SymbolicIntegration.jl (my GSoC project)

Author: Mattia Micheletta Merlin; Date: 02/09/2025

• Translation of integration rules: The integration rules are written in Mathematica syntax (very different from Julia syntax). I tackled this challenge by creating a translator script that automatically translates them into Julia syntax using regexes and other really fancy string manipulation functions that I wrote. Even with this script, I had to manually test the rules and this was time conusming.
• Utility functions: There are many integration rules in RUBI, but also many utility functions used in the rule conditions, including both base Mathematica functions (usually not present in Symbolics.jl) and custom functions made for the RUBI package (the file where they are defined contains 7842 lines of code). The translation of these could not be automated and was a lot time consumning and error prone as I had to both understand what each utility function did (not always easy), rewrite it in Julia, and test its behaviour with lots of different rules.
• Rule application: Julia's Symbolics.jl already had a pattern matching functionality, with the `@rule` macro, but it was not sufficiently good for this package to work well, so I improved it. There was not one single big improvement but many small ones.

julia> integrate(sqrt(4 - 12*x + 9*x^2)+sqrt(1+x),x)
┌-------Applied rule 0_1_0 on ∫(sqrt(1 + x) + sqrt(4 - 12x + 9(x^2)), x)
| ∫( a + b + ..., x) => ∫(a,x) + ∫(b,x) + ...
└-------with result: ∫(sqrt(4 - 12x + 9(x^2)), x) + ∫(sqrt(1 + x), x)
┌-------Applied rule 1_1_1_1_4 on ∫(sqrt(1 + x), x)
| ∫((a + b * x) ^ m, x) => if 
|       !(contains_var(a, b, m, x)) &&
|       m !== -1
| (a + b * x) ^ (m + 1) / (b * (m + 1))
└-------with result: (2//3)*((1 + x)^(3//2))
┌-------Applied rule 1_2_1_1_3 on ∫(sqrt(4 - 12x + 9(x^2)), x)
| ∫((a + b * x + c * x ^ 2) ^ p, x) => if 
|       !(contains_var(a, b, c, p, x)) &&
|       (
|             b ^ 2 - 4 * a * c == 0 &&
|             p !== -1 / 2
|       )
| ((b + 2 * c * x) * (a + b * x + c * x ^ 2) ^ p) / (2 * c * (2 * p + 1))
└-------with result: (1//36)*(-12 + 18x)*((4 - 12x + 9(x^2))^(1//2))
(2//3)*((1 + x)^(3//2)) + (1//36)*(-12 + 18x)*sqrt(4 - 12x + 9(x^2))


julia> integrate((2+2x+2x^2)/(1+x^3);verbose=false)
(2//3)*log(1 + x^3) + 2.3094010767585034atan(0.14433756729740646(-4 + 8x))

julia> integrate((1 - x)^2*(1 + x)^(2.34);verbose=false)
1.1976047904191618((1 + x)^3.34) - 0.9216589861751152((1 + x)^4.34) + 0.18726591760299627((1 + x)^5.34)

julia> integrate(1/(a+b*x^2),x;verbose=false)
(atan(x / sqrt(a / b))*sqrt(a / b)) / a

julia> integrate(x^2/(1+a*x^3),x;verbose=false)
log(1 + a*(x^3)) / (3a)

julia> integrate(exp(x)/(exp(2x)-1);verbose=false)
-atanh(exp(x))

julia> integrate(sqrt(x)*exp(x);verbose=false)
-0.8862269254527579SpecialFunctions.erfi(sqrt(x)) + sqrt(x)*exp(x)

julia> integrate(log(x)*x;verbose=false)
-(1//4)*(x^2) + (1//2)*(x^2)*log(x)

julia> integrate(log(x)/sqrt(x);verbose=false)
-(4//1)*sqrt(x) + (2//1)*sqrt(x)*log(x)

julia> integrate(log(log(x));verbose=false)
-SpecialFunctions.expinti(log(x)) + x*log(log(x))

julia> integrate(sin(x)^3*cos(x)^2;verbose=false)
-(1//3)*(cos(x)^3) + (1//5)*(cos(x)^5)

julia> integrate(sqrt(sin(x));verbose=false)
2Elliptic.E((1//2)*(-1.5707963267948966 + x), 2)

julia> integrate(sqrt(sin(x)+1);verbose=false)
(-2cos(x)) / sqrt(1 + sin(x))

julia> integrate(sin(log(2x^2))/x;verbose=false)
(-1//2)*cos(log(2(x^2)))

julia> integrate(sin(x^2)/x;verbose=false)
(1//2)*SpecialFunctions.sinint(x^2)

julia> integrate(acosh(x+1);verbose=false)
(1 + x)*acosh(1 + x) - sqrt(x)*sqrt(2 + x)