}}+{{\left({{r_{3}}^{2}}-{{r_{1}}^{2}}\right)}\ {{e}^{r_{2}}}}+{{\left(-{{r_{3}}^{2}}+{{r_{2}}^{2}}\right)}\ {{e}^{r_{1}}}}}{{{\left({r_{2}}-{r_{1}}\right)}\ {{r_{3}}^{2}}}+{{\left(-{{r_{2}}^{2}}+{{r_{1}}^{2}}\right)}\ {r_{3}}}+{{r_{1}}\ {{r_{2}}^{2}}}-{{{r_{1}}^{2}}\ {r_{2}}}}}$ (13)

Type: Equation(Expression(Integer))

fricas
```
eq3_3c:= g[2]=groupPolyCoeff(2)
```
$\label{eq14}\begin{array}{@{}l} \displaystyle {g_{2}}= \ \ \displaystyle {\frac{{{\left({r_{2}}-{r_{1}}\right)}\ {{e}^{r_{3}}}}+{{\left(-{r_{3}}+{r_{1}}\right)}\ {{e}^{r_{2}}}}+{{\left({r_{3}}-{r_{2}}\right)}\ {{e}^{r_{1}}}}}{{{\left({r_{2}}-{r_{1}}\right)}\ {{r_{3}}^{2}}}+{{\left(-{{r_{2}}^{2}}+{{r_{1}}^{2}}\right)}\ {r_{3}}}+{{r_{1}}\ {{r_{2}}^{2}}}-{{{r_{1}}^{2}}\ {r_{2}}}}}$ (14)

Type: Equation(Expression(Integer))
Example 3.1
fricas
```
eval(eq3_1,[r[2]=-r[1],r[3]=0])
```
$\label{eq15}{m_{X}}={{{x}^{3}}-{{{r_{1}}^{2}}\ x}}$ (15)

Type: Equation(Polynomial(Integer))

fricas
```
eq3_4:= eval(eval(eq3_2,[eq3_3a,eq3_3b,eq3_3c]),[r[2]=-r[1],r[3]=0])
```
$\label{eq16}\begin{array}{@{}l} \displaystyle {{e}^{X}}= \ \ \displaystyle {\frac{{{\left({{X}^{2}}+{{r_{1}}\ X}\right)}\ {{e}^{r_{1}}}}+{{\left({{X}^{2}}-{{r_{1}}\ X}\right)}\ {{e}^{-{r_{1}}}}}-{2 \ {{X}^{2}}}+{2 \ {{r_{1}}^{2}}}}{2 \ {{r_{1}}^{2}}}}$ (16)

Type: Equation(Expression(Integer))

fricas
```
htrigs rhs eq3_4
```
$\label{eq17}\frac{{{r_{1}}\ X \ {\sinh \left({r_{1}}\right)}}+{{{X}^{2}}\ {\cosh \left({r_{1}}\right)}}-{{X}^{2}}+{{r_{1}}^{2}}}{{r_{1}}^{2}}$ (17)

Type: Expression(Integer)

Comment 3.2 (Rescaled enomorphism)
fricas
```
eq3_6:= X' = sinh(r[1])/r[1]*X
```
$\label{eq18}X' ={\frac{X \ {\sinh \left({r_{1}}\right)}}{r_{1}}}$ (18)

Type: Equation(Expression(Integer))

fricas
```
eq3_7:= γ = cosh(r[1])
```
$\label{eq19}�� ={\cosh \left({r_{1}}\right)}$ (19)

Type: Equation(Expression(Integer))

fricas
```
eq3_8:= exp(X) = 1+X'+X'^2/(1+γ)
```
$\label{eq20}{{e}^{X}}={\frac{{{\left(X' + 1 \right)}\ ��}+{{X'}^{2}}+ X' + 1}{�� + 1}}$ (20)

Type: Equation(Expression(Integer))

fricas
```
test(normalize(rhs eval(eq3_8,[eq3_6,eq3_7]) - rhs eq3_4)=0)
```
$\label{eq21} \mbox{\rm true}$ (21)

Type: Boolean

Exercise 3.3
fricas
```
eval(eq3_1,[r[3]=r[2]])
```
$\label{eq22}{m_{X}}={{{x}^{3}}+{{\left(-{2 \ {r_{2}}}-{r_{1}}\right)}\ {{x}^{2}}}+{{\left({{r_{2}}^{2}}+{2 \ {r_{1}}\ {r_{2}}}\right)}\ x}-{{r_{1}}\ {{r_{2}}^{2}}}}$ (22)

Type: Equation(Polynomial(Integer))

fricas
```
eq3_9a:=lhs eq3_3a = limit(rhs eq3_3a,r[3]=r[2])
```
$\label{eq23}\begin{array}{@{}l} \displaystyle {g_{0}}= \ \ \displaystyle {\frac{{{\left({{r_{1}}\ {{r_{2}}^{2}}}+{{\left(-{{r_{1}}^{2}}-{2 \ {r_{1}}}\right)}\ {r_{2}}}+{{r_{1}}^{2}}\right)}\ {{e}^{r_{2}}}}+{{{r_{2}}^{2}}\ {{e}^{r_{1}}}}}{{{r_{2}}^{2}}-{2 \ {r_{1}}\ {r_{2}}}+{{r_{1}}^{2}}}}$ (23)

Type: Equation(OrderedCompletion?(Expression(Integer)))

fricas
```
(numer rhs eq3_9a)/factor(denom rhs eq3_9a)
```
$\label{eq24}\begin{array}{@{}l} \displaystyle {{\left({ \begin{array}{@{}l} \displaystyle {{\frac{1}{{\left({r_{2}}-{r_{1}}\right)}^{2}}}\ {r_{1}}\ {{r_{2}}^{2}}}+ \ \ \displaystyle {{\left({ \begin{array}{@{}l} \displaystyle -{{\frac{1}{{\left({r_{2}}-{r_{1}}\right)}^{2}}}\ {{r_{1}}^{2}}}- \ \ \displaystyle {{\frac{2}{{\left({r_{2}}-{r_{1}}\right)}^{2}}}\ {r_{1}}}$ (24)

Type: SparseMultivariatePolynomial?(Fraction(Factored(SparseMultivariatePolynomial?(Integer,Kernel(Expression(Integer))))),Kernel(Expression(Integer)))

fricas
```
eq3_9b:=lhs eq3_3b = limit(rhs eq3_3b,r[3]=r[2])
```
$\label{eq25}\begin{array}{@{}l} \displaystyle {g_{1}}= \ \ \displaystyle {\frac{{{\left(-{{r_{2}}^{2}}+{2 \ {r_{2}}}+{{r_{1}}^{2}}\right)}\ {{e}^{r_{2}}}}-{2 \ {r_{2}}\ {{e}^{r_{1}}}}}{{{r_{2}}^{2}}-{2 \ {r_{1}}\ {r_{2}}}+{{r_{1}}^{2}}}}$ (25)

Type: Equation(OrderedCompletion?(Expression(Integer)))

fricas
```
(numer rhs eq3_9b)/factor(denom rhs eq3_9b)
```
$\label{eq26}\begin{array}{@{}l} \displaystyle {{\left({ \begin{array}{@{}l} \displaystyle -{{\frac{1}{{\left({r_{2}}-{r_{1}}\right)}^{2}}}\ {{r_{2}}^{2}}}+{{\frac{2}{{\left({r_{2}}-{r_{1}}\right)}^{2}}}\ {r_{2}}}+ \ \ \displaystyle {{\frac{1}{{\left({r_{2}}-{r_{1}}\right)}^{2}}}\ {{r_{1}}^{2}}}$ (26)

Type: SparseMultivariatePolynomial?(Fraction(Factored(SparseMultivariatePolynomial?(Integer,Kernel(Expression(Integer))))),Kernel(Expression(Integer)))

fricas
```
eq3_9c:=lhs eq3_3c = limit(rhs eq3_3c,r[3]=r[2])
```
$\label{eq27}{g_{2}}={\frac{{{\left({r_{2}}-{r_{1}}- 1 \right)}\ {{e}^{r_{2}}}}+{{e}^{r_{1}}}}{{{r_{2}}^{2}}-{2 \ {r_{1}}\ {r_{2}}}+{{r_{1}}^{2}}}}$ (27)

Type: Equation(OrderedCompletion?(Expression(Integer)))

fricas
```
(numer rhs eq3_9c)/factor(denom rhs eq3_9c)
```
$\label{eq28}\begin{array}{@{}l} \displaystyle {{\left({ \begin{array}{@{}l} \displaystyle {{\frac{1}{{\left({r_{2}}-{r_{1}}\right)}^{2}}}\ {r_{2}}}-{{\frac{1}{{\left({r_{2}}-{r_{1}}\right)}^{2}}}\ {r_{1}}}- \ \ \displaystyle {\frac{1}{{\left({r_{2}}-{r_{1}}\right)}^{2}}}$ (28)

Type: SparseMultivariatePolynomial?(Fraction(Factored(SparseMultivariatePolynomial?(Integer,Kernel(Expression(Integer))))),Kernel(Expression(Integer)))
Polynomial of degree 4
fricas
```
n:=4
```
$\label{eq29}4$ (29)

Type: PositiveInteger?

fricas
```
eq4_1:= m[X]=(x-r[1])*(x-r[2])*(x-r[3])*(x-r[4])
```
$\label{eq30}\begin{array}{@{}l} \displaystyle {m_{X}}={ \begin{array}{@{}l} \displaystyle {{x}^{4}}+{{\left(-{r_{4}}-{r_{3}}-{r_{2}}-{r_{1}}\right)}\ {{x}^{3}}}+ \ \ \displaystyle {{\left({{\left({r_{3}}+{r_{2}}+{r_{1}}\right)}\ {r_{4}}}+{{\left({r_{2}}+{r_{1}}\right)}\ {r_{3}}}+{{r_{1}}\ {r_{2}}}\right)}\ {{x}^{2}}}+ \ \ \displaystyle {{\left({{\left({{\left(-{r_{2}}-{r_{1}}\right)}\ {r_{3}}}-{{r_{1}}\ {r_{2}}}\right)}\ {r_{4}}}-{{r_{1}}\ {r_{2}}\ {r_{3}}}\right)}\ x}+ \ \ \displaystyle {{r_{1}}\ {r_{2}}\ {r_{3}}\ {r_{4}}}$ (30)

Type: Equation(Polynomial(Integer))

fricas
```
eq4_2:= g[0]=groupPolyCoeff(0)
```
$\label{eq31}\begin{array}{@{}l} \displaystyle {g_{0}}= \ \ \displaystyle {\frac{{{\left({{\left(-{{r_{1}}\ {{r_{2}}^{2}}}+{{{r_{1}}^{2}}\ {r_{2}}}\right)}\ {{r_{3}}^{3}}}+{{\left({{r_{1}}\ {{r_{2}}^{3}}}-{{{r_{1}}^{3}}\ {r_{2}}}\right)}\ {{r_{3}}^{2}}}+{{\left(-{{{r_{1}}^{2}}\ {{r_{2}}^{3}}}+{{{r_{1}}^{3}}\ {{r_{2}}^{2}}}\right)}\ {r_{3}}}\right)}\ {{e}^{r_{4}}}}+{{\left({{\left({{r_{1}}\ {{r_{2}}^{2}}}-{{{r_{1}}^{2}}\ {r_{2}}}\right)}\ {{r_{4}}^{3}}}+{{\left(-{{r_{1}}\ {{r_{2}}^{3}}}+{{{r_{1}}^{3}}\ {r_{2}}}\right)}\ {{r_{4}}^{2}}}+{{\left({{{r_{1}}^{2}}\ {{r_{2}}^{3}}}-{{{r_{1}}^{3}}\ {{r_{2}}^{2}}}\right)}\ {r_{4}}}\right)}\ {{e}^{r_{3}}}}+{{\left({{\left(-{{r_{1}}\ {{r_{3}}^{2}}}+{{{r_{1}}^{2}}\ {r_{3}}}\right)}\ {{r_{4}}^{3}}}+{{\left({{r_{1}}\ {{r_{3}}^{3}}}-{{{r_{1}}^{3}}\ {r_{3}}}\right)}\ {{r_{4}}^{2}}}+{{\left(-{{{r_{1}}^{2}}\ {{r_{3}}^{3}}}+{{{r_{1}}^{3}}\ {{r_{3}}^{2}}}\right)}\ {r_{4}}}\right)}\ {{e}^{r_{2}}}}+{{\left({{\left({{r_{2}}\ {{r_{3}}^{2}}}-{{{r_{2}}^{2}}\ {r_{3}}}\right)}\ {{r_{4}}^{3}}}+{{\left(-{{r_{2}}\ {{r_{3}}^{3}}}+{{{r_{2}}^{3}}\ {r_{3}}}\right)}\ {{r_{4}}^{2}}}+{{\left({{{r_{2}}^{2}}\ {{r_{3}}^{3}}}-{{{r_{2}}^{3}}\ {{r_{3}}^{2}}}\right)}\ {r_{4}}}\right)}\ {{e}^{r_{1}}}}}{{{\left({{\left({r_{2}}-{r_{1}}\right)}\ {{r_{3}}^{2}}}+{{\left(-{{r_{2}}^{2}}+{{r_{1}}^{2}}\right)}\ {r_{3}}}+{{r_{1}}\ {{r_{2}}^{2}}}-{{{r_{1}}^{2}}\ {r_{2}}}\right)}\ {{r_{4}}^{3}}}+{{\left({{\left(-{r_{2}}+{r_{1}}\right)}\ {{r_{3}}^{3}}}+{{\left({{r_{2}}^{3}}-{{r_{1}}^{3}}\right)}\ {r_{3}}}-{{r_{1}}\ {{r_{2}}^{3}}}+{{{r_{1}}^{3}}\ {r_{2}}}\right)}\ {{r_{4}}^{2}}}+{{\left({{\left({{r_{2}}^{2}}-{{r_{1}}^{2}}\right)}\ {{r_{3}}^{3}}}+{{\left(-{{r_{2}}^{3}}+{{r_{1}}^{3}}\right)}\ {{r_{3}}^{2}}}+{{{r_{1}}^{2}}\ {{r_{2}}^{3}}}-{{{r_{1}}^{3}}\ {{r_{2}}^{2}}}\right)}\ {r_{4}}}+{{\left(-{{r_{1}}\ {{r_{2}}^{2}}}+{{{r_{1}}^{2}}\ {r_{2}}}\right)}\ {{r_{3}}^{3}}}+{{\left({{r_{1}}\ {{r_{2}}^{3}}}-{{{r_{1}}^{3}}\ {r_{2}}}\right)}\ {{r_{3}}^{2}}}+{{\left(-{{{r_{1}}^{2}}\ {{r_{2}}^{3}}}+{{{r_{1}}^{3}}\ {{r_{2}}^{2}}}\right)}\ {r_{3}}}}}$ (31)

Type: Equation(Expression(Integer))

fricas
```
eq4_3:= g[1]=groupPolyCoeff(1)
```
$\label{eq32}\begin{array}{@{}l} \displaystyle {g_{1}}= \ \ \displaystyle {\frac{{{\left({{\left({{r_{2}}^{2}}-{{r_{1}}^{2}}\right)}\ {{r_{3}}^{3}}}+{{\left(-{{r_{2}}^{3}}+{{r_{1}}^{3}}\right)}\ {{r_{3}}^{2}}}+{{{r_{1}}^{2}}\ {{r_{2}}^{3}}}-{{{r_{1}}^{3}}\ {{r_{2}}^{2}}}\right)}\ {{e}^{r_{4}}}}+{{\left({{\left(-{{r_{2}}^{2}}+{{r_{1}}^{2}}\right)}\ {{r_{4}}^{3}}}+{{\left({{r_{2}}^{3}}-{{r_{1}}^{3}}\right)}\ {{r_{4}}^{2}}}-{{{r_{1}}^{2}}\ {{r_{2}}^{3}}}+{{{r_{1}}^{3}}\ {{r_{2}}^{2}}}\right)}\ {{e}^{r_{3}}}}+{{\left({{\left({{r_{3}}^{2}}-{{r_{1}}^{2}}\right)}\ {{r_{4}}^{3}}}+{{\left(-{{r_{3}}^{3}}+{{r_{1}}^{3}}\right)}\ {{r_{4}}^{2}}}+{{{r_{1}}^{2}}\ {{r_{3}}^{3}}}-{{{r_{1}}^{3}}\ {{r_{3}}^{2}}}\right)}\ {{e}^{r_{2}}}}+{{\left({{\left(-{{r_{3}}^{2}}+{{r_{2}}^{2}}\right)}\ {{r_{4}}^{3}}}+{{\left({{r_{3}}^{3}}-{{r_{2}}^{3}}\right)}\ {{r_{4}}^{2}}}-{{{r_{2}}^{2}}\ {{r_{3}}^{3}}}+{{{r_{2}}^{3}}\ {{r_{3}}^{2}}}\right)}\ {{e}^{r_{1}}}}}{{{\left({{\left({r_{2}}-{r_{1}}\right)}\ {{r_{3}}^{2}}}+{{\left(-{{r_{2}}^{2}}+{{r_{1}}^{2}}\right)}\ {r_{3}}}+{{r_{1}}\ {{r_{2}}^{2}}}-{{{r_{1}}^{2}}\ {r_{2}}}\right)}\ {{r_{4}}^{3}}}+{{\left({{\left(-{r_{2}}+{r_{1}}\right)}\ {{r_{3}}^{3}}}+{{\left({{r_{2}}^{3}}-{{r_{1}}^{3}}\right)}\ {r_{3}}}-{{r_{1}}\ {{r_{2}}^{3}}}+{{{r_{1}}^{3}}\ {r_{2}}}\right)}\ {{r_{4}}^{2}}}+{{\left({{\left({{r_{2}}^{2}}-{{r_{1}}^{2}}\right)}\ {{r_{3}}^{3}}}+{{\left(-{{r_{2}}^{3}}+{{r_{1}}^{3}}\right)}\ {{r_{3}}^{2}}}+{{{r_{1}}^{2}}\ {{r_{2}}^{3}}}-{{{r_{1}}^{3}}\ {{r_{2}}^{2}}}\right)}\ {r_{4}}}+{{\left(-{{r_{1}}\ {{r_{2}}^{2}}}+{{{r_{1}}^{2}}\ {r_{2}}}\right)}\ {{r_{3}}^{3}}}+{{\left({{r_{1}}\ {{r_{2}}^{3}}}-{{{r_{1}}^{3}}\ {r_{2}}}\right)}\ {{r_{3}}^{2}}}+{{\left(-{{{r_{1}}^{2}}\ {{r_{2}}^{3}}}+{{{r_{1}}^{3}}\ {{r_{2}}^{2}}}\right)}\ {r_{3}}}}}$ (32)

Type: Equation(Expression(Integer))

fricas
```
eq4_4:= g[2]=groupPolyCoeff(2)
```
$\label{eq33}\begin{array}{@{}l} \displaystyle {g_{2}}= \ \ \displaystyle {\frac{{{\left({{\left(-{r_{2}}+{r_{1}}\right)}\ {{r_{3}}^{3}}}+{{\left({{r_{2}}^{3}}-{{r_{1}}^{3}}\right)}\ {r_{3}}}-{{r_{1}}\ {{r_{2}}^{3}}}+{{{r_{1}}^{3}}\ {r_{2}}}\right)}\ {{e}^{r_{4}}}}+{{\left({{\left({r_{2}}-{r_{1}}\right)}\ {{r_{4}}^{3}}}+{{\left(-{{r_{2}}^{3}}+{{r_{1}}^{3}}\right)}\ {r_{4}}}+{{r_{1}}\ {{r_{2}}^{3}}}-{{{r_{1}}^{3}}\ {r_{2}}}\right)}\ {{e}^{r_{3}}}}+{{\left({{\left(-{r_{3}}+{r_{1}}\right)}\ {{r_{4}}^{3}}}+{{\left({{r_{3}}^{3}}-{{r_{1}}^{3}}\right)}\ {r_{4}}}-{{r_{1}}\ {{r_{3}}^{3}}}+{{{r_{1}}^{3}}\ {r_{3}}}\right)}\ {{e}^{r_{2}}}}+{{\left({{\left({r_{3}}-{r_{2}}\right)}\ {{r_{4}}^{3}}}+{{\left(-{{r_{3}}^{3}}+{{r_{2}}^{3}}\right)}\ {r_{4}}}+{{r_{2}}\ {{r_{3}}^{3}}}-{{{r_{2}}^{3}}\ {r_{3}}}\right)}\ {{e}^{r_{1}}}}}{{{\left({{\left({r_{2}}-{r_{1}}\right)}\ {{r_{3}}^{2}}}+{{\left(-{{r_{2}}^{2}}+{{r_{1}}^{2}}\right)}\ {r_{3}}}+{{r_{1}}\ {{r_{2}}^{2}}}-{{{r_{1}}^{2}}\ {r_{2}}}\right)}\ {{r_{4}}^{3}}}+{{\left({{\left(-{r_{2}}+{r_{1}}\right)}\ {{r_{3}}^{3}}}+{{\left({{r_{2}}^{3}}-{{r_{1}}^{3}}\right)}\ {r_{3}}}-{{r_{1}}\ {{r_{2}}^{3}}}+{{{r_{1}}^{3}}\ {r_{2}}}\right)}\ {{r_{4}}^{2}}}+{{\left({{\left({{r_{2}}^{2}}-{{r_{1}}^{2}}\right)}\ {{r_{3}}^{3}}}+{{\left(-{{r_{2}}^{3}}+{{r_{1}}^{3}}\right)}\ {{r_{3}}^{2}}}+{{{r_{1}}^{2}}\ {{r_{2}}^{3}}}-{{{r_{1}}^{3}}\ {{r_{2}}^{2}}}\right)}\ {r_{4}}}+{{\left(-{{r_{1}}\ {{r_{2}}^{2}}}+{{{r_{1}}^{2}}\ {r_{2}}}\right)}\ {{r_{3}}^{3}}}+{{\left({{r_{1}}\ {{r_{2}}^{3}}}-{{{r_{1}}^{3}}\ {r_{2}}}\right)}\ {{r_{3}}^{2}}}+{{\left(-{{{r_{1}}^{2}}\ {{r_{2}}^{3}}}+{{{r_{1}}^{3}}\ {{r_{2}}^{2}}}\right)}\ {r_{3}}}}}$ (33)

Type: Equation(Expression(Integer))

fricas
```
eq4_5:= g[3]=groupPolyCoeff(3)
```
$\label{eq34}\begin{array}{@{}l} \displaystyle {g_{3}}= \ \ \displaystyle {\frac{{{\left({{\left({r_{2}}-{r_{1}}\right)}\ {{r_{3}}^{2}}}+{{\left(-{{r_{2}}^{2}}+{{r_{1}}^{2}}\right)}\ {r_{3}}}+{{r_{1}}\ {{r_{2}}^{2}}}-{{{r_{1}}^{2}}\ {r_{2}}}\right)}\ {{e}^{r_{4}}}}+{{\left({{\left(-{r_{2}}+{r_{1}}\right)}\ {{r_{4}}^{2}}}+{{\left({{r_{2}}^{2}}-{{r_{1}}^{2}}\right)}\ {r_{4}}}-{{r_{1}}\ {{r_{2}}^{2}}}+{{{r_{1}}^{2}}\ {r_{2}}}\right)}\ {{e}^{r_{3}}}}+{{\left({{\left({r_{3}}-{r_{1}}\right)}\ {{r_{4}}^{2}}}+{{\left(-{{r_{3}}^{2}}+{{r_{1}}^{2}}\right)}\ {r_{4}}}+{{r_{1}}\ {{r_{3}}^{2}}}-{{{r_{1}}^{2}}\ {r_{3}}}\right)}\ {{e}^{r_{2}}}}+{{\left({{\left(-{r_{3}}+{r_{2}}\right)}\ {{r_{4}}^{2}}}+{{\left({{r_{3}}^{2}}-{{r_{2}}^{2}}\right)}\ {r_{4}}}-{{r_{2}}\ {{r_{3}}^{2}}}+{{{r_{2}}^{2}}\ {r_{3}}}\right)}\ {{e}^{r_{1}}}}}{{{\left({{\left({r_{2}}-{r_{1}}\right)}\ {{r_{3}}^{2}}}+{{\left(-{{r_{2}}^{2}}+{{r_{1}}^{2}}\right)}\ {r_{3}}}+{{r_{1}}\ {{r_{2}}^{2}}}-{{{r_{1}}^{2}}\ {r_{2}}}\right)}\ {{r_{4}}^{3}}}+{{\left({{\left(-{r_{2}}+{r_{1}}\right)}\ {{r_{3}}^{3}}}+{{\left({{r_{2}}^{3}}-{{r_{1}}^{3}}\right)}\ {r_{3}}}-{{r_{1}}\ {{r_{2}}^{3}}}+{{{r_{1}}^{3}}\ {r_{2}}}\right)}\ {{r_{4}}^{2}}}+{{\left({{\left({{r_{2}}^{2}}-{{r_{1}}^{2}}\right)}\ {{r_{3}}^{3}}}+{{\left(-{{r_{2}}^{3}}+{{r_{1}}^{3}}\right)}\ {{r_{3}}^{2}}}+{{{r_{1}}^{2}}\ {{r_{2}}^{3}}}-{{{r_{1}}^{3}}\ {{r_{2}}^{2}}}\right)}\ {r_{4}}}+{{\left(-{{r_{1}}\ {{r_{2}}^{2}}}+{{{r_{1}}^{2}}\ {r_{2}}}\right)}\ {{r_{3}}^{3}}}+{{\left({{r_{1}}\ {{r_{2}}^{3}}}-{{{r_{1}}^{3}}\ {r_{2}}}\right)}\ {{r_{3}}^{2}}}+{{\left(-{{{r_{1}}^{2}}\ {{r_{2}}^{3}}}+{{{r_{1}}^{3}}\ {{r_{2}}^{2}}}\right)}\ {r_{3}}}}}$ (34)

Type: Equation(Expression(Integer))
fricas
```
eq5_1:=eval(eq4_1,[r[3]=-r[1],r[4]=-r[2]])
```
$\label{eq35}{m_{X}}={{{x}^{4}}+{{\left(-{{r_{2}}^{2}}-{{r_{1}}^{2}}\right)}\ {{x}^{2}}}+{{{r_{1}}^{2}}\ {{r_{2}}^{2}}}}$ (35)

Type: Equation(Polynomial(Integer))
Corollary 5.1
fricas
```
eq5_2:= exp(X)=g[0]+g[1]*X+g[2]*X^2+g[3]*X^3
```
$\label{eq36}{{e}^{X}}={{{g_{3}}\ {{X}^{3}}}+{{g_{2}}\ {{X}^{2}}}+{{g_{1}}\ X}+{g_{0}}}$ (36)

Type: Equation(Expression(Integer))

fricas
```
eq5_3a:= eval(eq4_2,[r[3]=-r[1],r[4]=-r[2]])
```
$\label{eq37}\begin{array}{@{}l} \displaystyle {g_{0}}= \ \ \displaystyle {\frac{-{{{r_{1}}^{2}}\ {{e}^{r_{2}}}}+{{{r_{2}}^{2}}\ {{e}^{r_{1}}}}+{{{r_{2}}^{2}}\ {{e}^{-{r_{1}}}}}-{{{r_{1}}^{2}}\ {{e}^{-{r_{2}}}}}}{{2 \ {{r_{2}}^{2}}}-{2 \ {{r_{1}}^{2}}}}}$ (37)

Type: Equation(Expression(Integer))

fricas
```
htrigs rhs %
```
$\label{eq38}\frac{-{{{r_{1}}^{2}}\ {\cosh \left({r_{2}}\right)}}+{{{r_{2}}^{2}}\ {\cosh \left({r_{1}}\right)}}}{{{r_{2}}^{2}}-{{r_{1}}^{2}}}$ (38)

Type: Expression(Integer)

fricas
```
eq5_3b:= eval(eq4_3,[r[3]=-r[1],r[4]=-r[2]])
```
$\label{eq39}\begin{array}{@{}l} \displaystyle {g_{1}}= \ \ \displaystyle {\frac{-{{{r_{1}}^{3}}\ {{e}^{r_{2}}}}+{{{r_{2}}^{3}}\ {{e}^{r_{1}}}}-{{{r_{2}}^{3}}\ {{e}^{-{r_{1}}}}}+{{{r_{1}}^{3}}\ {{e}^{-{r_{2}}}}}}{{2 \ {r_{1}}\ {{r_{2}}^{3}}}-{2 \ {{r_{1}}^{3}}\ {r_{2}}}}}$ (39)

Type: Equation(Expression(Integer))

fricas
```
htrigs rhs %
```
$\label{eq40}\frac{-{{{r_{1}}^{3}}\ {\sinh \left({r_{2}}\right)}}+{{{r_{2}}^{3}}\ {\sinh \left({r_{1}}\right)}}}{{{r_{1}}\ {{r_{2}}^{3}}}-{{{r_{1}}^{3}}\ {r_{2}}}}$ (40)

Type: Expression(Integer)

fricas
```
eq5_3c:= eval(eq4_4,[r[3]=-r[1],r[4]=-r[2]])
```
$\label{eq41}{g_{2}}={\frac{{{e}^{r_{2}}}-{{e}^{r_{1}}}-{{e}^{-{r_{1}}}}+{{e}^{-{r_{2}}}}}{{2 \ {{r_{2}}^{2}}}-{2 \ {{r_{1}}^{2}}}}}$ (41)

Type: Equation(Expression(Integer))

fricas
```
htrigs rhs %
```
$\label{eq42}\frac{{\cosh \left({r_{2}}\right)}-{\cosh \left({r_{1}}\right)}}{{{r_{2}}^{2}}-{{r_{1}}^{2}}}$ (42)

Type: Expression(Integer)

fricas
```
eq5_3d:= eval(eq4_5,[r[3]=-r[1],r[4]=-r[2]])
```
$\label{eq43}\begin{array}{@{}l} \displaystyle {g_{3}}= \ \ \displaystyle {\frac{{{r_{1}}\ {{e}^{r_{2}}}}-{{r_{2}}\ {{e}^{r_{1}}}}+{{r_{2}}\ {{e}^{-{r_{1}}}}}-{{r_{1}}\ {{e}^{-{r_{2}}}}}}{{2 \ {r_{1}}\ {{r_{2}}^{3}}}-{2 \ {{r_{1}}^{3}}\ {r_{2}}}}}$ (43)

Type: Equation(Expression(Integer))

fricas
```
htrigs rhs %
```
$\label{eq44}\frac{{{r_{1}}\ {\sinh \left({r_{2}}\right)}}-{{r_{2}}\ {\sinh \left({r_{1}}\right)}}}{{{r_{1}}\ {{r_{2}}^{3}}}-{{{r_{1}}^{3}}\ {r_{2}}}}$ (44)

Type: Expression(Integer)

fricas
```
eq5_4:= eval(eval(eq5_2,[eq4_2,eq4_3,eq4_4,eq4_5]),[r[3]=-r[1],r[4]=-r[2]])
```
$\label{eq45}\begin{array}{@{}l} \displaystyle {{e}^{X}}= \ \ \displaystyle {\frac{{{\left({{r_{1}}\ {{X}^{3}}}+{{r_{1}}\ {r_{2}}\ {{X}^{2}}}-{{{r_{1}}^{3}}\ X}-{{{r_{1}}^{3}}\ {r_{2}}}\right)}\ {{e}^{r_{2}}}}+{{\left(-{{r_{2}}\ {{X}^{3}}}-{{r_{1}}\ {r_{2}}\ {{X}^{2}}}+{{{r_{2}}^{3}}\ X}+{{r_{1}}\ {{r_{2}}^{3}}}\right)}\ {{e}^{r_{1}}}}+{{\left({{r_{2}}\ {{X}^{3}}}-{{r_{1}}\ {r_{2}}\ {{X}^{2}}}-{{{r_{2}}^{3}}\ X}+{{r_{1}}\ {{r_{2}}^{3}}}\right)}\ {{e}^{-{r_{1}}}}}+{{\left(-{{r_{1}}\ {{X}^{3}}}+{{r_{1}}\ {r_{2}}\ {{X}^{2}}}+{{{r_{1}}^{3}}\ X}-{{{r_{1}}^{3}}\ {r_{2}}}\right)}\ {{e}^{-{r_{2}}}}}}{{2 \ {r_{1}}\ {{r_{2}}^{3}}}-{2 \ {{r_{1}}^{3}}\ {r_{2}}}}}$ (45)

Type: Equation(Expression(Integer))

fricas
```
htrigs rhs %
```
$\label{eq46}\frac{{{\left({{r_{1}}\ {{X}^{3}}}-{{{r_{1}}^{3}}\ X}\right)}\ {\sinh \left({r_{2}}\right)}}+{{\left(-{{r_{2}}\ {{X}^{3}}}+{{{r_{2}}^{3}}\ X}\right)}\ {\sinh \left({r_{1}}\right)}}+{{\left({{r_{1}}\ {r_{2}}\ {{X}^{2}}}-{{{r_{1}}^{3}}\ {r_{2}}}\right)}\ {\cosh \left({r_{2}}\right)}}+{{\left(-{{r_{1}}\ {r_{2}}\ {{X}^{2}}}+{{r_{1}}\ {{r_{2}}^{3}}}\right)}\ {\cosh \left({r_{1}}\right)}}}{{{r_{1}}\ {{r_{2}}^{3}}}-{{{r_{1}}^{3}}\ {r_{2}}}}$ (46)

Type: Expression(Integer)

Definition 5.2
fricas
```
eq5_5a:= Y[1] = 1/(2*r[1]^2-r[1]^2-r[2]^2)*(X^3-(r[1]^2+r[2]^2-r[1]^2)*X)
```
$\label{eq47}{Y_{1}}={\frac{-{{X}^{3}}+{{{r_{2}}^{2}}\ X}}{{{r_{2}}^{2}}-{{r_{1}}^{2}}}}$ (47)

Type: Equation(Fraction(Polynomial(Integer)))

fricas
```
eq5_5b:= Y[2] = 1/(2*r[2]^2-r[1]^2-r[2]^2)*(X^3-(r[1]^2+r[2]^2-r[2]^2)*X)
```
$\label{eq48}{Y_{2}}={\frac{{{X}^{3}}-{{{r_{1}}^{2}}\ X}}{{{r_{2}}^{2}}-{{r_{1}}^{2}}}}$ (48)

Type: Equation(Fraction(Polynomial(Integer)))

Exercise 5.3
fricas
```
eq5_6a:= X = Y[1]+Y[2]
```
$\label{eq49}X ={{Y_{2}}+{Y_{1}}}$ (49)

Type: Equation(Polynomial(Integer))

fricas
```
test(eval(eq5_6a,[eq5_5a,eq5_5b]))
```
$\label{eq50} \mbox{\rm true}$ (50)

Type: Boolean

fricas
```
eq5_6b:= eval(Y[1]*Y[2]=0,[eq5_5a,eq5_5b])
```
$\label{eq51}\begin{array}{@{}l} \displaystyle {\frac{-{{X}^{6}}+{{\left({{r_{2}}^{2}}+{{r_{1}}^{2}}\right)}\ {{X}^{4}}}-{{{r_{1}}^{2}}\ {{r_{2}}^{2}}\ {{X}^{2}}}}{{{r_{2}}^{4}}-{2 \ {{r_{1}}^{2}}\ {{r_{2}}^{2}}}+{{r_{1}}^{4}}}}= \ \ \displaystyle 0$ (51)

Type: Equation(Expression(Integer))

fricas
```
eq5_6c:= X^4 = X^4-eval(rhs eq5_1,x=X)
```
$\label{eq52}{{X}^{4}}={{{\left({{r_{2}}^{2}}+{{r_{1}}^{2}}\right)}\ {{X}^{2}}}-{{{r_{1}}^{2}}\ {{r_{2}}^{2}}}}$ (52)

Type: Equation(Polynomial(Integer))

fricas
```
eq5_6d:= X^2*(lhs %)=X^2*(rhs %)
```
$\label{eq53}{{X}^{6}}={{{\left({{r_{2}}^{2}}+{{r_{1}}^{2}}\right)}\ {{X}^{4}}}-{{{r_{1}}^{2}}\ {{r_{2}}^{2}}\ {{X}^{2}}}}$ (53)

Type: Equation(Polynomial(Integer))

fricas
```
test(_rule(lhs eq5_6d,rhs eq5_6d)(lhs eq5_6b)=rhs eq5_6b)
```
$\label{eq54} \mbox{\rm true}$ (54)

Type: Boolean

Comment 5.4 (Rescaling)
fricas
```
eq5_7:= sinh(r[1])/r[1]*Y[1]+sinh(r[2])/r[2]*Y[2]
```
$\label{eq55}\frac{{{Y_{2}}\ {r_{1}}\ {\sinh \left({r_{2}}\right)}}+{{Y_{1}}\ {r_{2}}\ {\sinh \left({r_{1}}\right)}}}{{r_{1}}\ {r_{2}}}$ (55)

Type: Expression(Integer)

fricas
```
eq5_8a:= [ Y'[1]=sinh(r[1])/r[1]*Y[1], Y'[2]=sinh(r[2])/r[2]*Y[2] ]
```
$\label{eq56}\left[{{Y'_{1}}={\frac{{Y_{1}}\ {\sinh \left({r_{1}}\right)}}{r_{1}}}}, \:{{Y'_{2}}={\frac{{Y_{2}}\ {\sinh \left({r_{2}}\right)}}{r_{2}}}}\right]$ (56)

Type: List(Equation(Expression(Integer)))

fricas
```
eq5_8b:= [ γ[1] = cosh(r[1]), γ[2] = cosh(r[2]) ]
```
$\label{eq57}\left[{{��_{1}}={\cosh \left({r_{1}}\right)}}, \:{{��_{2}}={\cosh \left({r_{2}}\right)}}\right]$ (57)

Type: List(Equation(Expression(Integer)))

fricas
```
eq5_9a:= exp(X) = exp(Y[1])*exp(Y[2])
```
$\label{eq58}{{e}^{X}}={{{e}^{Y_{1}}}\ {{e}^{Y_{2}}}}$ (58)

Type: Equation(Expression(Integer))

fricas
```
eval(eq5_9a,[eq5_5a,eq5_5b])
```
$\label{eq59}{{e}^{X}}={{{e}^{\frac{-{{X}^{3}}+{{{r_{2}}^{2}}\ X}}{{{r_{2}}^{2}}-{{r_{1}}^{2}}}}}\ {{e}^{\frac{{{X}^{3}}-{{{r_{1}}^{2}}\ X}}{{{r_{2}}^{2}}-{{r_{1}}^{2}}}}}}$ (59)

Type: Equation(Expression(Integer))

fricas
```
test(lhs % = simplify expand rhs %)
```
$\label{eq60} \mbox{\rm true}$ (60)

Type: Boolean

fricas
```
eq5_9b:= exp(X) = 1 + Y'[1] +Y'[2] + Y'[1]^2/(1+γ[1]) + Y'[2]^2/(1+γ[2])
```
$\label{eq61}\begin{array}{@{}l} \displaystyle {{e}^{X}}= \ \ \displaystyle {\frac{{{\left({{\left({Y'_{2}}+{Y'_{1}}+ 1 \right)}\ {��_{1}}}+{Y'_{2}}+{{Y'_{1}}^{2}}+{Y'_{1}}+ 1 \right)}\ {��_{2}}}+{{\left({{Y'_{2}}^{2}}+{Y'_{2}}+{Y'_{1}}+ 1 \right)}\ {��_{1}}}+{{Y'_{2}}^{2}}+{Y'_{2}}+{{Y'_{1}}^{2}}+{Y'_{1}}+ 1}{{{\left({��_{1}}+ 1 \right)}\ {��_{2}}}+{��_{1}}+ 1}}$ (61)

Type: Equation(Expression(Integer))

fricas
```
normalize eval(eval(rhs eq5_9b,concat [eq5_8a,eq5_8b]),[eq5_5a,eq5_5b])
```
$\label{eq62}\frac{{{\left({{{r_{1}}^{2}}\ {{X}^{6}}}-{2 \ {{r_{1}}^{4}}\ {{X}^{4}}}+{{\left({{{r_{1}}^{2}}\ {{r_{2}}^{3}}}-{{{r_{1}}^{4}}\ {r_{2}}}\right)}\ {{X}^{3}}}+{{{r_{1}}^{6}}\ {{X}^{2}}}+{{\left(-{{{r_{1}}^{4}}\ {{r_{2}}^{3}}}+{{{r_{1}}^{6}}\ {r_{2}}}\right)}\ X}\right)}\ {{e}^{r_{1}}}\ {{{e}^{r_{2}}}^{2}}}+{{\left({{\left({{{r_{2}}^{2}}\ {{X}^{6}}}-{2 \ {{r_{2}}^{4}}\ {{X}^{4}}}+{{\left(-{{r_{1}}\ {{r_{2}}^{4}}}+{{{r_{1}}^{3}}\ {{r_{2}}^{2}}}\right)}\ {{X}^{3}}}+{{{r_{2}}^{6}}\ {{X}^{2}}}+{{\left({{r_{1}}\ {{r_{2}}^{6}}}-{{{r_{1}}^{3}}\ {{r_{2}}^{4}}}\right)}\ X}\right)}\ {{{e}^{r_{1}}}^{2}}}+{{\left({{\left(-{2 \ {{r_{2}}^{2}}}-{2 \ {{r_{1}}^{2}}}\right)}\ {{X}^{6}}}+{{\left({4 \ {{r_{2}}^{4}}}+{4 \ {{r_{1}}^{4}}}\right)}\ {{X}^{4}}}+{{\left(-{2 \ {{r_{2}}^{6}}}-{2 \ {{r_{1}}^{6}}}\right)}\ {{X}^{2}}}+{2 \ {{r_{1}}^{2}}\ {{r_{2}}^{6}}}-{4 \ {{r_{1}}^{4}}\ {{r_{2}}^{4}}}+{2 \ {{r_{1}}^{6}}\ {{r_{2}}^{2}}}\right)}\ {{e}^{r_{1}}}}+{{{r_{2}}^{2}}\ {{X}^{6}}}-{2 \ {{r_{2}}^{4}}\ {{X}^{4}}}+{{\left({{r_{1}}\ {{r_{2}}^{4}}}-{{{r_{1}}^{3}}\ {{r_{2}}^{2}}}\right)}\ {{X}^{3}}}+{{{r_{2}}^{6}}\ {{X}^{2}}}+{{\left(-{{r_{1}}\ {{r_{2}}^{6}}}+{{{r_{1}}^{3}}\ {{r_{2}}^{4}}}\right)}\ X}\right)}\ {{e}^{r_{2}}}}+{{\left({{{r_{1}}^{2}}\ {{X}^{6}}}-{2 \ {{r_{1}}^{4}}\ {{X}^{4}}}+{{\left(-{{{r_{1}}^{2}}\ {{r_{2}}^{3}}}+{{{r_{1}}^{4}}\ {r_{2}}}\right)}\ {{X}^{3}}}+{{{r_{1}}^{6}}\ {{X}^{2}}}+{{\left({{{r_{1}}^{4}}\ {{r_{2}}^{3}}}-{{{r_{1}}^{6}}\ {r_{2}}}\right)}\ X}\right)}\ {{e}^{r_{1}}}}}{{\left({2 \ {{r_{1}}^{2}}\ {{r_{2}}^{6}}}-{4 \ {{r_{1}}^{4}}\ {{r_{2}}^{4}}}+{2 \ {{r_{1}}^{6}}\ {{r_{2}}^{2}}}\right)}\ {{e}^{r_{1}}}\ {{e}^{r_{2}}}}$ (62)

Type: Expression(Integer)

fricas
```
_rule(lhs eq5_6d,rhs eq5_6d)(%)
```
$\label{eq63}\frac{{{\left({{{r_{1}}^{2}}\ {{X}^{4}}}+{{{r_{1}}^{2}}\ {r_{2}}\ {{X}^{3}}}-{{{r_{1}}^{4}}\ {{X}^{2}}}-{{{r_{1}}^{4}}\ {r_{2}}\ X}\right)}\ {{e}^{r_{1}}}\ {{{e}^{r_{2}}}^{2}}}+{{\left({{\left(-{{{r_{2}}^{2}}\ {{X}^{4}}}-{{r_{1}}\ {{r_{2}}^{2}}\ {{X}^{3}}}+{{{r_{2}}^{4}}\ {{X}^{2}}}+{{r_{1}}\ {{r_{2}}^{4}}\ X}\right)}\ {{{e}^{r_{1}}}^{2}}}+{{\left({{\left({2 \ {{r_{2}}^{2}}}-{2 \ {{r_{1}}^{2}}}\right)}\ {{X}^{4}}}+{{\left(-{2 \ {{r_{2}}^{4}}}+{2 \ {{r_{1}}^{4}}}\right)}\ {{X}^{2}}}+{2 \ {{r_{1}}^{2}}\ {{r_{2}}^{4}}}-{2 \ {{r_{1}}^{4}}\ {{r_{2}}^{2}}}\right)}\ {{e}^{r_{1}}}}-{{{r_{2}}^{2}}\ {{X}^{4}}}+{{r_{1}}\ {{r_{2}}^{2}}\ {{X}^{3}}}+{{{r_{2}}^{4}}\ {{X}^{2}}}-{{r_{1}}\ {{r_{2}}^{4}}\ X}\right)}\ {{e}^{r_{2}}}}+{{\left({{{r_{1}}^{2}}\ {{X}^{4}}}-{{{r_{1}}^{2}}\ {r_{2}}\ {{X}^{3}}}-{{{r_{1}}^{4}}\ {{X}^{2}}}+{{{r_{1}}^{4}}\ {r_{2}}\ X}\right)}\ {{e}^{r_{1}}}}}{{\left({2 \ {{r_{1}}^{2}}\ {{r_{2}}^{4}}}-{2 \ {{r_{1}}^{4}}\ {{r_{2}}^{2}}}\right)}\ {{e}^{r_{1}}}\ {{e}^{r_{2}}}}$ (63)

Type: Expression(Integer)

fricas
```
_rule(lhs eq5_6c,rhs eq5_6c)(%)
```
$\label{eq64}\frac{{{\left({{r_{1}}\ {{X}^{3}}}+{{r_{1}}\ {r_{2}}\ {{X}^{2}}}-{{{r_{1}}^{3}}\ X}-{{{r_{1}}^{3}}\ {r_{2}}}\right)}\ {{e}^{r_{1}}}\ {{{e}^{r_{2}}}^{2}}}+{{\left({{\left(-{{r_{2}}\ {{X}^{3}}}-{{r_{1}}\ {r_{2}}\ {{X}^{2}}}+{{{r_{2}}^{3}}\ X}+{{r_{1}}\ {{r_{2}}^{3}}}\right)}\ {{{e}^{r_{1}}}^{2}}}+{{r_{2}}\ {{X}^{3}}}-{{r_{1}}\ {r_{2}}\ {{X}^{2}}}-{{{r_{2}}^{3}}\ X}+{{r_{1}}\ {{r_{2}}^{3}}}\right)}\ {{e}^{r_{2}}}}+{{\left(-{{r_{1}}\ {{X}^{3}}}+{{r_{1}}\ {r_{2}}\ {{X}^{2}}}+{{{r_{1}}^{3}}\ X}-{{{r_{1}}^{3}}\ {r_{2}}}\right)}\ {{e}^{r_{1}}}}}{{\left({2 \ {r_{1}}\ {{r_{2}}^{3}}}-{2 \ {{r_{1}}^{3}}\ {r_{2}}}\right)}\ {{e}^{r_{1}}}\ {{e}^{r_{2}}}}$ (64)

Type: Expression(Integer)

fricas
```
test(normalize(% - rhs eq5_4) = 0)
```
$\label{eq65} \mbox{\rm true}$ (65)

Type: Boolean