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Source: Risch algorithm

In symbolic computation, the Risch algorithm is a method of indefinite integration used in some computer algebra systems to find antiderivatives. It is named after the American mathematician Robert Henry Risch, a specialist in computer algebra who developed it in 1968.
The algorithm transforms the problem of integration into a problem in algebra. It is based on the form of the function being integrated and on methods for integrating rational functions, radicals, logarithms, and exponential functions. Risch called it a decision procedure, because it is a method for deciding whether a function has an elementary function as an indefinite integral, and if it does, for determining that indefinite integral. However, the algorithm does not always succeed in identifying whether or not the antiderivative of a given function in fact can be expressed in terms of elementary functions.
The complete description of the Risch algorithm takes over 100 pages. The Risch–Norman algorithm is a simpler, faster, but less powerful variant that was developed in 1976 by Arthur Norman.
Some significant progress has been made in computing the logarithmic part of a mixed transcendental-algebraic integral by Brian L. Miller.

Description

Risch

algorithm

Risch

algorithm

Risch developed a method that allows one to consider only a finite set of functions of Liouville's form.
The intuition for the Risch algorithm comes from the behavior of the exponential and logarithm functions under differentiation. For the function f eg, where f and g are differentiable functions, we have

(

f
⋅

e

g

)

′

=

(

f

′

+
f
⋅

g

′

)

⋅

e

g

,

{\displaystyle \left(f\cdot e^{g}\right)^{\prime }=\left(f^{\prime }+f\cdot g^{\prime }\right)\cdot e^{g},\,}

so if eg were in the result of an indefinite integration, it should be expected to be inside the integral. Also, as

(

f
⋅
(
ln
⁡
g

)

n

)

′

=

f

′

(

ln
⁡
g

)

n

+
n
f

g

′

g

(

ln
⁡
g

)

n
−
1

{\displaystyle \left(f\cdot (\ln g)^{n}\right)^{\prime }=f^{\prime }\left(\ln g\right)^{n}+nf{\frac {g^{\prime }}{g}}\left(\ln g\right)^{n-1}}

then if (ln g)n were in the result of an integration, then only a few powers of the logarithm should be expected.

Problem examples

Finding an elementary antiderivative is very sensitive to details. For instance, the following algebraic function (posted to sci.math.symbolic by Henri Cohen in 1993) has an elementary antiderivative, as Wolfram Mathematica since version 13 shows (however, Mathematica does not use the Risch algorithm to compute this integral):

f
(
x
)
=

x

x

4

+
10

x

2

−
96
x
−
71

,

{\displaystyle f(x)={\frac {x}{\sqrt {x^{4}+10x^{2}-96x-71}}},}

namely:

F
(
x
)
=
−

1
8

ln

(

(

x

6

+
15

x

4

−
80

x

3

+
27

x

2

−
528
x
+
781
)

x

4

+
10

x

2

−
96
x
−
71

−

(

x

8

+
20

x

6

−
128

x

5

+
54

x

4

−
1408

x

3

+
3124

x

2

+
10001
)

)

+
C
.

{\displaystyle {\begin{aligned}F(x)=-{\frac {1}{8}}\ln &\,{\Big (}(x^{6}+15x^{4}-80x^{3}+27x^{2}-528x+781){\sqrt {x^{4}+10x^{2}-96x-71}}{\Big .}\\&{}-{\Big .}(x^{8}+20x^{6}-128x^{5}+54x^{4}-1408x^{3}+3124x^{2}+10001){\Big )}+C.\end{aligned}}}

But if the constant term 71 is changed to 72, it is not possible to represent the antiderivative in terms of elementary functions, as FriCAS also shows. Some computer algebra systems may here return an antiderivative in terms of non-elementary functions (i.e. elliptic integrals), which are outside the scope of the Risch algorithm. For example, Mathematica returns a result with the functions EllipticPi and EllipticF. This integral was solved by Chebyshev (and in what cases it is elementary), but the strict proof for it was ultimately done by Zolotarev.
The following is a more complex example that involves both algebraic and transcendental functions:

f
(
x
)
=

x

2

+
2
x
+
1
+
(
3
x
+
1
)

x
+
ln
⁡
x

x

x
+
ln
⁡
x

(

x
+

x
+
ln
⁡
x

)

.

{\displaystyle f(x)={\frac {x^{2}+2x+1+(3x+1){\sqrt {x+\ln x}}}{x\,{\sqrt {x+\ln x}}\left(x+{\sqrt {x+\ln x}}\right)}}.}

In fact, the antiderivative of this function has a fairly short form that can be found using substitution

u
=
x
+

x
+
ln
⁡
x

{\displaystyle u=x+{\sqrt {x+\ln x}}}

(SymPy can solve it while FriCAS fails with "implementation incomplete (constant residues)" error in Risch algorithm):

F
(
x
)
=
2

(

x
+
ln
⁡
x

+
ln
⁡

(

x
+

x
+
ln
⁡
x

)

)

+
C
.

{\displaystyle F(x)=2\left({\sqrt {x+\ln x}}+\ln \left(x+{\sqrt {x+\ln x}}\right)\right)+C.}

Some Davenport "theorems" are still being clarified. For example in 2020 a counterexample to such a "theorem" was found, where it turns out that an elementary antiderivative exists after all.

Implementation

Transforming Risch's theoretical algorithm into an algorithm that can be effectively executed by a computer was a complex task which took a long time.
The case of the purely transcendental functions (which do not involve roots of polynomials) is relatively easy and was implemented early in most computer algebra systems. The first implementation was done by Joel Moses in Macsyma soon after the publication of Risch's paper.
The case of purely algebraic functions was partially solved and implemented in Reduce by James H. Davenport - for simplicity it could only deal with square roots and repeated square roots and not general Radicals or other non-quadratic algebraic relations between variables.
The general case was solved and almost fully implemented in Scratchpad, a precursor of Axiom, by Manuel Bronstein, and is now being developed in Axiom's fork, FriCAS. However, the implementation did not include some of the branches for special cases completely. Currently, there is no known full implementation of the Risch algorithm.

Decidability

The Risch algorithm applied to general elementary functions is not an algorithm but a semi-algorithm because it needs to check, as a part of its operation, if certain expressions are equivalent to zero (constant problem), in particular in the constant field. For expressions that involve only functions commonly taken to be elementary it is not known whether an algorithm performing such a check exists or not (current computer algebra systems use heuristics); moreover, if one adds the absolute value function to the list of elementary functions, it is known that no such algorithm exists; see Richardson's theorem.
Note that this issue also arises in the polynomial division algorithm; this algorithm will fail if it cannot correctly determine whether coefficients vanish identically. Virtually every non-trivial algorithm relating to polynomials uses the polynomial division algorithm, the Risch algorithm included. If the constant field is computable, i.e., for elements not dependent on x, the problem of zero-equivalence is decidable, then the Risch algorithm is a complete algorithm. Examples of computable constant fields are Q and Q(y), i.e., rational numbers and rational functions in y with rational number coefficients, respectively, where y is an indeterminate that does not depend on x.
This is also an issue in the Gaussian elimination matrix algorithm (or any algorithm that can compute the nullspace of a matrix), which is also necessary for many parts of the Risch algorithm. Gaussian elimination will produce incorrect results if it cannot correctly determine if a pivot is identically zero.

Notes

References

Bronstein, Manuel (1990). "Integration of elementary functions". Journal of Symbolic Computation. 9 (2): 117–173. doi:10.1016/s0747-7171(08)80027-2.
Bronstein, Manuel (1998). "Symbolic Integration Tutorial" (PDF). ISSAC'98, Rostock (August 1998) and Differential Algebra Workshop, Rutgers.
Bronstein, Manuel (2005). Symbolic Integration I. Springer. ISBN 3-540-21493-3.
Davenport, James H. (1981). On the integration of algebraic functions. Lecture Notes in Computer Science. Vol. 102. Springer. ISBN 978-3-540-10290-8.
Geddes, Keith O.; Czapor, Stephen R.; Labahn, George (1992). Algorithms for computer algebra. Boston, MA: Kluwer Academic Publishers. pp. xxii+585. Bibcode:1992afca.book.....G. doi:10.1007/b102438. ISBN 0-7923-9259-0.
Moses, Joel (2012). "Macsyma: A personal history". Journal of Symbolic Computation. 47 (2): 123–130. doi:10.1016/j.jsc.2010.08.018.
Risch, R. H. (1969). "The problem of integration in finite terms". Transactions of the American Mathematical Society. 139. American Mathematical Society: 167–189. doi:10.2307/1995313. JSTOR 1995313.
Risch, R. H. (1970). "The solution of the problem of integration in finite terms". Bulletin of the American Mathematical Society. 76 (3): 605–608. doi:10.1090/S0002-9904-1970-12454-5.
Rosenlicht, Maxwell (1972). "Integration in finite terms". American Mathematical Monthly. 79 (9). Mathematical Association of America: 963–972. doi:10.2307/2318066. JSTOR 2318066.