Revisions to Does there exist a complete implementation of the Risch algorithm?

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edited Oct 18, 2020 at 11:21

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My experiments suggest Maple has the best implementation of the Risch-Trager-Bronstein algorithm for the integration of purely algebraic integrals in terms of elementary functions (ref: table 1, section 3 of Sam Blake, https://arxiv.org/pdf/2004.04910.pdfA Simple Method for Computing Some Pseudo-Elliptic Integrals in Terms of Elementary Functions, arXiv:2004.04910). However, Maple's implementation does not integrate expressions containing parameters or nested radicals (both of which has some support in AXIOM and FriCAS).

$$\int\frac{\left(p x^2-q\right) \left(p x^2-x+q\right)dx}{x \left(p x^2+2 x+q\right) \sqrt{2 p^2x^4+2 p x^3+(4 p q+1) x^2+2 q x+2 q^2}} = - \frac{1}{\sqrt{2}}\log (x) + \frac{1}{\sqrt{2}}\log \left(\sqrt{2} y +2 p x^2+x+2q\right) - \frac{3}{\sqrt{5}}\tanh ^{-1}\left(\frac{\sqrt{5} y}{3 p x^2+3 q+x}\right),$$ where $y=\sqrt{2 p^2 x^4+2 p x^3+(4 pq+1)x^2+2 q x+2 q^2}$. My heuristic in the heuristic previously-linked paper computes this integral quickly with the substitution $u=\frac{px^2+q}{p x}$.

My experiments suggest Maple has the best implementation of the Risch-Trager-Bronstein algorithm for the integration of purely algebraic integrals in terms of elementary functions (ref: table 1, section 3 of https://arxiv.org/pdf/2004.04910.pdf). However, Maple's implementation does not integrate expressions containing parameters or nested radicals (both of which has some support in AXIOM and FriCAS).

$$\int\frac{\left(p x^2-q\right) \left(p x^2-x+q\right)dx}{x \left(p x^2+2 x+q\right) \sqrt{2 p^2x^4+2 p x^3+(4 p q+1) x^2+2 q x+2 q^2}} = - \frac{1}{\sqrt{2}}\log (x) + \frac{1}{\sqrt{2}}\log \left(\sqrt{2} y +2 p x^2+x+2q\right) - \frac{3}{\sqrt{5}}\tanh ^{-1}\left(\frac{\sqrt{5} y}{3 p x^2+3 q+x}\right),$$ where $y=\sqrt{2 p^2 x^4+2 p x^3+(4 pq+1)x^2+2 q x+2 q^2}$. My heuristic computes this integral quickly with the substitution $u=\frac{px^2+q}{p x}$.

My experiments suggest Maple has the best implementation of the Risch-Trager-Bronstein algorithm for the integration of purely algebraic integrals in terms of elementary functions (ref: table 1, section 3 of Sam Blake, A Simple Method for Computing Some Pseudo-Elliptic Integrals in Terms of Elementary Functions, arXiv:2004.04910). However, Maple's implementation does not integrate expressions containing parameters or nested radicals (both of which has some support in AXIOM and FriCAS).

$$\int\frac{\left(p x^2-q\right) \left(p x^2-x+q\right)dx}{x \left(p x^2+2 x+q\right) \sqrt{2 p^2x^4+2 p x^3+(4 p q+1) x^2+2 q x+2 q^2}} = - \frac{1}{\sqrt{2}}\log (x) + \frac{1}{\sqrt{2}}\log \left(\sqrt{2} y +2 p x^2+x+2q\right) - \frac{3}{\sqrt{5}}\tanh ^{-1}\left(\frac{\sqrt{5} y}{3 p x^2+3 q+x}\right),$$ where $y=\sqrt{2 p^2 x^4+2 p x^3+(4 pq+1)x^2+2 q x+2 q^2}$. My heuristic in the previously-linked paper computes this integral quickly with the substitution $u=\frac{px^2+q}{p x}$.

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edited Oct 18, 2020 at 3:35

Sam Blake

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$$\int \frac{\left(\sqrt{x}+1\right) \left(e^{2x \sqrt{x}} -a\right) \sqrt{a^2+2 a x e^{2 \sqrt{x}} +cx e^{2 \sqrt{x}} +x^2 e^{4 \sqrt{x}}}}{2 x \sqrt{x}e^{\sqrt{x}} \left(a+x e^{2 \sqrt{x}} \right)} dx.$$$$\int \frac{\left(\sqrt{x}+1\right) \left(e^{2x \sqrt{x}} -a\right) \sqrt{a^2+2 a x e^{2 \sqrt{x}} +cx e^{2 \sqrt{x}} +x^2 e^{4 \sqrt{x}}}}{x \sqrt{x}e^{\sqrt{x}} \left(a+x e^{2 \sqrt{x}} \right)} dx.$$

(1) -> integrate(((-a+exp(2*x^(1/2))*x)*x^(-3/2)*(1+x^(1/2))*(a^2+2*a*exp(2*x^(1/2))*x+c*exp(2*x^(1/2))*x+exp(4*x^(1/2))*x^2)^(1/2))/(2*expexp(x^(1/2))*(a+exp(2*x^(1/2))*x)),x)
                                                                                                        
   >> Error detected within library code:                                                               
   integrate: implementation incomplete (has polynomial part)

(1) -> integrate(((-a+exp(2*x^(1/2))*x)*x^(-3/2)*(1+x^(1/2))*(a^2+2*a*exp(2*x^(1/2))*x+c*exp(2*x^(1/2))*x+exp(4*x^(1/2))*x^2)^(1/2))/(2*expexp(x^(1/2))*(a+exp(2*x^(1/2))*x)),x)
                                                                                                        
   >> Error detected within library code:
   integrate: implementation incomplete (constant residues)

$$\int \frac{\left(\sqrt{x}+1\right) \left(e^{2x \sqrt{x}} -a\right) \sqrt{a^2+2 a x e^{2 \sqrt{x}} +cx e^{2 \sqrt{x}} +x^2 e^{4 \sqrt{x}}}}{2 x \sqrt{x}e^{\sqrt{x}} \left(a+x e^{2 \sqrt{x}} \right)} dx.$$

(1) -> integrate(((-a+exp(2*x^(1/2))*x)*x^(-3/2)*(1+x^(1/2))*(a^2+2*a*exp(2*x^(1/2))*x+c*exp(2*x^(1/2))*x+exp(4*x^(1/2))*x^2)^(1/2))/(2*exp(x^(1/2))*(a+exp(2*x^(1/2))*x)),x)
                                                                                                        
   >> Error detected within library code:                                                               
   integrate: implementation incomplete (has polynomial part)

(1) -> integrate(((-a+exp(2*x^(1/2))*x)*x^(-3/2)*(1+x^(1/2))*(a^2+2*a*exp(2*x^(1/2))*x+c*exp(2*x^(1/2))*x+exp(4*x^(1/2))*x^2)^(1/2))/(2*exp(x^(1/2))*(a+exp(2*x^(1/2))*x)),x)
                                                                                                        
   >> Error detected within library code:
   integrate: implementation incomplete (constant residues)

$$\int \frac{\left(\sqrt{x}+1\right) \left(e^{2x \sqrt{x}} -a\right) \sqrt{a^2+2 a x e^{2 \sqrt{x}} +cx e^{2 \sqrt{x}} +x^2 e^{4 \sqrt{x}}}}{x \sqrt{x}e^{\sqrt{x}} \left(a+x e^{2 \sqrt{x}} \right)} dx.$$

(1) -> integrate(((-a+exp(2*x^(1/2))*x)*x^(-3/2)*(1+x^(1/2))*(a^2+2*a*exp(2*x^(1/2))*x+c*exp(2*x^(1/2))*x+exp(4*x^(1/2))*x^2)^(1/2))/(exp(x^(1/2))*(a+exp(2*x^(1/2))*x)),x)
                                                                                                        
   >> Error detected within library code:                                                               
   integrate: implementation incomplete (has polynomial part)

(1) -> integrate(((-a+exp(2*x^(1/2))*x)*x^(-3/2)*(1+x^(1/2))*(a^2+2*a*exp(2*x^(1/2))*x+c*exp(2*x^(1/2))*x+exp(4*x^(1/2))*x^2)^(1/2))/(exp(x^(1/2))*(a+exp(2*x^(1/2))*x)),x)
                                                                                                        
   >> Error detected within library code:
   integrate: implementation incomplete (constant residues)

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edited Oct 18, 2020 at 2:33

Sam Blake

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In regards to the mixed algebraic-transcendental case of the Risch-Trager-Bronstein algorithm, an integral which cannot be solved with Maple, Mathematica, AXIOM or FriCAS (and possibly other CAS) is

$$\int \frac{\left(\sqrt{x}+1\right) \left(e^{2x \sqrt{x}} -a\right) \sqrt{a^2+2 a x e^{2 \sqrt{x}} +cx e^{2 \sqrt{x}} +x^2 e^{4 \sqrt{x}}}}{2 x \sqrt{x}e^{\sqrt{x}} \left(a+x e^{2 \sqrt{x}} \right)} dx.$$

This integral is interesting as it returns two distinct messages from AXIOM and FriCAS suggesting their respective implementations are incomplete. FriCAS returns

(1) -> integrate(((-a+exp(2*x^(1/2))*x)*x^(-3/2)*(1+x^(1/2))*(a^2+2*a*exp(2*x^(1/2))*x+c*exp(2*x^(1/2))*x+exp(4*x^(1/2))*x^2)^(1/2))/(2*exp(x^(1/2))*(a+exp(2*x^(1/2))*x)),x)
                                                                                                        
   >> Error detected within library code:                                                               
   integrate: implementation incomplete (has polynomial part)

While AXIOM returns

(1) -> integrate(((-a+exp(2*x^(1/2))*x)*x^(-3/2)*(1+x^(1/2))*(a^2+2*a*exp(2*x^(1/2))*x+c*exp(2*x^(1/2))*x+exp(4*x^(1/2))*x^2)^(1/2))/(2*exp(x^(1/2))*(a+exp(2*x^(1/2))*x)),x)
                                                                                                        
   >> Error detected within library code:
   integrate: implementation incomplete (constant residues)

[1] Miller, B. (2012). “On the Integration of Elementary Functions: Computing the Logarithmic Part”. Thesis (Ph.D.) Texas Tech University, Dept. of Mathematics and Statistics.

In regards to the mixed algebraic-transcendental case of the Risch-Trager-Bronstein algorithm, an integral which cannot be solved with Maple, Mathematica, AXIOM or FriCAS (and possibly other CAS) is

$$\int \frac{\left(\sqrt{x}+1\right) \left(e^{2x \sqrt{x}} -a\right) \sqrt{a^2+2 a x e^{2 \sqrt{x}} +cx e^{2 \sqrt{x}} +x^2 e^{4 \sqrt{x}}}}{2 x \sqrt{x}e^{\sqrt{x}} \left(a+x e^{2 \sqrt{x}} \right)} dx.$$

This integral is interesting as it returns two distinct messages from AXIOM and FriCAS suggesting their respective implementations are incomplete. FriCAS returns

(1) -> integrate(((-a+exp(2*x^(1/2))*x)*x^(-3/2)*(1+x^(1/2))*(a^2+2*a*exp(2*x^(1/2))*x+c*exp(2*x^(1/2))*x+exp(4*x^(1/2))*x^2)^(1/2))/(2*exp(x^(1/2))*(a+exp(2*x^(1/2))*x)),x)
                                                                                                        
   >> Error detected within library code:                                                               
   integrate: implementation incomplete (has polynomial part)

While AXIOM returns

(1) -> integrate(((-a+exp(2*x^(1/2))*x)*x^(-3/2)*(1+x^(1/2))*(a^2+2*a*exp(2*x^(1/2))*x+c*exp(2*x^(1/2))*x+exp(4*x^(1/2))*x^2)^(1/2))/(2*exp(x^(1/2))*(a+exp(2*x^(1/2))*x)),x)
                                                                                                        
   >> Error detected within library code:
   integrate: implementation incomplete (constant residues)

[1] Miller, B. (2012). “On the Integration of Elementary Functions: Computing the Logarithmic Part”. Thesis (Ph.D.) Texas Tech University, Dept. of Mathematics and Statistics.

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edited Oct 17, 2020 at 13:58

Sam Blake

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edited Oct 15, 2020 at 13:22

Sam Blake

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edited Oct 15, 2020 at 12:31

Timothy Chow

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created Oct 15, 2020 at 1:43

Sam Blake

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