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integration example

acer 31076 Maple
May 06 2020
5

There is an interesting preprint here, on a method for integration of some pseudo-elliptic integrals.

It was mentioned in a (sci.math.symbolic) usenet thread, which can be accessed via Google Groups here.

Inside the paper, the author mentions that the following is possible for some cases -- to first convert to RootOf form.

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restart;
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kernelopts(version);

`Maple 2020.0, X86 64 LINUX, Mar 4 2020, Build ID 1455132`

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ig:=((-3+x^2)*(1-6*x^2+x^4)^(-1/4))/(-1+x^2);

(x^2-3)/((x^4-6*x^2+1)^(1/4)*(x^2-1))

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int(ig,x);

int((x^2-3)/((x^4-6*x^2+1)^(1/4)*(x^2-1)), x)

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infolevel[int] := 2:
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S:=simplify([allvalues(int(convert(ig,RootOf),x))],size):

Stage1: first-stage indefinite integration

Stage2: second-stage indefinite integration

Norman: enter Risch-Norman integrator

Norman: exit Risch-Norman integrator

int/algrisch/int: Risch/Trager's algorithm for algebraic function

int/algrisch/int: entering integrator at time 9.029

int/algrisch/int: function field has degree 4

int/algrisch/int: computation of an integral basis: start time 9.031

int/algrisch/int: computation of an integral basis: end time 9.038

int/algrisch/int: normalization at infinity: start time 9.039

int/algrisch/int: normalization at infinity: end time 9.048

int/algrisch/int: genus of the function field 3

int/algrisch/int: computation of the algebraic part: start time 9.059

int/algrisch/int: computation of the algebraic part: end time 9.060

int/algrisch/int: computation of the transcendental part: start time 9.063

int/algrisch/transcpar: computing a basis for the residues at time 9.068

int/algrisch/residues: computing a splitting field at time 9.068

int/algrisch/transcpar: basis for the residues computed at time 9.103

int/algrisch/transcpar: dimension is 2

int/algrisch/transcpar: building divisors at time 9.300

int/algrisch/transcpar: testing divisors for principality at time 9.605

int/algrisch/goodprime: searching for a good prime at time 9.606

int/algrisch/goodprime: good prime found at time 9.704

int/algrisch/goodprime: searching for a good prime at time 9.704

int/algrisch/goodprime: good prime found at time 9.762

int/algrisch/areprinc: the divisor is principal: time 10.084

int/algrisch/areprinc: the divisor is principal: time 11.833

int/algrisch/transcpar: divisors proven pincipal at time at time 11.833

int/algrisch/transcpar: generators computed at time 11.834

int/algrisch/transcpar: orders are [1 1]

int/algrisch/transcpar: check that the candidate is an actual antiderivative

int/algrisch/transcpar: the antiderivative is elementary

int/algrisch/transcpar: antiderivative is (1/2)*ln((RootOf(_Z^4*_z^4-_z^4+6*_z^2-1 index = 1)^3*_z^3-RootOf(_Z^4*_z^4-_z^4+6*_z^2-1 index = 1)^2*_z^4+RootOf(_Z^4*_z^4-_z^4+6*_z^2-1 index = 1)^2*_z^2-3*_z^3*RootOf(_Z^4*_z^4-_z^4+6*_z^2-1 index = 1)+RootOf(_Z^4*_z^4-_z^4+6*_z^2-1 index = 1)*_z-5*_z^2+1)/((_z+1)*(_z-1)*_z^2))+(1/2)*RootOf(_Z^2+1)*ln((RootOf(_Z^4*_z^4-_z^4+6*_z^2-1 index = 1)^2*RootOf(_Z^2+1)*_z^4+RootOf(_Z^4*_z^4-_z^4+6*_z^2-1 index = 1)^3*_z^3-RootOf(_Z^2+1)*RootOf(_Z^4*_z^4-_z^4+6*_z^2-1 index = 1)^2*_z^2+3*_z^3*RootOf(_Z^4*_z^4-_z^4+6*_z^2-1 index = 1)-5*RootOf(_Z^2+1)*_z^2-RootOf(_Z^4*_z^4-_z^4+6*_z^2-1 index = 1)*_z+RootOf(_Z^2+1))/((_z+1)*(_z-1)*_z^2))

int/algrisch/int: computation of the transcendental part: end time 12.040

int/algrisch/int: exiting integrator for algebraic functions at time 12.041

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S[1];

(1/2)*ln(-((x^4-6*x^2+1)^(3/4)*x+(x^4-6*x^2+1)^(1/2)*x^2+(x^4-6*x^2+1)^(1/4)*x^3+x^4-(x^4-6*x^2+1)^(1/2)-3*(x^4-6*x^2+1)^(1/4)*x-5*x^2)/((x+1)*(x-1)))+((1/2)*I)*ln((-(x^4-6*x^2+1)^(3/4)*x+(I*x^2-I)*(x^4-6*x^2+1)^(1/2)+(x^3-3*x)*(x^4-6*x^2+1)^(1/4)-I*x^2*(x^2-5))/((x+1)*(x-1)))

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S[2];

(1/2)*ln(-((x^4-6*x^2+1)^(3/4)*x+(x^4-6*x^2+1)^(1/2)*x^2+(x^4-6*x^2+1)^(1/4)*x^3+x^4-(x^4-6*x^2+1)^(1/2)-3*(x^4-6*x^2+1)^(1/4)*x-5*x^2)/((x+1)*(x-1)))-((1/2)*I)*ln((-(x^4-6*x^2+1)^(3/4)*x+(-I*x^2+I)*(x^4-6*x^2+1)^(1/2)+(x^3-3*x)*(x^4-6*x^2+1)^(1/4)+I*x^2*(x^2-5))/((x+1)*(x-1)))

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simplify( diff(S[1],x) - ig ), simplify( diff(S[2],x) - ig );

0, 0

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