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optimal antiderivative for 1/((x^2+x+1)*SQRT(x^2-x+1))

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clicl...@freenet.de

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Oct 15, 2011, 4:09:32 AM10/15/11
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Derive 6.10 returns these elementary integrals unevaluated:

INT(1/((x^2 + x + 1)*SQRT(x^2 - x + 1)), x)

INT(x/((x^2 + x + 1)*SQRT(x^2 - x + 1)), x)

The optimal antiderivatives are:

ATANH(SQRT(6)*(x - 1)/(3*SQRT(x^2 - x + 1)))/SQRT(6)
+ ATAN(SQRT(2)*(x + 1)/SQRT(x^2 - x + 1))/SQRT(2)

ATANH(SQRT(6)*(x - 1)/(3*SQRT(x^2 - x + 1)))/SQRT(6)
- ATAN(SQRT(2)*(x + 1)/SQRT(x^2 - x + 1))/SQRT(2)

Is there an integrator clever enough to find them?

Martin.

Axel Vogt

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Oct 15, 2011, 6:23:36 AM10/15/11
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Maple 15 answers in terms of arctanh and arctan as well,
but does not simplify the algebraic terms (except using
the risky command 'simplify(term, symbolic)', which will
ignore some rules for sqrt(negative)).

G. A. Edgar

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Oct 15, 2011, 12:53:39 PM10/15/11
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In article <9ft59s...@mid.individual.net>, Axel Vogt
Instead, you could 'simplify(term) assuming real' to get
a simplified answer.

--
G. A. Edgar http://www.math.ohio-state.edu/~edgar/

clicl...@freenet.de

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Oct 15, 2011, 3:10:47 PM10/15/11
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"G. A. Edgar" schrieb:
>
> In article <9ft59s...@mid.individual.net>, Axel Vogt
> <&nor...@axelvogt.de> wrote:
>
> > On 15.10.2011 10:09, clicl...@freenet.de wrote:
> > >
> > > Derive 6.10 returns these elementary integrals unevaluated:
> > >
> > > INT(1/((x^2 + x + 1)*SQRT(x^2 - x + 1)), x)
> > >
> > > INT(x/((x^2 + x + 1)*SQRT(x^2 - x + 1)), x)
> > >
> > > The optimal antiderivatives are:
> > >
> > > ATANH(SQRT(6)*(x - 1)/(3*SQRT(x^2 - x + 1)))/SQRT(6)
> > > + ATAN(SQRT(2)*(x + 1)/SQRT(x^2 - x + 1))/SQRT(2)
> > >
> > > ATANH(SQRT(6)*(x - 1)/(3*SQRT(x^2 - x + 1)))/SQRT(6)
> > > - ATAN(SQRT(2)*(x + 1)/SQRT(x^2 - x + 1))/SQRT(2)
> > >
> > > Is there an integrator clever enough to find them?
> >
> > Maple 15 answers in terms of arctanh and arctan as well,
> > but does not simplify the algebraic terms (except using
> > the risky command 'simplify(term, symbolic)', which will
> > ignore some rules for sqrt(negative)).
>
> Instead, you could 'simplify(term) assuming real' to get
> a simplified answer.

To prevent a possible misunderstanding: The above antiderivatives apply
on the entire complex plane.

Martin.

Richard Fateman

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Oct 16, 2011, 11:17:53 AM10/16/11
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Macsyma returns results in terms of asinh. Then trigsimp(%) returns a
sum of just two.

The argument to asinh is more complicated than the illustrated result
in terms of atanh.

if atanh(y)=asinh(x) then x= y/sqrt(1-y^2). I don't know of any magic
Macsyma command that will try heuristic simplification to see which is
simpler, x or y, although you can program this. I haven't thought
about the implications of the real domain, but a glance at the
input shows that the ambiguity of the sqrt should lead to two
possibly different answers for each of the two problems.


Maxima returns the unevaluated integral.

Waldek Hebisch

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Oct 18, 2011, 12:53:06 PM10/18/11
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FriCAS 1.1.4 finds somewhat complicated expressions with
two atans and two logs containing nested root. However,
the result returned from core integrator is simple:

+----------+
--+ 3 2 | 2
> %F log(24%F + 6%F + 2%F + \|x - x + 1 - x)
--+
4 1 2 1
%F + - %F + --= 0
6 36

This reuslt is then processed to eliminate sum over roots
and in the process FriCAS messes it up.

--
Waldek Hebisch
heb...@math.uni.wroc.pl

dimitris

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Oct 18, 2011, 5:08:49 PM10/18/11
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Hello.
Every attempt in Mathematica failed to produce something simpler than

In[1]:= Integrate[1/((x^2 + x + 1)*Sqrt[x^2 - x + 1]), x]

Out[1]= ArcTan[(3*(Sqrt[3] + I*x - I*x^2)*(1 - x + x^2))/(-3 +
I*Sqrt[3] - 3*x - 3*I*Sqrt[3]*x + 3*x^2 + I*Sqrt[3]*x^2 -
6*x^3 -
2*I*Sqrt[3]*x^4 -
I*Sqrt[3*(1 - I*Sqrt[3])]*Sqrt[1 - x + x^2] +
I*Sqrt[3*(1 - I*Sqrt[3])]*x*Sqrt[1 - x + x^2] +
I*Sqrt[3*(1 - I*Sqrt[3])]*x^2*Sqrt[1 - x + x^2] +
2*I*Sqrt[3*(1 - I*Sqrt[3])]*x^3*Sqrt[1 - x + x^2])]/
Sqrt[3*(1 - I*Sqrt[3])] -
ArcTan[(3*(Sqrt[3] - I*x + I*x^2)*(1 - x + x^2))/(3 + I*Sqrt[3] +
3*x - 3*I*Sqrt[3]*x - 3*x^2 +
I*Sqrt[3]*x^2 + 6*x^3 - 2*I*Sqrt[3]*x^4 -
I*Sqrt[3*(1 + I*Sqrt[3])]*Sqrt[1 - x + x^2] +
I*Sqrt[3*(1 + I*Sqrt[3])]*x*Sqrt[1 - x + x^2] +
I*Sqrt[3*(1 + I*Sqrt[3])]*x^2*Sqrt[1 - x + x^2] +
2*I*Sqrt[3*(1 + I*Sqrt[3])]*x^3*Sqrt[1 - x + x^2])]/
Sqrt[3*(1 + I*Sqrt[3])] -
(I*Log[(-I + Sqrt[3] - 2*I*x)^2*(I + Sqrt[3] + 2*I*x)^2])/(2*
Sqrt[3*(1 - I*Sqrt[3])]) +
(I*Log[(-I + Sqrt[3] - 2*I*x)^2*(I + Sqrt[3] + 2*I*x)^2])/(2*
Sqrt[3*(1 + I*Sqrt[3])]) +
(I*Log[(1 + x + x^2)*(11*I + 4*Sqrt[3] - 17*I*x - 4*Sqrt[3]*x +
11*I*x^2 + 4*Sqrt[3]*x^2 +
10*I*Sqrt[1 - I*Sqrt[3]]*Sqrt[1 - x + x^2] -
8*I*Sqrt[1 - I*Sqrt[3]]*x*Sqrt[1 - x + x^2])])/(2*
Sqrt[3*(1 - I*Sqrt[3])]) -
(I*Log[(1 + x + x^2)*(-11*I + 4*Sqrt[3] + 17*I*x - 4*Sqrt[3]*x -
11*I*x^2 + 4*Sqrt[3]*x^2 -
10*I*Sqrt[1 + I*Sqrt[3]]*Sqrt[1 - x + x^2] +
8*I*Sqrt[1 + I*Sqrt[3]]*x*Sqrt[1 - x + x^2])])/(2*
Sqrt[3*(1 + I*Sqrt[3])])

Moreover, this expression is not continuous in the whole real plane.

BTW, With what procedure did you find your antiderivatives?

Dimitris Anagnostou

Richard Fateman

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Oct 18, 2011, 7:25:31 PM10/18/11
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On 10/18/2011 2:08 PM, dimitris wrote:
> On Oct 15, 10:09 am, cliclic...@freenet.de wrote:
>> Derive 6.10 returns these elementary integrals unevaluated:
>>
>> INT(1/((x^2 + x + 1)*SQRT(x^2 - x + 1)), x)
>>
.....
To add to my previous note...

Macsyma produces a result much smaller than Mathematica.
I believe it is a correct antiderivative at least some places.

((%i * asinh((4/(2 * sqrt(3) * x - 3 * %i + sqrt(3))) - ((4 * %i)/(2 * x
- sqrt(3) * %i + 1)) + %i - (2/(sqrt(3)))))/(sqrt(3) * sqrt(1 - sqrt(3)
* %i)))
- ((%i * asinh((4/(2 * sqrt(3) * x + 3 * %i + sqrt(3))) + ((4 * %i)/(2
* x + sqrt(3) * %i + 1)) - %i - (2/(sqrt(3)))))/(sqrt(3) * sqrt(sqrt(3)
* %i + 1)))

RJF

clicl...@freenet.de

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Oct 21, 2011, 1:31:33 AM10/21/11
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dimitris schrieb:
>
> On Oct 15, 10:09 am, cliclic...@freenet.de wrote:
> >
> > Derive 6.10 returns these elementary integrals unevaluated:
> >
> > INT(1/((x^2 + x + 1)*SQRT(x^2 - x + 1)), x)
> >
> > INT(x/((x^2 + x + 1)*SQRT(x^2 - x + 1)), x)
> >
> > The optimal antiderivatives are:
> >
> > ATANH(SQRT(6)*(x - 1)/(3*SQRT(x^2 - x + 1)))/SQRT(6)
> > + ATAN(SQRT(2)*(x + 1)/SQRT(x^2 - x + 1))/SQRT(2)
> >
> > ATANH(SQRT(6)*(x - 1)/(3*SQRT(x^2 - x + 1)))/SQRT(6)
> > - ATAN(SQRT(2)*(x + 1)/SQRT(x^2 - x + 1))/SQRT(2)
> >
> > Is there an integrator clever enough to find them?
> >
>
Hey, this is turning out to be interesting! The MMA antiderivative is
correct on the entire complex plane (differentiation leads back to the
integrand), but it has about thirteen times the size of my result, it
unnecessarily involves the imaginary unit, and it unnecessarily jumps
near x = -0.725 on the real axis. Is at least INT(..., x, -1, 0) =
ACOT(SQRT(2)/2)/SQRT(2) + LN(2*SQRT(6) + 3*SQRT(3) - 3*SQRT(2) -
4)/SQRT(6) = 0.9272087241 calculated correctly?

The Macsyma antiderivative is much more compact, but incorrect in some
part of the complex plane. Differentiation produces:

((SQRT(6) - 3*SQRT(2)*#i)*SQRT((1 - SQRT(3)*#i)*(x^2 - x + 1)
/(2*x + 1 - SQRT(3)*#i)^2)
+ (SQRT(6) + 3*SQRT(2)*#i)*SQRT((1 + SQRT(3)*#i)*(x^2 - x + 1)
/(2*x + 1 + SQRT(3)*#i)^2))/(6*(x^2 - x + 1))

which on the real axis reduces to SIGN(x + 1)/((x^2 + x + 1)*SQRT(x^2 -
x + 1)); so here the sign is flipped below the jump at x = -1.

The real antiderivative of 1/((x^2 + x + 1)*SQRT(x^2 - x + 1)) is hard
to obtain because x^2 + x + 1 = (2*x + 1 + SQRT(3)*#i) * (2*x + 1 -
SQRT(3)*#i)/4 does not factor over the rational numbers, whence a
partial-fraction expansion of 1/(x^2 + x + 1) leads to such a seemingly
irreducible complex mess. Here is a complementary example where the root
is not real everywhere on the real axis:

INT((2 - x)/((3*x^2 - 2*x + 1)*SQRT(- x^2 + 2*x + 3)), x) =

SQRT(66*SQRT(33) - 286)/44*ATANH(SQRT(6*SQRT(33) - 34)
*(2*x + SQRT(33) + 5)/(4*SQRT(- x^2 + 2*x + 3)))
+ SQRT(66*SQRT(33) + 286)/44*ATAN(SQRT(6*SQRT(33) + 34)
*(2*x - SQRT(33) + 5)/(4*SQRT(- x^2 + 2*x + 3)))

I will describe a general recipe for finding the compact antiderivative
of (A + B*x)/((a + b*x + c*x^2)*SQRT(d + e*x + f*x^2)) in a forthcoming
manuscript; I feel that I should leave something apparently worth
describing undescribed until then.

Many thanks for the feedback.

Martin.

PS: I am having intermittent hardware trouble, so may be unable to
respond timely.

dimitris

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Oct 21, 2011, 4:47:05 AM10/21/11
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As you said Mathematica's lengthy expression is nevertheless correct
in the whole complex plane.

In[3]:= intMath = Integrate[1/((x^2 + x + 1)*Sqrt[x^2 - x + 1]), x]
FullSimplify[D[intMath, x] - 1/((x^2 + x + 1)*Sqrt[x^2 - x + 1])]

Out[3]= (1/(4*Sqrt[3]))*(-2*Sqrt[1 + I*Sqrt[3]]*
ArcTan[(3*(1 - x + x^2)*(I*Sqrt[3] - x + x^2))/
(3*I + Sqrt[3] - 2*Sqrt[3]*x^4 -
Sqrt[3 - 3*I*Sqrt[3]]*Sqrt[1 - x + x^2] +
2*x^3*(3*I + Sqrt[3 - 3*I*Sqrt[3]]*Sqrt[1 - x + x^2]) +
x*(3*I - 3*Sqrt[3] +
Sqrt[3 - 3*I*Sqrt[3]]*Sqrt[1 - x + x^2]) +
x^2*(-3*I + Sqrt[3] +
Sqrt[3 - 3*I*Sqrt[3]]*Sqrt[1 - x + x^2]))] -
2*Sqrt[1 - I*Sqrt[3]]*
ArcTan[(3*(I*Sqrt[3] + x - x^2)*(1 - x + x^2))/(3*I - Sqrt[3] +
2*Sqrt[3]*x^4 +
Sqrt[3 + 3*I*Sqrt[3]]*Sqrt[1 - x + x^2] +
x^3*(6*I - 2*Sqrt[3 + 3*I*Sqrt[3]]*Sqrt[1 - x + x^2])
+
x*(3*I + 3*Sqrt[3] -
Sqrt[3 + 3*I*Sqrt[3]]*Sqrt[1 - x + x^2]) -
x^2*(3*I + Sqrt[3] +
Sqrt[3 + 3*I*Sqrt[3]]*Sqrt[1 - x + x^2]))] +
I*((Sqrt[1 - I*Sqrt[3]] - Sqrt[1 + I*Sqrt[3]])*
Log[16*(1 + x + x^2)^2] + Sqrt[1 + I*Sqrt[3]]*
Log[(1 + x + x^2)*(11*I + 4*Sqrt[3] + (11*I + 4*Sqrt[3])*x^2 +
10*I*Sqrt[1 - I*Sqrt[3]]*
Sqrt[1 - x + x^2] -
x*(17*I + 4*Sqrt[3] +
8*I*Sqrt[1 - I*Sqrt[3]]*Sqrt[1 - x + x^2]))] - Sqrt[1 -
I*Sqrt[3]]*
Log[(1 + x + x^2)*(-11*I + 4*Sqrt[3] + (-11*I + 4*Sqrt[3])*x^2
-
10*I*Sqrt[1 + I*Sqrt[3]]*
Sqrt[1 - x + x^2] +
x*(17*I - 4*Sqrt[3] +
8*I*Sqrt[1 + I*Sqrt[3]]*Sqrt[1 - x + x^2]))]))
Out[4]= 0

Unfortunately the definite integral in the range (-1,0) is evaluated
incorrectly.

In[10]:= Integrate[1/((x^2 + x + 1)*Sqrt[x^2 - x + 1]), {x, -1, 0}]
Out[10]= -((1/(4*
Sqrt[3]))*(I*(Sqrt[
1 + I*Sqrt[3]]*(-2*I*
ArcTan[(6 + 3*I*Sqrt[3])/(3*I - Sqrt[3] +
3*Sqrt[1 - I*Sqrt[3]])] +
2*ArcTanh[(3*Sqrt[3])/(3*I + Sqrt[3] -
Sqrt[3 - 3*I*Sqrt[3]])] -
Log[11*I + 4*Sqrt[3] + 10*I*Sqrt[1 - I*Sqrt[3]]]
+
Log[3*(13*I + 4*Sqrt[3] + 6*I*Sqrt[3 - 3*I*Sqrt[3]])]) +
Sqrt[1 - I*Sqrt[3]]*
(2*
ArcTanh[(3*(2*I + Sqrt[3]))/(3*I + Sqrt[3] -
3*Sqrt[1 + I*Sqrt[3]])] +

2*ArcTanh[(3*Sqrt[3])/(3*I - Sqrt[3] +
Sqrt[3 + 3*I*Sqrt[3]])] +
Log[(1/16)*(-11*I + 4*Sqrt[3] - 10*I*Sqrt[1 + I*Sqrt[3]])]
-
Log[(3/16)*(-13*I + 4*Sqrt[3] -
6*I*Sqrt[3 + 3*I*Sqrt[3]])]))))

In[11]:= N[%]
Out[11]= -1.294232744953466 + 0.*I

In[12]:= NIntegrate[1/((x^2 + x + 1)*Sqrt[x^2 - x + 1]), {x, -1, 0}]
Out[12]= 0.9272087241257184

I guess that Mathematica does the following

In[16]:= N[(intMath /. x -> 0) - (intMath /. x -> -1)]
Out[16]= -1.2942327449534659 + 0.*I

failing to take into account the jump near x = -0.725.

Dimitris

Albert Rich

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Oct 22, 2011, 8:41:34 PM10/22/11
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Martin's first integrand equals

(1+x)/(2*(1+x+x^2)*Sqrt[1-x+x^2]) + (1-x)/(2*(1+x+x^2)*Sqrt[1-x+x^2])

and his second integrand equals

(1+x)/(2*(1+x+x^2)*Sqrt[1-x+x^2]) - (1-x)/(2*(1+x+x^2)*Sqrt[1-x+x^2]).


Int[(1+x)/((1+x+x^2)*Sqrt[1-x+x^2]),x] is an instance of the rule:

If c*d^2-a*e^2=0, c*f=a*h and (b*h-c*g)/(e*(e*g+2*d*h))>0, then

Int[(d+e*x)/((f+g*x+h*x^2)*Sqrt[a+b*x+c*x^2]),x] -->
-2*e/((e*g+2*d*h)*Sqrt[(b*h-c*g)/(e*(e*g+2*d*h))])*
ArcTan[Sqrt[(b*h-c*g)/(e*(e*g+2*d*h))]*((d-e*x)/Sqrt[a+b*x
+c*x^2])]

Int[(1-x)/((1+x+x^2)*Sqrt[1-x+x^2]),x] is an instance of the rule:

If c*d^2-a*e^2=0, c*f=a*h and (b*h-c*g)/(e*(e*g+2*d*h))<0, then

Int[(d+e*x)/((f+g*x+h*x^2)*Sqrt[a+b*x+c*x^2]),x] -->
-2*e/((e*g+2*d*h)*Sqrt[(b*h-c*g)/(e*(e*g+2*d*h))])*
ArcTan[Sqrt[(b*h-c*g)/(e*(e*g+2*d*h))]*((d-e*x)/Sqrt[a+b*x
+c*x^2])]

Of course, these two terminal rules are of not much value unless
recursive rules can be found for reducing integrands of the form (d
+e*x)*(a+b*x+c*x^2)^m*(f+g*x+h*x^2)^n to these terminal cases...

Albert

clicl...@freenet.de

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Oct 23, 2011, 3:03:24 AM10/23/11
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Waldek Hebisch schrieb:
This is the antiderivative of x/((x^2+x+1)*SQRT(x^2-x+1)). Derive 6.10
handles the postprocessing reasonably well, in 26+1 steps:

SUM(t*LN(24*t^3+6*t^2+2*t+SQRT(x^2-x+1)-x),t,SOLUTIONS(r^4+r^2/6~
+1/36,r))

" SUM(F(x),x,[x1,x2,...]) -> F(x1)+F(x2)+... "

(-SQRT(6)/12-SQRT(2)*#i/4)*LN(24*(-SQRT(6)/12-SQRT(2)*#i/4)^3+6*~
(-SQRT(6)/12-SQRT(2)*#i/4)^2+SQRT(x^2-x+1)-x+2*(-SQRT(6)/12-SQRT~
(2)*#i/4))+(-SQRT(6)/12+SQRT(2)*#i/4)*LN(24*(-SQRT(6)/12+SQRT(2)~
*#i/4)^3+6*(-SQRT(6)/12+SQRT(2)*#i/4)^2+SQRT(x^2-x+1)-x+2*(-SQRT~
(6)/12+SQRT(2)*#i/4))+(SQRT(6)/12-SQRT(2)*#i/4)*LN(24*(SQRT(6)/1~
2-SQRT(2)*#i/4)^3+6*(SQRT(6)/12-SQRT(2)*#i/4)^2+SQRT(x^2-x+1)-x+~
2*(SQRT(6)/12-SQRT(2)*#i/4))+(SQRT(6)/12+SQRT(2)*#i/4)*LN(24*(SQ~
RT(6)/12+SQRT(2)*#i/4)^3+6*(SQRT(6)/12+SQRT(2)*#i/4)^2+SQRT(x^2-~
x+1)-x+2*(SQRT(6)/12+SQRT(2)*#i/4))

" LN(x+y*#i) -> LN(|x+y*#i|) + #i*ATAN(y,x) "

(-SQRT(6)/12-SQRT(2)*#i/4)*(LN(ABS(SQRT(x^2-x+1)-x+SQRT(6)/2-1/2~
+#i*(SQRT(3)/2-SQRT(2)/2)))+#i*ATAN(SQRT(3)/2-SQRT(2)/2,SQRT(x^2~
-x+1)-x+SQRT(6)/2-1/2))+(-SQRT(6)/12+SQRT(2)*#i/4)*LN(24*(-SQRT(~
6)/12+SQRT(2)*#i/4)^3+6*(-SQRT(6)/12+SQRT(2)*#i/4)^2+SQRT(x^2-x+~
1)-x+2*(-SQRT(6)/12+SQRT(2)*#i/4))+(SQRT(6)/12-SQRT(2)*#i/4)*LN(~
24*(SQRT(6)/12-SQRT(2)*#i/4)^3+6*(SQRT(6)/12-SQRT(2)*#i/4)^2+SQR~
T(x^2-x+1)-x+2*(SQRT(6)/12-SQRT(2)*#i/4))+(SQRT(6)/12+SQRT(2)*#i~
/4)*LN(24*(SQRT(6)/12+SQRT(2)*#i/4)^3+6*(SQRT(6)/12+SQRT(2)*#i/4~
)^2+SQRT(x^2-x+1)-x+2*(SQRT(6)/12+SQRT(2)*#i/4))

" ABS(x+#i*y) -> SQRT(x^2+y^2) "

(-SQRT(6)/12-SQRT(2)*#i/4)*(LN(SQRT((SQRT(x^2-x+1)-x+SQRT(6)/2-1~
/2)^2+(SQRT(3)/2-SQRT(2)/2)^2))+#i*ATAN(SQRT(3)/2-SQRT(2)/2,SQRT~
(x^2-x+1)-x+SQRT(6)/2-1/2))+(-SQRT(6)/12+SQRT(2)*#i/4)*LN(24*(-S~
QRT(6)/12+SQRT(2)*#i/4)^3+6*(-SQRT(6)/12+SQRT(2)*#i/4)^2+SQRT(x^~
2-x+1)-x+2*(-SQRT(6)/12+SQRT(2)*#i/4))+(SQRT(6)/12-SQRT(2)*#i/4)~
*LN(24*(SQRT(6)/12-SQRT(2)*#i/4)^3+6*(SQRT(6)/12-SQRT(2)*#i/4)^2~
+SQRT(x^2-x+1)-x+2*(SQRT(6)/12-SQRT(2)*#i/4))+(SQRT(6)/12+SQRT(2~
)*#i/4)*LN(24*(SQRT(6)/12+SQRT(2)*#i/4)^3+6*(SQRT(6)/12+SQRT(2)*~
#i/4)^2+SQRT(x^2-x+1)-x+2*(SQRT(6)/12+SQRT(2)*#i/4))

" If -1<n<1, LN(z^n) -> n*LN(z) "

(-SQRT(6)/12-SQRT(2)*#i/4)*(LN(-SQRT(x^2-x+1)*(2*x-SQRT(6)+1)+2*~
x^2-SQRT(6)*x-SQRT(6)+4)/2+#i*ATAN(SQRT(3)/2-SQRT(2)/2,SQRT(x^2-~
x+1)-x+SQRT(6)/2-1/2))+(-SQRT(6)/12+SQRT(2)*#i/4)*LN(24*(-SQRT(6~
)/12+SQRT(2)*#i/4)^3+6*(-SQRT(6)/12+SQRT(2)*#i/4)^2+SQRT(x^2-x+1~
)-x+2*(-SQRT(6)/12+SQRT(2)*#i/4))+(SQRT(6)/12-SQRT(2)*#i/4)*LN(2~
4*(SQRT(6)/12-SQRT(2)*#i/4)^3+6*(SQRT(6)/12-SQRT(2)*#i/4)^2+SQRT~
(x^2-x+1)-x+2*(SQRT(6)/12-SQRT(2)*#i/4))+(SQRT(6)/12+SQRT(2)*#i/~
4)*LN(24*(SQRT(6)/12+SQRT(2)*#i/4)^3+6*(SQRT(6)/12+SQRT(2)*#i/4)~
^2+SQRT(x^2-x+1)-x+2*(SQRT(6)/12+SQRT(2)*#i/4))

" ATAN(y,x) -> pi*SIGN(y)/2-ATAN(x/y) "

(-SQRT(6)/12-SQRT(2)*#i/4)*(LN(-SQRT(x^2-x+1)*(2*x-SQRT(6)+1)+2*~
x^2-SQRT(6)*x-SQRT(6)+4)/2+#i*(pi*SIGN(SQRT(3)/2-SQRT(2)/2)/2-AT~
AN((SQRT(x^2-x+1)-x+SQRT(6)/2-1/2)/(SQRT(3)/2-SQRT(2)/2))))+(-SQ~
RT(6)/12+SQRT(2)*#i/4)*LN(24*(-SQRT(6)/12+SQRT(2)*#i/4)^3+6*(-SQ~
RT(6)/12+SQRT(2)*#i/4)^2+SQRT(x^2-x+1)-x+2*(-SQRT(6)/12+SQRT(2)*~
#i/4))+(SQRT(6)/12-SQRT(2)*#i/4)*LN(24*(SQRT(6)/12-SQRT(2)*#i/4)~
^3+6*(SQRT(6)/12-SQRT(2)*#i/4)^2+SQRT(x^2-x+1)-x+2*(SQRT(6)/12-S~
QRT(2)*#i/4))+(SQRT(6)/12+SQRT(2)*#i/4)*LN(24*(SQRT(6)/12+SQRT(2~
)*#i/4)^3+6*(SQRT(6)/12+SQRT(2)*#i/4)^2+SQRT(x^2-x+1)-x+2*(SQRT(~
6)/12+SQRT(2)*#i/4))

" 1/(z+w) -> (z-w)/(z^2-w^2) "

(-SQRT(6)/12-SQRT(2)*#i/4)*(LN(-SQRT(x^2-x+1)*(2*x-SQRT(6)+1)+2*~
x^2-SQRT(6)*x-SQRT(6)+4)/2+#i*(pi/2-ATAN(4*(SQRT(x^2-x+1)-x+SQRT~
(6)/2-1/2)*(SQRT(3)/2+SQRT(2)/2))))+(-SQRT(6)/12+SQRT(2)*#i/4)*L~
N(24*(-SQRT(6)/12+SQRT(2)*#i/4)^3+6*(-SQRT(6)/12+SQRT(2)*#i/4)^2~
+SQRT(x^2-x+1)-x+2*(-SQRT(6)/12+SQRT(2)*#i/4))+(SQRT(6)/12-SQRT(~
2)*#i/4)*LN(24*(SQRT(6)/12-SQRT(2)*#i/4)^3+6*(SQRT(6)/12-SQRT(2)~
*#i/4)^2+SQRT(x^2-x+1)-x+2*(SQRT(6)/12-SQRT(2)*#i/4))+(SQRT(6)/1~
2+SQRT(2)*#i/4)*LN(24*(SQRT(6)/12+SQRT(2)*#i/4)^3+6*(SQRT(6)/12+~
SQRT(2)*#i/4)^2+SQRT(x^2-x+1)-x+2*(SQRT(6)/12+SQRT(2)*#i/4))

" (a+b*#i)*(c+d*#i) -> (a*c-b*d)+(a*d+b*c)*#i "

-SQRT(6)*LN(-SQRT(x^2-x+1)*(2*x-SQRT(6)+1)+2*x^2-SQRT(6)*x-SQRT(~
6)+4)/(2*12)-SQRT(2)*(ATAN((SQRT(3)+SQRT(2))*(2*SQRT(x^2-x+1)-2*~
x+SQRT(6)-1))-pi/2)/4+#i*(SQRT(6)*(ATAN((SQRT(3)+SQRT(2))*(2*SQR~
T(x^2-x+1)-2*x+SQRT(6)-1))-pi/2)/12-SQRT(2)*LN(-SQRT(x^2-x+1)*(2~
*x-SQRT(6)+1)+2*x^2-SQRT(6)*x-SQRT(6)+4)/(2*4))+(-SQRT(6)/12+SQR~
T(2)*#i/4)*LN(24*(-SQRT(6)/12+SQRT(2)*#i/4)^3+6*(-SQRT(6)/12+SQR~
T(2)*#i/4)^2+SQRT(x^2-x+1)-x+2*(-SQRT(6)/12+SQRT(2)*#i/4))+(SQRT~
(6)/12-SQRT(2)*#i/4)*LN(24*(SQRT(6)/12-SQRT(2)*#i/4)^3+6*(SQRT(6~
)/12-SQRT(2)*#i/4)^2+SQRT(x^2-x+1)-x+2*(SQRT(6)/12-SQRT(2)*#i/4)~
)+(SQRT(6)/12+SQRT(2)*#i/4)*LN(24*(SQRT(6)/12+SQRT(2)*#i/4)^3+6*~
(SQRT(6)/12+SQRT(2)*#i/4)^2+SQRT(x^2-x+1)-x+2*(SQRT(6)/12+SQRT(2~
)*#i/4))

" LN(x+y*#i) -> LN(|x+y*#i|) + #i*ATAN(y,x) "

-SQRT(2)*ATAN((SQRT(3)+SQRT(2))*(2*SQRT(x^2-x+1)-2*x+SQRT(6)-1))~
/4-SQRT(6)*LN(-SQRT(x^2-x+1)*(2*x-SQRT(6)+1)+2*x^2-SQRT(6)*x-SQR~
T(6)+4)/24+SQRT(2)*pi/8+#i*(SQRT(6)*ATAN((SQRT(3)+SQRT(2))*(2*SQ~
RT(x^2-x+1)-2*x+SQRT(6)-1))/12-SQRT(2)*LN(-SQRT(x^2-x+1)*(2*x-SQ~
RT(6)+1)+2*x^2-SQRT(6)*x-SQRT(6)+4)/8-SQRT(6)*pi/24)+(-SQRT(6)/1~
2+SQRT(2)*#i/4)*(LN(ABS(SQRT(x^2-x+1)-x+SQRT(6)/2-1/2+#i*(SQRT(2~
)/2-SQRT(3)/2)))+#i*ATAN(-SQRT(3)/2+SQRT(2)/2,SQRT(x^2-x+1)-x+SQ~
RT(6)/2-1/2))+(SQRT(6)/12-SQRT(2)*#i/4)*LN(24*(SQRT(6)/12-SQRT(2~
)*#i/4)^3+6*(SQRT(6)/12-SQRT(2)*#i/4)^2+SQRT(x^2-x+1)-x+2*(SQRT(~
6)/12-SQRT(2)*#i/4))+(SQRT(6)/12+SQRT(2)*#i/4)*LN(24*(SQRT(6)/12~
+SQRT(2)*#i/4)^3+6*(SQRT(6)/12+SQRT(2)*#i/4)^2+SQRT(x^2-x+1)-x+2~
*(SQRT(6)/12+SQRT(2)*#i/4))

" ABS(x+#i*y) -> SQRT(x^2+y^2) "

-SQRT(2)*ATAN((SQRT(3)+SQRT(2))*(2*SQRT(x^2-x+1)-2*x+SQRT(6)-1))~
/4-SQRT(6)*LN(-SQRT(x^2-x+1)*(2*x-SQRT(6)+1)+2*x^2-SQRT(6)*x-SQR~
T(6)+4)/24+SQRT(2)*pi/8+#i*(SQRT(6)*ATAN((SQRT(3)+SQRT(2))*(2*SQ~
RT(x^2-x+1)-2*x+SQRT(6)-1))/12-SQRT(2)*LN(-SQRT(x^2-x+1)*(2*x-SQ~
RT(6)+1)+2*x^2-SQRT(6)*x-SQRT(6)+4)/8-SQRT(6)*pi/24)+(-SQRT(6)/1~
2+SQRT(2)*#i/4)*(LN(SQRT((SQRT(x^2-x+1)-x+SQRT(6)/2-1/2)^2+(-SQR~
T(3)/2+SQRT(2)/2)^2))+#i*ATAN(-SQRT(3)/2+SQRT(2)/2,SQRT(x^2-x+1)~
-x+SQRT(6)/2-1/2))+(SQRT(6)/12-SQRT(2)*#i/4)*LN(24*(SQRT(6)/12-S~
QRT(2)*#i/4)^3+6*(SQRT(6)/12-SQRT(2)*#i/4)^2+SQRT(x^2-x+1)-x+2*(~
SQRT(6)/12-SQRT(2)*#i/4))+(SQRT(6)/12+SQRT(2)*#i/4)*LN(24*(SQRT(~
6)/12+SQRT(2)*#i/4)^3+6*(SQRT(6)/12+SQRT(2)*#i/4)^2+SQRT(x^2-x+1~
)-x+2*(SQRT(6)/12+SQRT(2)*#i/4))

" If -1<n<1, LN(z^n) -> n*LN(z) "

-SQRT(2)*ATAN((SQRT(3)+SQRT(2))*(2*SQRT(x^2-x+1)-2*x+SQRT(6)-1))~
/4-SQRT(6)*LN(-SQRT(x^2-x+1)*(2*x-SQRT(6)+1)+2*x^2-SQRT(6)*x-SQR~
T(6)+4)/24+SQRT(2)*pi/8+#i*(SQRT(6)*ATAN((SQRT(3)+SQRT(2))*(2*SQ~
RT(x^2-x+1)-2*x+SQRT(6)-1))/12-SQRT(2)*LN(-SQRT(x^2-x+1)*(2*x-SQ~
RT(6)+1)+2*x^2-SQRT(6)*x-SQRT(6)+4)/8-SQRT(6)*pi/24)+(-SQRT(6)/1~
2+SQRT(2)*#i/4)*(LN(-SQRT(x^2-x+1)*(2*x-SQRT(6)+1)+2*x^2-SQRT(6)~
*x-SQRT(6)+4)/2+#i*ATAN(-SQRT(3)/2+SQRT(2)/2,SQRT(x^2-x+1)-x+SQR~
T(6)/2-1/2))+(SQRT(6)/12-SQRT(2)*#i/4)*LN(24*(SQRT(6)/12-SQRT(2)~
*#i/4)^3+6*(SQRT(6)/12-SQRT(2)*#i/4)^2+SQRT(x^2-x+1)-x+2*(SQRT(6~
)/12-SQRT(2)*#i/4))+(SQRT(6)/12+SQRT(2)*#i/4)*LN(24*(SQRT(6)/12+~
SQRT(2)*#i/4)^3+6*(SQRT(6)/12+SQRT(2)*#i/4)^2+SQRT(x^2-x+1)-x+2*~
(SQRT(6)/12+SQRT(2)*#i/4))

" ATAN(y,x) -> pi*SIGN(y)/2-ATAN(x/y) "

-SQRT(2)*ATAN((SQRT(3)+SQRT(2))*(2*SQRT(x^2-x+1)-2*x+SQRT(6)-1))~
/4-SQRT(6)*LN(-SQRT(x^2-x+1)*(2*x-SQRT(6)+1)+2*x^2-SQRT(6)*x-SQR~
T(6)+4)/24+SQRT(2)*pi/8+#i*(SQRT(6)*ATAN((SQRT(3)+SQRT(2))*(2*SQ~
RT(x^2-x+1)-2*x+SQRT(6)-1))/12-SQRT(2)*LN(-SQRT(x^2-x+1)*(2*x-SQ~
RT(6)+1)+2*x^2-SQRT(6)*x-SQRT(6)+4)/8-SQRT(6)*pi/24)+(-SQRT(6)/1~
2+SQRT(2)*#i/4)*(LN(-SQRT(x^2-x+1)*(2*x-SQRT(6)+1)+2*x^2-SQRT(6)~
*x-SQRT(6)+4)/2+#i*(pi*SIGN(-SQRT(3)/2+SQRT(2)/2)/2-ATAN((SQRT(x~
^2-x+1)-x+SQRT(6)/2-1/2)/(SQRT(2)/2-SQRT(3)/2))))+(SQRT(6)/12-SQ~
RT(2)*#i/4)*LN(24*(SQRT(6)/12-SQRT(2)*#i/4)^3+6*(SQRT(6)/12-SQRT~
(2)*#i/4)^2+SQRT(x^2-x+1)-x+2*(SQRT(6)/12-SQRT(2)*#i/4))+(SQRT(6~
)/12+SQRT(2)*#i/4)*LN(24*(SQRT(6)/12+SQRT(2)*#i/4)^3+6*(SQRT(6)/~
12+SQRT(2)*#i/4)^2+SQRT(x^2-x+1)-x+2*(SQRT(6)/12+SQRT(2)*#i/4))

" 1/(z+w) -> (z-w)/(z^2-w^2) "

-SQRT(2)*ATAN((SQRT(3)+SQRT(2))*(2*SQRT(x^2-x+1)-2*x+SQRT(6)-1))~
/4-SQRT(6)*LN(-SQRT(x^2-x+1)*(2*x-SQRT(6)+1)+2*x^2-SQRT(6)*x-SQR~
T(6)+4)/24+SQRT(2)*pi/8+#i*(SQRT(6)*ATAN((SQRT(3)+SQRT(2))*(2*SQ~
RT(x^2-x+1)-2*x+SQRT(6)-1))/12-SQRT(2)*LN(-SQRT(x^2-x+1)*(2*x-SQ~
RT(6)+1)+2*x^2-SQRT(6)*x-SQRT(6)+4)/8-SQRT(6)*pi/24)+(-SQRT(6)/1~
2+SQRT(2)*#i/4)*(LN(-SQRT(x^2-x+1)*(2*x-SQRT(6)+1)+2*x^2-SQRT(6)~
*x-SQRT(6)+4)/2+#i*(-pi/2-ATAN(4*(SQRT(x^2-x+1)-x+SQRT(6)/2-1/2)~
*(-SQRT(3)/2-SQRT(2)/2))))+(SQRT(6)/12-SQRT(2)*#i/4)*LN(24*(SQRT~
(6)/12-SQRT(2)*#i/4)^3+6*(SQRT(6)/12-SQRT(2)*#i/4)^2+SQRT(x^2-x+~
1)-x+2*(SQRT(6)/12-SQRT(2)*#i/4))+(SQRT(6)/12+SQRT(2)*#i/4)*LN(2~
4*(SQRT(6)/12+SQRT(2)*#i/4)^3+6*(SQRT(6)/12+SQRT(2)*#i/4)^2+SQRT~
(x^2-x+1)-x+2*(SQRT(6)/12+SQRT(2)*#i/4))

" ATAN(-z) -> -ATAN(z) "

-SQRT(2)*ATAN((SQRT(3)+SQRT(2))*(2*SQRT(x^2-x+1)-2*x+SQRT(6)-1))~
/4-SQRT(6)*LN(-SQRT(x^2-x+1)*(2*x-SQRT(6)+1)+2*x^2-SQRT(6)*x-SQR~
T(6)+4)/24+SQRT(2)*pi/8+#i*(SQRT(6)*ATAN((SQRT(3)+SQRT(2))*(2*SQ~
RT(x^2-x+1)-2*x+SQRT(6)-1))/12-SQRT(2)*LN(-SQRT(x^2-x+1)*(2*x-SQ~
RT(6)+1)+2*x^2-SQRT(6)*x-SQRT(6)+4)/8-SQRT(6)*pi/24)+(-SQRT(6)/1~
2+SQRT(2)*#i/4)*(LN(-SQRT(x^2-x+1)*(2*x-SQRT(6)+1)+2*x^2-SQRT(6)~
*x-SQRT(6)+4)/2+#i*(-pi/2+ATAN((SQRT(3)+SQRT(2))*(2*SQRT(x^2-x+1~
)-2*x+SQRT(6)-1))))+(SQRT(6)/12-SQRT(2)*#i/4)*LN(24*(SQRT(6)/12-~
SQRT(2)*#i/4)^3+6*(SQRT(6)/12-SQRT(2)*#i/4)^2+SQRT(x^2-x+1)-x+2*~
(SQRT(6)/12-SQRT(2)*#i/4))+(SQRT(6)/12+SQRT(2)*#i/4)*LN(24*(SQRT~
(6)/12+SQRT(2)*#i/4)^3+6*(SQRT(6)/12+SQRT(2)*#i/4)^2+SQRT(x^2-x+~
1)-x+2*(SQRT(6)/12+SQRT(2)*#i/4))

" (a+b*#i)*(c+d*#i) -> (a*c-b*d)+(a*d+b*c)*#i "

-SQRT(2)*ATAN((SQRT(3)+SQRT(2))*(2*SQRT(x^2-x+1)-2*x+SQRT(6)-1))~
/4-SQRT(6)*LN(-SQRT(x^2-x+1)*(2*x-SQRT(6)+1)+2*x^2-SQRT(6)*x-SQR~
T(6)+4)/24+SQRT(2)*pi/8+#i*(SQRT(6)*ATAN((SQRT(3)+SQRT(2))*(2*SQ~
RT(x^2-x+1)-2*x+SQRT(6)-1))/12-SQRT(2)*LN(-SQRT(x^2-x+1)*(2*x-SQ~
RT(6)+1)+2*x^2-SQRT(6)*x-SQRT(6)+4)/8-SQRT(6)*pi/24)-SQRT(6)*LN(~
-SQRT(x^2-x+1)*(2*x-SQRT(6)+1)+2*x^2-SQRT(6)*x-SQRT(6)+4)/(2*12)~
+SQRT(2)*(pi/2-ATAN((SQRT(3)+SQRT(2))*(2*SQRT(x^2-x+1)-2*x+SQRT(~
6)-1)))/4+#i*(SQRT(2)*LN(-SQRT(x^2-x+1)*(2*x-SQRT(6)+1)+2*x^2-SQ~
RT(6)*x-SQRT(6)+4)/(2*4)-SQRT(6)*(ATAN((SQRT(3)+SQRT(2))*(2*SQRT~
(x^2-x+1)-2*x+SQRT(6)-1))-pi/2)/12)+(SQRT(6)/12-SQRT(2)*#i/4)*LN~
(24*(SQRT(6)/12-SQRT(2)*#i/4)^3+6*(SQRT(6)/12-SQRT(2)*#i/4)^2+SQ~
RT(x^2-x+1)-x+2*(SQRT(6)/12-SQRT(2)*#i/4))+(SQRT(6)/12+SQRT(2)*#~
i/4)*LN(24*(SQRT(6)/12+SQRT(2)*#i/4)^3+6*(SQRT(6)/12+SQRT(2)*#i/~
4)^2+SQRT(x^2-x+1)-x+2*(SQRT(6)/12+SQRT(2)*#i/4))

" LN(x+y*#i) -> LN(|x+y*#i|) + #i*ATAN(y,x) "

-SQRT(2)*ATAN((SQRT(3)+SQRT(2))*(2*SQRT(x^2-x+1)-2*x+SQRT(6)-1))~
/2-SQRT(6)*LN(-SQRT(x^2-x+1)*(2*x-SQRT(6)+1)+2*x^2-SQRT(6)*x-SQR~
T(6)+4)/12+SQRT(2)*pi/4+(SQRT(6)/12-SQRT(2)*#i/4)*(LN(ABS(SQRT(x~
^2-x+1)-x-SQRT(6)/2-1/2-#i*(SQRT(3)/2+SQRT(2)/2)))+#i*ATAN(-SQRT~
(3)/2-SQRT(2)/2,SQRT(x^2-x+1)-x-SQRT(6)/2-1/2))+(SQRT(6)/12+SQRT~
(2)*#i/4)*LN(24*(SQRT(6)/12+SQRT(2)*#i/4)^3+6*(SQRT(6)/12+SQRT(2~
)*#i/4)^2+SQRT(x^2-x+1)-x+2*(SQRT(6)/12+SQRT(2)*#i/4))

" ABS(x+#i*y) -> SQRT(x^2+y^2) "

-SQRT(2)*ATAN((SQRT(3)+SQRT(2))*(2*SQRT(x^2-x+1)-2*x+SQRT(6)-1))~
/2-SQRT(6)*LN(-SQRT(x^2-x+1)*(2*x-SQRT(6)+1)+2*x^2-SQRT(6)*x-SQR~
T(6)+4)/12+SQRT(2)*pi/4+(SQRT(6)/12-SQRT(2)*#i/4)*(LN(SQRT((SQRT~
(x^2-x+1)-x-SQRT(6)/2-1/2)^2+(-SQRT(3)/2-SQRT(2)/2)^2))+#i*ATAN(~
-SQRT(3)/2-SQRT(2)/2,SQRT(x^2-x+1)-x-SQRT(6)/2-1/2))+(SQRT(6)/12~
+SQRT(2)*#i/4)*LN(24*(SQRT(6)/12+SQRT(2)*#i/4)^3+6*(SQRT(6)/12+S~
QRT(2)*#i/4)^2+SQRT(x^2-x+1)-x+2*(SQRT(6)/12+SQRT(2)*#i/4))

" If -1<n<1, LN(z^n) -> n*LN(z) "

-SQRT(2)*ATAN((SQRT(3)+SQRT(2))*(2*SQRT(x^2-x+1)-2*x+SQRT(6)-1))~
/2-SQRT(6)*LN(-SQRT(x^2-x+1)*(2*x-SQRT(6)+1)+2*x^2-SQRT(6)*x-SQR~
T(6)+4)/12+SQRT(2)*pi/4+(SQRT(6)/12-SQRT(2)*#i/4)*(LN(-SQRT(x^2-~
x+1)*(2*x+SQRT(6)+1)+2*x^2+SQRT(6)*x+SQRT(6)+4)/2+#i*ATAN(-SQRT(~
3)/2-SQRT(2)/2,SQRT(x^2-x+1)-x-SQRT(6)/2-1/2))+(SQRT(6)/12+SQRT(~
2)*#i/4)*LN(24*(SQRT(6)/12+SQRT(2)*#i/4)^3+6*(SQRT(6)/12+SQRT(2)~
*#i/4)^2+SQRT(x^2-x+1)-x+2*(SQRT(6)/12+SQRT(2)*#i/4))

" ATAN(y,x) -> pi*SIGN(y)/2-ATAN(x/y) "

-SQRT(2)*ATAN((SQRT(3)+SQRT(2))*(2*SQRT(x^2-x+1)-2*x+SQRT(6)-1))~
/2-SQRT(6)*LN(-SQRT(x^2-x+1)*(2*x-SQRT(6)+1)+2*x^2-SQRT(6)*x-SQR~
T(6)+4)/12+SQRT(2)*pi/4+(SQRT(6)/12-SQRT(2)*#i/4)*(LN(-SQRT(x^2-~
x+1)*(2*x+SQRT(6)+1)+2*x^2+SQRT(6)*x+SQRT(6)+4)/2+#i*(pi*SIGN(-S~
QRT(3)/2-SQRT(2)/2)/2+ATAN((SQRT(x^2-x+1)-x-SQRT(6)/2-1/2)/(SQRT~
(3)/2+SQRT(2)/2))))+(SQRT(6)/12+SQRT(2)*#i/4)*LN(24*(SQRT(6)/12+~
SQRT(2)*#i/4)^3+6*(SQRT(6)/12+SQRT(2)*#i/4)^2+SQRT(x^2-x+1)-x+2*~
(SQRT(6)/12+SQRT(2)*#i/4))

" 1/(z+w) -> (z-w)/(z^2-w^2) "

-SQRT(2)*ATAN((SQRT(3)+SQRT(2))*(2*SQRT(x^2-x+1)-2*x+SQRT(6)-1))~
/2-SQRT(6)*LN(-SQRT(x^2-x+1)*(2*x-SQRT(6)+1)+2*x^2-SQRT(6)*x-SQR~
T(6)+4)/12+SQRT(2)*pi/4+(SQRT(6)/12-SQRT(2)*#i/4)*(LN(-SQRT(x^2-~
x+1)*(2*x+SQRT(6)+1)+2*x^2+SQRT(6)*x+SQRT(6)+4)/2+#i*(-pi/2+ATAN~
(4*(SQRT(x^2-x+1)-x-SQRT(6)/2-1/2)*(SQRT(3)/2-SQRT(2)/2))))+(SQR~
T(6)/12+SQRT(2)*#i/4)*LN(24*(SQRT(6)/12+SQRT(2)*#i/4)^3+6*(SQRT(~
6)/12+SQRT(2)*#i/4)^2+SQRT(x^2-x+1)-x+2*(SQRT(6)/12+SQRT(2)*#i/4~
))

" (a+b*#i)*(c+d*#i) -> (a*c-b*d)+(a*d+b*c)*#i "

-SQRT(2)*ATAN((SQRT(3)+SQRT(2))*(2*SQRT(x^2-x+1)-2*x+SQRT(6)-1))~
/2-SQRT(6)*LN(-SQRT(x^2-x+1)*(2*x-SQRT(6)+1)+2*x^2-SQRT(6)*x-SQR~
T(6)+4)/12+SQRT(2)*pi/4+SQRT(6)*LN(-SQRT(x^2-x+1)*(2*x+SQRT(6)+1~
)+2*x^2+SQRT(6)*x+SQRT(6)+4)/(2*12)-SQRT(2)*(pi/2-ATAN((SQRT(3)-~
SQRT(2))*(2*SQRT(x^2-x+1)-2*x-SQRT(6)-1)))/4-#i*(SQRT(2)*LN(-SQR~
T(x^2-x+1)*(2*x+SQRT(6)+1)+2*x^2+SQRT(6)*x+SQRT(6)+4)/(2*4)+SQRT~
(6)*(pi/2-ATAN((SQRT(3)-SQRT(2))*(2*SQRT(x^2-x+1)-2*x-SQRT(6)-1)~
))/12)+(SQRT(6)/12+SQRT(2)*#i/4)*LN(24*(SQRT(6)/12+SQRT(2)*#i/4)~
^3+6*(SQRT(6)/12+SQRT(2)*#i/4)^2+SQRT(x^2-x+1)-x+2*(SQRT(6)/12+S~
QRT(2)*#i/4))

" LN(x+y*#i) -> LN(|x+y*#i|) + #i*ATAN(y,x) "

-SQRT(2)*ATAN((SQRT(3)+SQRT(2))*(2*SQRT(x^2-x+1)-2*x+SQRT(6)-1))~
/2-SQRT(6)*LN(-SQRT(x^2-x+1)*(2*x-SQRT(6)+1)+2*x^2-SQRT(6)*x-SQR~
T(6)+4)/12+SQRT(2)*ATAN((SQRT(3)-SQRT(2))*(2*SQRT(x^2-x+1)-2*x-S~
QRT(6)-1))/4+SQRT(6)*LN(-SQRT(x^2-x+1)*(2*x+SQRT(6)+1)+2*x^2+SQR~
T(6)*x+SQRT(6)+4)/24+SQRT(2)*pi/8+#i*(SQRT(6)*ATAN((SQRT(3)-SQRT~
(2))*(2*SQRT(x^2-x+1)-2*x-SQRT(6)-1))/12-SQRT(2)*LN(-SQRT(x^2-x+~
1)*(2*x+SQRT(6)+1)+2*x^2+SQRT(6)*x+SQRT(6)+4)/8-SQRT(6)*pi/24)+(~
SQRT(6)/12+SQRT(2)*#i/4)*(LN(ABS(SQRT(x^2-x+1)-x-SQRT(6)/2-1/2+#~
i*(SQRT(3)/2+SQRT(2)/2)))+#i*ATAN(SQRT(3)/2+SQRT(2)/2,SQRT(x^2-x~
+1)-x-SQRT(6)/2-1/2))

" ABS(x+#i*y) -> SQRT(x^2+y^2) "

-SQRT(2)*ATAN((SQRT(3)+SQRT(2))*(2*SQRT(x^2-x+1)-2*x+SQRT(6)-1))~
/2-SQRT(6)*LN(-SQRT(x^2-x+1)*(2*x-SQRT(6)+1)+2*x^2-SQRT(6)*x-SQR~
T(6)+4)/12+SQRT(2)*ATAN((SQRT(3)-SQRT(2))*(2*SQRT(x^2-x+1)-2*x-S~
QRT(6)-1))/4+SQRT(6)*LN(-SQRT(x^2-x+1)*(2*x+SQRT(6)+1)+2*x^2+SQR~
T(6)*x+SQRT(6)+4)/24+SQRT(2)*pi/8+#i*(SQRT(6)*ATAN((SQRT(3)-SQRT~
(2))*(2*SQRT(x^2-x+1)-2*x-SQRT(6)-1))/12-SQRT(2)*LN(-SQRT(x^2-x+~
1)*(2*x+SQRT(6)+1)+2*x^2+SQRT(6)*x+SQRT(6)+4)/8-SQRT(6)*pi/24)+(~
SQRT(6)/12+SQRT(2)*#i/4)*(LN(SQRT((SQRT(x^2-x+1)-x-SQRT(6)/2-1/2~
)^2+(SQRT(3)/2+SQRT(2)/2)^2))+#i*ATAN(SQRT(3)/2+SQRT(2)/2,SQRT(x~
^2-x+1)-x-SQRT(6)/2-1/2))

" If -1<n<1, LN(z^n) -> n*LN(z) "

-SQRT(2)*ATAN((SQRT(3)+SQRT(2))*(2*SQRT(x^2-x+1)-2*x+SQRT(6)-1))~
/2-SQRT(6)*LN(-SQRT(x^2-x+1)*(2*x-SQRT(6)+1)+2*x^2-SQRT(6)*x-SQR~
T(6)+4)/12+SQRT(2)*ATAN((SQRT(3)-SQRT(2))*(2*SQRT(x^2-x+1)-2*x-S~
QRT(6)-1))/4+SQRT(6)*LN(-SQRT(x^2-x+1)*(2*x+SQRT(6)+1)+2*x^2+SQR~
T(6)*x+SQRT(6)+4)/24+SQRT(2)*pi/8+#i*(SQRT(6)*ATAN((SQRT(3)-SQRT~
(2))*(2*SQRT(x^2-x+1)-2*x-SQRT(6)-1))/12-SQRT(2)*LN(-SQRT(x^2-x+~
1)*(2*x+SQRT(6)+1)+2*x^2+SQRT(6)*x+SQRT(6)+4)/8-SQRT(6)*pi/24)+(~
SQRT(6)/12+SQRT(2)*#i/4)*(LN(-SQRT(x^2-x+1)*(2*x+SQRT(6)+1)+2*x^~
2+SQRT(6)*x+SQRT(6)+4)/2+#i*ATAN(SQRT(3)/2+SQRT(2)/2,SQRT(x^2-x+~
1)-x-SQRT(6)/2-1/2))

" ATAN(y,x) -> pi*SIGN(y)/2-ATAN(x/y) "

-SQRT(2)*ATAN((SQRT(3)+SQRT(2))*(2*SQRT(x^2-x+1)-2*x+SQRT(6)-1))~
/2-SQRT(6)*LN(-SQRT(x^2-x+1)*(2*x-SQRT(6)+1)+2*x^2-SQRT(6)*x-SQR~
T(6)+4)/12+SQRT(2)*ATAN((SQRT(3)-SQRT(2))*(2*SQRT(x^2-x+1)-2*x-S~
QRT(6)-1))/4+SQRT(6)*LN(-SQRT(x^2-x+1)*(2*x+SQRT(6)+1)+2*x^2+SQR~
T(6)*x+SQRT(6)+4)/24+SQRT(2)*pi/8+#i*(SQRT(6)*ATAN((SQRT(3)-SQRT~
(2))*(2*SQRT(x^2-x+1)-2*x-SQRT(6)-1))/12-SQRT(2)*LN(-SQRT(x^2-x+~
1)*(2*x+SQRT(6)+1)+2*x^2+SQRT(6)*x+SQRT(6)+4)/8-SQRT(6)*pi/24)+(~
SQRT(6)/12+SQRT(2)*#i/4)*(LN(-SQRT(x^2-x+1)*(2*x+SQRT(6)+1)+2*x^~
2+SQRT(6)*x+SQRT(6)+4)/2+#i*(pi*SIGN(SQRT(3)/2+SQRT(2)/2)/2-ATAN~
((SQRT(x^2-x+1)-x-SQRT(6)/2-1/2)/(SQRT(3)/2+SQRT(2)/2))))

" 1/(z+w) -> (z-w)/(z^2-w^2) "

-SQRT(2)*ATAN((SQRT(3)+SQRT(2))*(2*SQRT(x^2-x+1)-2*x+SQRT(6)-1))~
/2-SQRT(6)*LN(-SQRT(x^2-x+1)*(2*x-SQRT(6)+1)+2*x^2-SQRT(6)*x-SQR~
T(6)+4)/12+SQRT(2)*ATAN((SQRT(3)-SQRT(2))*(2*SQRT(x^2-x+1)-2*x-S~
QRT(6)-1))/4+SQRT(6)*LN(-SQRT(x^2-x+1)*(2*x+SQRT(6)+1)+2*x^2+SQR~
T(6)*x+SQRT(6)+4)/24+SQRT(2)*pi/8+#i*(SQRT(6)*ATAN((SQRT(3)-SQRT~
(2))*(2*SQRT(x^2-x+1)-2*x-SQRT(6)-1))/12-SQRT(2)*LN(-SQRT(x^2-x+~
1)*(2*x+SQRT(6)+1)+2*x^2+SQRT(6)*x+SQRT(6)+4)/8-SQRT(6)*pi/24)+(~
SQRT(6)/12+SQRT(2)*#i/4)*(LN(-SQRT(x^2-x+1)*(2*x+SQRT(6)+1)+2*x^~
2+SQRT(6)*x+SQRT(6)+4)/2+#i*(pi/2-ATAN(4*(SQRT(x^2-x+1)-x-SQRT(6~
)/2-1/2)*(SQRT(3)/2-SQRT(2)/2))))

" (a+b*#i)*(c+d*#i) -> (a*c-b*d)+(a*d+b*c)*#i "

-SQRT(2)*ATAN((SQRT(3)+SQRT(2))*(2*SQRT(x^2-x+1)-2*x+SQRT(6)-1))~
/2-SQRT(6)*LN(-SQRT(x^2-x+1)*(2*x-SQRT(6)+1)+2*x^2-SQRT(6)*x-SQR~
T(6)+4)/12+SQRT(2)*ATAN((SQRT(3)-SQRT(2))*(2*SQRT(x^2-x+1)-2*x-S~
QRT(6)-1))/4+SQRT(6)*LN(-SQRT(x^2-x+1)*(2*x+SQRT(6)+1)+2*x^2+SQR~
T(6)*x+SQRT(6)+4)/24+SQRT(2)*pi/8+#i*(SQRT(6)*ATAN((SQRT(3)-SQRT~
(2))*(2*SQRT(x^2-x+1)-2*x-SQRT(6)-1))/12-SQRT(2)*LN(-SQRT(x^2-x+~
1)*(2*x+SQRT(6)+1)+2*x^2+SQRT(6)*x+SQRT(6)+4)/8-SQRT(6)*pi/24)+S~
QRT(6)*LN(-SQRT(x^2-x+1)*(2*x+SQRT(6)+1)+2*x^2+SQRT(6)*x+SQRT(6)~
+4)/(2*12)+SQRT(2)*(ATAN((SQRT(3)-SQRT(2))*(2*SQRT(x^2-x+1)-2*x-~
SQRT(6)-1))-pi/2)/4+#i*(SQRT(2)*LN(-SQRT(x^2-x+1)*(2*x+SQRT(6)+1~
)+2*x^2+SQRT(6)*x+SQRT(6)+4)/(2*4)+SQRT(6)*(pi/2-ATAN((SQRT(3)-S~
QRT(2))*(2*SQRT(x^2-x+1)-2*x-SQRT(6)-1)))/12)

" one final step "

SQRT(2)*ATAN((SQRT(3)-SQRT(2))*(2*SQRT(x^2-x+1)-2*x-SQRT(6)-1))/~
2-SQRT(2)*ATAN((SQRT(3)+SQRT(2))*(2*SQRT(x^2-x+1)-2*x+SQRT(6)-1)~
)/2-SQRT(6)*LN((SQRT(x^2-x+1)*(2*x-SQRT(6)+1)-2*x^2+SQRT(6)*x+SQ~
RT(6)-4)/(SQRT(x^2-x+1)*(2*x+SQRT(6)+1)-2*x^2-SQRT(6)*x-SQRT(6)-~
4))/12

However, the combined logarithmic term can be manually simplified
further to:

-SQRT(6)*LN((5-2*SQRT(6))*(SQRT(3)*SQRT(x^2-x+1)-SQRT(2)*(x-1))
/(SQRT(3)*SQRT(x^2-x+1)+SQRT(2)*(x-1)))/12

where the factor 5-2*SQRT(6) represents an additive constant that can be
omitted.

Martin.

clicl...@freenet.de

unread,
Oct 23, 2011, 8:03:54 AM10/23/11
to

Albert Rich schrieb:
>
> On Oct 14, 10:09 pm, cliclic...@freenet.de wrote:
> >
> > Derive 6.10 returns these elementary integrals unevaluated:
> >
> > INT(1/((x^2 + x + 1)*SQRT(x^2 - x + 1)), x)
> >
> > INT(x/((x^2 + x + 1)*SQRT(x^2 - x + 1)), x)
> >
> > The optimal antiderivatives are:
> >
> > ATANH(SQRT(6)*(x - 1)/(3*SQRT(x^2 - x + 1)))/SQRT(6)
> > + ATAN(SQRT(2)*(x + 1)/SQRT(x^2 - x + 1))/SQRT(2)
> >
> > ATANH(SQRT(6)*(x - 1)/(3*SQRT(x^2 - x + 1)))/SQRT(6)
> > - ATAN(SQRT(2)*(x + 1)/SQRT(x^2 - x + 1))/SQRT(2)
> >
> > Is there an integrator clever enough to find them?
> >
>
> [...]
>
> Int[(1+x)/((1+x+x^2)*Sqrt[1-x+x^2]),x] is an instance of the rule:
>
> If c*d^2-a*e^2=0, c*f=a*h and (b*h-c*g)/(e*(e*g+2*d*h))>0, then
>
> Int[(d+e*x)/((f+g*x+h*x^2)*Sqrt[a+b*x+c*x^2]),x] -->
> -2*e/((e*g+2*d*h)*Sqrt[(b*h-c*g)/(e*(e*g+2*d*h))])*
> ArcTan[Sqrt[(b*h-c*g)/(e*(e*g+2*d*h))]*((d-e*x)/Sqrt[a+b*x
> +c*x^2])]
>
> Int[(1-x)/((1+x+x^2)*Sqrt[1-x+x^2]),x] is an instance of the rule:
>
> If c*d^2-a*e^2=0, c*f=a*h and (b*h-c*g)/(e*(e*g+2*d*h))<0, then
>
> Int[(d+e*x)/((f+g*x+h*x^2)*Sqrt[a+b*x+c*x^2]),x] -->
> -2*e/((e*g+2*d*h)*Sqrt[(b*h-c*g)/(e*(e*g+2*d*h))])*
> ArcTan[Sqrt[(b*h-c*g)/(e*(e*g+2*d*h))]*((d-e*x)/Sqrt[a+b*x
> +c*x^2])]
>
> Of course, these two terminal rules are of not much value unless
> recursive rules can be found for reducing integrands of the form (d
> +e*x)*(a+b*x+c*x^2)^m*(f+g*x+h*x^2)^n to these terminal cases...
>

It must be emphasized that these rules do not apply for arbitrary
parameters a,b,c,f,g,h.

I disagree with the last statement. Integrals of the type INT((A + B*x)
/ ((a + b*x + c*x^2) * SQRT(d + e*x + f*x^2)), x) routinely arise when
an integrand R(x, SQRT(d + e*x + f*x^2)) is rewritten as R1(x) + R2(x) *
SQRT(d + e*x + f*x^2) and R2(x) is subsequently expanded into partial
fractions without introducing complex factors; here R, R1, and R2 denote
rational functions of their arguments. I suspect that Rubi was handling
such integrals by means of Euler substitutions.

Martin.

Waldek Hebisch

unread,
Oct 23, 2011, 4:29:11 PM10/23/11
to
...
> " LN(x+y*#i) -> LN(|x+y*#i|) + #i*ATAN(y,x) "
...
> " ABS(x+#i*y) -> SQRT(x^2+y^2) "
...
> " If -1<n<1, LN(z^n) -> n*LN(z) "
...
> " ATAN(y,x) -> pi*SIGN(y)/2-ATAN(x/y) "
...
> " 1/(z+w) -> (z-w)/(z^2-w^2) "
...
> " (a+b*#i)*(c+d*#i) -> (a*c-b*d)+(a*d+b*c)*#i "

FriCAS proceeds in similar way, however it does not perform
the " If -1<n<1, LN(z^n) -> n*LN(z) " simplification. Also,
ATAN(x/y) may produce branch cuts on real line so FriCAS
uses more complicated formula (essentially this is
LN(z) -> 2*LN(SQRT(z))), which leads to nested roots.

--
Waldek Hebisch
heb...@math.uni.wroc.pl
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