### Install Symbolics.jl

Source: https://github.com/juliasymbolics/symbolics.jl/blob/master/README.md

Use the Julia package manager to add Symbolics.jl to your project.

```julia
using Pkg
julia> Pkg.add("Symbolics")
```

--------------------------------

### Example Function for Derivative Registration

Source: https://github.com/juliasymbolics/symbolics.jl/blob/master/docs/src/manual/derivatives.md

An example function `f(x,y,z)` used to demonstrate how derivatives can be automatically defined via tracing or explicitly registered.

```julia
using Symbolics
@variables x y z
f(x,y,z) = x^2 + sin(x+y) - z
```

--------------------------------

### Install Symbolics.jl

Source: https://github.com/juliasymbolics/symbolics.jl/blob/master/docs/src/index.md

Use the Julia package manager to add Symbolics.jl to your project. This command installs the package and its dependencies.

```julia
using Pkg
Pkg.add("Symbolics")
```

--------------------------------

### Mathematica Reduce Inequality

Source: https://github.com/juliasymbolics/symbolics.jl/blob/master/docs/src/manual/solver.md

Example of using Mathematica's `Reduce` function to solve an inequality.

```mathematica
Reduce[x^2 - x - 6 > 0, x]
```

--------------------------------

### Trace Lorenz Equations with Specific Symbolics Variables

Source: https://github.com/juliasymbolics/symbolics.jl/blob/master/docs/src/manual/functions.md

This example shows another way to trace the Lorenz equations, using specific Symbolics variables for clarity. Ensure Symbolics.jl is imported.

```julia
@variables t x(t) y(t) z(t) dx(t) dy(t) dz(t) σ ρ β
du = [dx,dy,dz]
u = [x,y,z]
p = [σ,ρ,β]
lorenz(du,u,p,t)
du
```

--------------------------------

### Registering Derivative Rule with Direct API

Source: https://github.com/juliasymbolics/symbolics.jl/blob/master/docs/src/manual/functions.md

Example of registering a derivative rule for an interpolation function using the `@register_derivative` macro.

```julia
@register_derivative (interp::AbstractInterpolation)(x) 1 begin
    # ...
end
```

--------------------------------

### Symbolic Integration with Multiple Variables and Options

Source: https://github.com/juliasymbolics/symbolics.jl/blob/master/docs/src/manual/integration.md

Illustrates integrating a trigonometric function with respect to a variable, using `symbolic = true` and `detailed = false` options. This example demonstrates handling more complex expressions.

```julia
integrate(sin(a * x) * cos(b * x), x; symbolic = true, detailed = false)
# (-a * cos(a * x) * cos(b * x) - b * sin(a * x) * sin(b * x)) / (a^2 - (b^2))
```

--------------------------------

### Mathematica Reduce Inequality with Parameter

Source: https://github.com/juliasymbolics/symbolics.jl/blob/master/docs/src/manual/solver.md

Example of using Mathematica's `Reduce` function to solve an inequality involving a parameter.

```mathematica
Reduce[x+a > 0, x]
```

--------------------------------

### Trace Lorenz Equations with Symbolics Variables

Source: https://github.com/juliasymbolics/symbolics.jl/blob/master/docs/src/manual/functions.md

This example demonstrates tracing a standard Julia function to generate a symbolic expression by using Symbolics variables as inputs. Ensure Symbolics.jl is imported.

```julia
using Symbolics
function lorenz(du,u,p,t)
 du[1] = 10.0(u[2]-u[1])
 du[2] = u[1]*(28.0-u[3]) - u[2]
 du[3] = u[1]*u[2] - (8/3)*u[3]
end
@variables t p[1:3] u(t)[1:3]
du = Array{Any}(undef, 3)
lorenz(du,u,p,t)
du
```

--------------------------------

### Inspect Arguments of a Symbolic Expression

Source: https://github.com/juliasymbolics/symbolics.jl/blob/master/docs/src/manual/variables.md

Use Symbolics.arguments on the unwrapped Num value of a symbolic expression to get its arguments. For example, for 'x + y', the arguments are 'x' and 'y'.

```julia
Symbolics.arguments(Symbolics.value(x + y))
```

--------------------------------

### Define Derivative Rule for min(x, y)

Source: https://github.com/juliasymbolics/symbolics.jl/blob/master/docs/src/manual/functions.md

This example demonstrates how to define a symbolic derivative rule for a registered function using the `@register_derivative` macro. This is essential for symbolic differentiation to work correctly.

```julia
@register_derivative min(x, y) 1 ifelse(x < y, 1, 0)
```

--------------------------------

### Factorial function example

Source: https://github.com/juliasymbolics/symbolics.jl/blob/master/docs/src/manual/faq.md

Illustrates a function that cannot be directly traced into a symbolic algorithm due to its adaptive nature (loop iterations depend on input value).

```julia
function factorial(x)
  out = x
  while x > 1
    x -= 1
    out *= x
  end
  out
end
```

--------------------------------

### Solve Perturbation Equations Sequentially

Source: https://github.com/juliasymbolics/symbolics.jl/blob/master/docs/src/tutorials/perturbation.md

Implements a cascading strategy to solve a set of perturbation equations sequentially. It starts with a known solution for the 0-th order and solves for higher orders.

```julia
function solve_cascade(eqs, xs, x₀, ϵ)
    sol = Dict(xs[begin] => x₀) # store solutions in a map

    # verify that x₀ is a solution of the first equation
    eq0 = substitute(eqs[1], sol; fold = Val(true))
    isequal(eq0.lhs, eq0.rhs) || error("$sol does not solve $(eqs[1])")

    # solve remaining equations sequentially
    for i in 2:length(eqs)
        eq = substitute(eqs[i], sol) # insert previous solutions
        x = xs[begin+i-1] # need not be 1-indexed
        xsol = Symbolics.symbolic_linear_solve(eq, x) # solve current equation
        sol = merge(sol, Dict(x => xsol)) # store solution
    end

    return sol
end
```

--------------------------------

### Resolving Provably False Expressions

Source: https://github.com/juliasymbolics/symbolics.jl/blob/master/docs/src/manual/constraint_satisfaction.md

Demonstrates using `resolve` to identify expressions that are provably false under given constraints. For example, `v < 0` is false when `v > 5`.

```Julia
# v < 0 is provably false when v > 5
resolve(v < 0, resolve_constraints)
```

--------------------------------

### Demonstrate overriding and fallback behavior for symbolic structs

Source: https://github.com/juliasymbolics/symbolics.jl/blob/master/docs/src/manual/structs.md

This example creates symbolic instances of `Record3` and `Record4` to show how specific `@symstruct` declarations override abstract ones, and how fields fall back to abstract type definitions when not specifically declared.

```julia
@variables rec3::Record3{Real} rec4::Record4{Real}
@assert size(rec3.x) == (4,) # Different from the one declared by `AbstractRecord`
@assert size(rec4.x) == (3,) # Falls back to the `AbstractRecord` definition
@assert size(rec4.f) == (5,) # Uses the more specific declaration
```

--------------------------------

### Convert Julia to Mathematica Expression

Source: https://github.com/juliasymbolics/symbolics.jl/blob/master/docs/src/manual/external.md

Convert Symbolics.jl expressions into a format compatible with Mathematica using SymbolicsMathLink.jl. Requires Mathematica or Wolfram Engine installation.

```julia
expr_to_mathematica(juliaSymbolicExpression)
```

--------------------------------

### Handle Symbolic Pi in Arithmetic

Source: https://github.com/juliasymbolics/symbolics.jl/blob/master/docs/src/manual/variables.md

To maintain a symbolic representation of irrational numbers like Pi, wrap them with Num. This example demonstrates redefining Pi within a function to match the input type for consistent symbolic arithmetic.

```julia
function f(x)
    let π=oftype(x, π)
        1 + (2//3 + 4π/5) * x
    end
end
f(t)
```

--------------------------------

### Define Quintic Equation

Source: https://github.com/juliasymbolics/symbolics.jl/blob/master/docs/src/tutorials/perturbation.md

Defines the quintic equation x^5 + x = 1 using Symbolics.jl. This serves as a basic example for perturbation problems.

```julia
using Symbolics
@variables x
quintic = x^5 + x ~ 1
```

--------------------------------

### Mathematica Solve Transcendental Equation

Source: https://github.com/juliasymbolics/symbolics.jl/blob/master/docs/src/manual/solver.md

Example of using Mathematica's `Solve` function to find a complex solution for a transcendental equation.

```mathematica
Solve[x^(x)  + 3 == 0, x]
```

--------------------------------

### Get Terms of Expression

Source: https://github.com/juliasymbolics/symbolics.jl/blob/master/docs/src/manual/expression_manipulation.md

Use `Symbolics.terms` to extract the individual terms that constitute a symbolic expression. This is helpful for analyzing the structure of polynomials or sums.

```julia
Symbolics.terms(x)
```

--------------------------------

### Set up Perturbative Solution (Series in M)

Source: https://github.com/juliasymbolics/symbolics.jl/blob/master/docs/src/tutorials/perturbation.md

Expand E as a series in M and extract coefficients to set up the equations for the perturbative solution.

```julia
E_taylor = series(E, M, 0:5) # this auto-creates coefficients E[0:5]
E_coeffs = taylor_coeff(E_taylor, M) # get a handle to the coefficients
kepler_eqs = taylor_coeff(substitute(kepler, E => E_taylor), M, 0:5)
kepler_eqs[1:4] # for readability
```

--------------------------------

### Set up Perturbative Solution (Series in e)

Source: https://github.com/juliasymbolics/symbolics.jl/blob/master/docs/src/tutorials/perturbation.md

Expand E as a series in e and extract coefficients to set up the equations for an alternative perturbative solution.

```julia
E_taylor′ = series(E, e, 0:5)
E_coeffs′ = taylor_coeff(E_taylor′, e)
kepler_eqs′ = taylor_coeff(substitute(kepler, E => E_taylor′), e, 0:5)
E_coeffs_sol′ = solve_cascade(kepler_eqs′, E_coeffs′, M, e)
E_pert′ = substitute(E_taylor′, E_coeffs_sol′)
```

--------------------------------

### Basic Symbolic Differentiation and Expansion

Source: https://github.com/juliasymbolics/symbolics.jl/blob/master/README.md

Demonstrates symbolic variable declaration, differentiation, and derivative expansion.

```julia
using Symbolics

@variables t x y
D = Differential(t)

z = t + t^2
D(z) # symbolic representation of derivative(t + t^2, t)

expand_derivatives(D(z))
```

--------------------------------

### Get Symbolic Matrix Size

Source: https://github.com/juliasymbolics/symbolics.jl/blob/master/docs/src/manual/arrays.md

Retrieves the dimensions of the symbolic matrix `A`.

```julia
size(A)
```

--------------------------------

### Basic Symbolic Integration

Source: https://github.com/juliasymbolics/symbolics.jl/blob/master/docs/src/manual/integration.md

Demonstrates the basic usage of the `integrate` function for a polynomial expression. The output shows the symbolic solution and coefficients.

```julia
using Symbolics
using SymbolicNumericIntegration

@variables x a b

integrate(3x^3 + 2x - 5)
# (x^2 + (3 // 4) * (x^4) - (5 // 1) * x, 0, 0)
```

--------------------------------

### Get Symbolic Vector Size

Source: https://github.com/juliasymbolics/symbolics.jl/blob/master/docs/src/manual/arrays.md

Retrieves the length of the symbolic vector `b`.

```julia
size(b)
```

--------------------------------

### Basic Symbolic Integrations with SymbolicIntegration.jl

Source: https://github.com/juliasymbolics/symbolics.jl/blob/master/docs/src/manual/integration.md

Demonstrates automatic method selection for common integration tasks. Ensure `Symbolics` and `SymbolicIntegration` are imported.

```julia
using Symbolics
using SymbolicIntegration

@variables x

# Basic integrations (automatic method selection)
integrate(x^2, x)           # Returns (1//3)*(x^3)
integrate(1/x, x)           # Returns log(x)
integrate(exp(x), x)        # Returns exp(x)
integrate(1/(x^2 + 1), x)   # Returns atan(x)
```

--------------------------------

### Create and use symbolic PointN with shape metadata

Source: https://github.com/juliasymbolics/symbolics.jl/blob/master/docs/src/manual/structs.md

Demonstrates creating a symbolic `PointN` with a specified dimension and accessing its fields. Includes an assertion for the field size and a try-catch block to show behavior on out-of-bounds access.

```julia
@variables point::PointN{Real, 3}
point.x[1]^2 + point.x[2]^2 + point.x[3]^2
@assert size(point.x) == (3,)
```

```julia
try
    point.x[4]
catch e
    Base.showerror(e)
end
```

--------------------------------

### Get Number of Dimensions for Symbolic Matrix

Source: https://github.com/juliasymbolics/symbolics.jl/blob/master/docs/src/manual/arrays.md

Returns the number of dimensions for the symbolic matrix `A`.

```julia
ndims(A)
```

--------------------------------

### Basic Satisfiability and Provability Check

Source: https://github.com/juliasymbolics/symbolics.jl/blob/master/docs/src/manual/constraint_satisfaction.md

Demonstrates defining symbolic variables with real number types and checking satisfiability and provability of expressions under basic positive constraints.

```julia
using Symbolics, SymbolicSMT

@variables x::Real y::Real

# Define constraints
constraints = Constraints([x > 0, y > 0])

# Check if an expression is satisfiable under constraints
issatisfiable(x + y > 1, constraints)
```

```julia
# Check if an expression is provable (always true) under constraints
isprovable(x >= 0, constraints)
```

```julia
# Check if x can be negative (it cannot, given x > 0)
issatisfiable(x < 0, constraints)
```

--------------------------------

### Get Size of Matrix-Vector Product

Source: https://github.com/juliasymbolics/symbolics.jl/blob/master/docs/src/manual/arrays.md

Retrieves the dimensions of the symbolic array expression resulting from `A * b`.

```julia
size(c)
```

--------------------------------

### Symbolic Integration with Variable and Options

Source: https://github.com/juliasymbolics/symbolics.jl/blob/master/docs/src/manual/integration.md

Shows how to specify the integration variable and use the `symbolic = true` option for integrating an exponential function. The output includes the symbolic solution.

```julia
integrate(exp(a * x), x; symbolic = true)
# (exp(a * x) / a, 0, 0)
```

--------------------------------

### Get Number of Dimensions for Symbolic Vector

Source: https://github.com/juliasymbolics/symbolics.jl/blob/master/docs/src/manual/arrays.md

Returns the number of dimensions for the symbolic vector `b`.

```julia
ndims(b)
```

--------------------------------

### Symbolics.NAMESPACE_SEPARATOR Constant

Source: https://github.com/juliasymbolics/symbolics.jl/blob/master/docs/src/manual/misc.md

Represents the separator used in namespaces within Symbolics.jl. No setup is required to use this constant.

```julia
Symbolics.NAMESPACE_SEPARATOR
```

--------------------------------

### Get Element Type of Matrix-Vector Product

Source: https://github.com/juliasymbolics/symbolics.jl/blob/master/docs/src/manual/arrays.md

Retrieves the element type of the symbolic array expression resulting from `A * b`.

```julia
eltype(c)
```

--------------------------------

### Get Variables from Expression

Source: https://github.com/juliasymbolics/symbolics.jl/blob/master/docs/src/manual/expression_manipulation.md

Use `Symbolics.get_variables` to extract all variables present in a symbolic expression. This is useful for analysis and further manipulation.

```julia
Symbolics.get_variables
```

--------------------------------

### Create and access symbolic Point2D

Source: https://github.com/juliasymbolics/symbolics.jl/blob/master/docs/src/manual/structs.md

Demonstrates creating a symbolic `Point2D` and accessing its fields, which are inferred as `Num`.

```julia
@variables point::Point2D{Real}
typeof(point)
@assert point.x isa Num
@assert point.y isa Num
```

```julia
point.x^2 + point.y^2
```

--------------------------------

### Symbolic Matrix Simplification and Substitution

Source: https://github.com/juliasymbolics/symbolics.jl/blob/master/README.md

Shows how to simplify symbolic matrices and perform substitutions with symbolic or numeric values.

```julia
B = simplify.([t^2 + t + t^2  2t + 4t
                  x + y + y + 2t  x^2 - x^2 + y^2])

simplify.(substitute.(B, (Dict(x => y^2),)))

substitute.(B, (Dict(x => 2.0, y => 3.0, t => 4.0),))
```

--------------------------------

### Get Size of Outer Product Result

Source: https://github.com/juliasymbolics/symbolics.jl/blob/master/docs/src/manual/arrays.md

Retrieves the dimensions of the matrix-valued expression resulting from the outer product `b * b'`.

```julia
size(b*b')
```

--------------------------------

### Get Size of Dot Product Result

Source: https://github.com/juliasymbolics/symbolics.jl/blob/master/docs/src/manual/arrays.md

Retrieves the dimensions of the scalar-valued expression resulting from the dot product `b'b`.

```julia
size(b'b)
```

--------------------------------

### Boolean Constraints and Satisfiability

Source: https://github.com/juliasymbolics/symbolics.jl/blob/master/docs/src/manual/constraint_satisfaction.md

Demonstrates using boolean variables and checking satisfiability of boolean expressions under defined constraints.

```julia
@variables flag1::Bool flag2::Bool

# Boolean constraints
bool_constraints = Constraints([flag1, flag2])

# Both flags are true, so their AND is satisfiable
issatisfiable(flag1 & flag2, bool_constraints)
```

```julia
# Can flag1 be false? No, it's constrained to be true
issatisfiable(!flag1, bool_constraints)
```

--------------------------------

### Quadratic Constraints with Integer Variables

Source: https://github.com/juliasymbolics/symbolics.jl/blob/master/docs/src/manual/constraint_satisfaction.md

Shows how to define and check satisfiability for integer variables under a quadratic constraint representing a circle.

```julia
@variables p::Integer q::Integer

# Circle-like constraint
circle = Constraints([p^2 + q^2 < 25])

# Can p be 3?
issatisfiable(p == 3, circle)
```

```julia
# Can p be 5? (No, because 5^2 = 25 is not < 25)
issatisfiable(p == 5, circle)
```

--------------------------------

### Get Element Type for Symbolic Matrix

Source: https://github.com/juliasymbolics/symbolics.jl/blob/master/docs/src/manual/arrays.md

Returns the element type of the symbolic matrix `A`. For symbolic arrays, this is typically `Num`.

```julia
eltype(A)
```

--------------------------------

### Using StaticArrays with Built Functions

Source: https://github.com/juliasymbolics/symbolics.jl/blob/master/docs/src/getting_started.md

Demonstrates how to use functions built with `build_function` with StaticArrays for performance. Ensure the function is evaluated using `eval` before passing StaticArray inputs.

```julia
using StaticArrays
myf = eval(f_expr[1])
myf(SA[2.0, 3.0])
```

--------------------------------

### Get Denominator of Expression

Source: https://github.com/juliasymbolics/symbolics.jl/blob/master/docs/src/manual/expression_manipulation.md

Extract the denominator of a symbolic expression that represents a fraction. This function is useful for rational function manipulation.

```julia
denominator(x::Union{Num, SymbolicUtils.BasicSymbolic})
```

--------------------------------

### Set up ODE Problems for DifferentialEquations.jl

Source: https://github.com/juliasymbolics/symbolics.jl/blob/master/docs/src/tutorials/auto_parallel.md

Configures both the standard and the sparse multithreaded ODE problems using DifferentialEquations.jl. The 'fastprob' utilizes the compiled parallel function and Jacobian for performance.

```julia
using OrdinaryDiffEq
u0 = zeros(N, N, 3)
MyA = zeros(N, N);
AMx = zeros(N, N);
DA = zeros(N, N);
prob = ODEProblem(f, u0, (0.0, 10.0))
fastprob = ODEProblem(ODEFunction((du, u, p, t) -> fastf(du, u),
                                   jac = (du, u, p, t) -> fjac(du, u),
                                   jac_prototype = similar(jac, Float64)),
                                   u0, (0.0, 10.0))
```

--------------------------------

### Get Numerator of Expression

Source: https://github.com/juliasymbolics/symbolics.jl/blob/master/docs/src/manual/expression_manipulation.md

Extract the numerator of a symbolic expression that represents a fraction. This function is useful for rational function manipulation.

```julia
numerator(x::Union{Num, SymbolicUtils.BasicSymbolic})
```

--------------------------------

### Bound Checking with Integer Variables

Source: https://github.com/juliasymbolics/symbolics.jl/blob/master/docs/src/manual/constraint_satisfaction.md

Illustrates checking satisfiability and provability for sums of integer variables within defined bounds.

```julia
@variables a::Integer b::Integer

# Define bounds
bounds = Constraints([a >= 0, a <= 10, b >= 0, b <= 10])

# Can the sum exceed 15?
issatisfiable(a + b > 15, bounds)
```

```julia
# Is the sum always at most 20?
isprovable(a + b <= 20, bounds)
```

--------------------------------

### Get Element Type for Symbolic Vector

Source: https://github.com/juliasymbolics/symbolics.jl/blob/master/docs/src/manual/arrays.md

Returns the element type of the symbolic vector `b`. For symbolic arrays, this is typically `Num`.

```julia
eltype(b)
```

--------------------------------

### Rule-Based Method for Algebraic Functions in SymbolicIntegration.jl

Source: https://github.com/juliasymbolics/symbolics.jl/blob/master/docs/src/manual/integration.md

Demonstrates using the rule-based method to handle integrals involving algebraic functions. Ensure `Symbolics` and `SymbolicIntegration` are imported.

```julia
using Symbolics
using SymbolicIntegration

@variables x

# Rule-based method for cases not handled by Risch
integrate(sqrt(x + 1), x)   # Handles algebraic functions via rules
```

--------------------------------

### Symbolic Initial Value Problem (IVP) Solver

Source: https://github.com/juliasymbolics/symbolics.jl/blob/master/docs/src/manual/ode.md

This snippet refers to the documentation for `Symbolics.solve_symbolic_IVP`. It is used for solving initial value problems of ODEs symbolically.

```julia
Symbolics.solve_symbolic_IVP
```

--------------------------------

### Get Factors of Expression

Source: https://github.com/juliasymbolics/symbolics.jl/blob/master/docs/src/manual/expression_manipulation.md

Use `Symbolics.factors` to decompose a symbolic expression into its multiplicative factors. This is useful for factorization and analysis of algebraic structures.

```julia
Symbolics.factors(x)
```

--------------------------------

### Use symbolic abstract types and subtypes

Source: https://github.com/juliasymbolics/symbolics.jl/blob/master/docs/src/manual/structs.md

Demonstrates creating symbolic instances of `Record1` and `Record2` which inherit symbolic behavior from `AbstractRecord`. It verifies the shape of the `x` field and performs a symbolic operation.

```julia
@variables rec1::Record1{Real} rec2::Record2{Real}
@assert size(rec1.x) == (3,)
@assert size(rec2.x) == (3,)
rec1.x[1] + rec2.x[2] + rec1.y + rec2.z
```

--------------------------------

### Get Degree of Expression

Source: https://github.com/juliasymbolics/symbolics.jl/blob/master/docs/src/manual/expression_manipulation.md

Use `Symbolics.degree` to determine the degree of a symbolic expression with respect to a specific variable. This is fundamental for polynomial analysis.

```julia
Symbolics.degree
```

--------------------------------

### Arrays of symbolic structs

Source: https://github.com/juliasymbolics/symbolics.jl/blob/master/docs/src/manual/structs.md

Demonstrates creating an array of symbolic structs, requiring a concrete `eltype`. It then performs a symbolic operation on elements of the array.

```julia
@variables pts[1:3]::PointN{Real, 3}
pts[1].x[1] + pts[2].x[2]
```

--------------------------------

### Build C Function from Symbolics IR

Source: https://github.com/juliasymbolics/symbolics.jl/blob/master/docs/src/tutorials/converting_to_C.md

This snippet demonstrates building a C function from the Symbolics IR using `build_function` with `Symbolics.CTarget()`. This generates the C code structure.

```julia
build_function(du, u, p, t, target=Symbolics.CTarget())
```

--------------------------------

### Get Coefficient of Term

Source: https://github.com/juliasymbolics/symbolics.jl/blob/master/docs/src/manual/expression_manipulation.md

Use `Symbolics.coeff` to extract the coefficient of a specific term within a symbolic expression. This is crucial for polynomial manipulation and analysis.

```julia
Symbolics.coeff
```

--------------------------------

### Display Version Information

Source: https://github.com/juliasymbolics/symbolics.jl/blob/master/docs/src/index.md

Displays detailed information about the Julia version, operating system, and hardware. This helps in reproducing the exact environment where the documentation was built.

```julia
using InteractiveUtils # hide
versioninfo() # hide
```

--------------------------------

### Explicit Risch Method Selection in SymbolicIntegration.jl

Source: https://github.com/juliasymbolics/symbolics.jl/blob/master/docs/src/manual/integration.md

Demonstrates how to explicitly select the Risch method for integration. Ensure `Symbolics` and `SymbolicIntegration` are imported.

```julia
using Symbolics
using SymbolicIntegration

@variables x

# Explicit Risch method selection
integrate(sin(x)*cos(x), x, RischMethod())
```

--------------------------------

### Define Symbolic Integral Operator

Source: https://github.com/juliasymbolics/symbolics.jl/blob/master/docs/src/manual/integration.md

Shows how to use the `Symbolics.Integral` documentation.

```julia
using Symbolics

Symbolics.Integral
```

--------------------------------

### Documenting groebner_basis Function

Source: https://github.com/juliasymbolics/symbolics.jl/blob/master/docs/src/manual/groebner.md

This snippet indicates that documentation for the `groebner_basis` function is available. It serves as a placeholder for the actual documentation content.

```julia
```@docs
groebner_basis
```
```

--------------------------------

### Introduce Expansion Variable for Perturbation

Source: https://github.com/juliasymbolics/symbolics.jl/blob/master/docs/src/tutorials/perturbation.md

Introduces an expansion variable 'ϵ' into the quintic equation to prepare for perturbation theory. Setting ϵ=1 recovers the original problem.

```julia
@variables ϵ # expansion variable
quintic = x^5 + ϵ*x ~ 1
```

--------------------------------

### Configurable Risch Method in SymbolicIntegration.jl

Source: https://github.com/juliasymbolics/symbolics.jl/blob/master/docs/src/manual/integration.md

Shows how to configure the Risch method with specific options for integration. Ensure `Symbolics` and `SymbolicIntegration` are imported.

```julia
using Symbolics
using SymbolicIntegration

@variables x

# Configurable Risch method
risch = RischMethod(use_algebraic_closure=true, catch_errors=false)
integrate(exp(x)/(1 + exp(x)), x, risch)
```

--------------------------------

### Display Full Manifest Status

Source: https://github.com/juliasymbolics/symbolics.jl/blob/master/docs/src/index.md

Shows the status of all dependencies, including indirect ones, as recorded in the Manifest.toml file. This provides a complete picture of the project's dependency tree.

```julia
using Pkg # hide
Pkg.status(; mode = PKGMODE_MANIFEST) # hide
```

--------------------------------

### Compute Perturbative Solution (Series in M)

Source: https://github.com/juliasymbolics/symbolics.jl/blob/master/docs/src/tutorials/perturbation.md

Solve the cascade of equations to find the coefficients for the perturbative series expansion of E in terms of M.

```julia
E_coeffs_sol = solve_cascade(kepler_eqs, E_coeffs, 0, M)
E_pert = substitute(E_taylor, E_coeffs_sol)
```

--------------------------------

### Define PDE Constants and Discretized Function

Source: https://github.com/juliasymbolics/symbolics.jl/blob/master/docs/src/tutorials/auto_parallel.md

Sets up constants and defines the discretized PDE as a Julia function 'f'. This function takes state variables, parameters, and time, and returns the derivatives.

```julia
using Symbolics, LinearAlgebra, SparseArrays, Plots

# Define the constants for the PDE
const α₂ = 1.0
const α₃ = 1.0
const β₁ = 1.0
const β₂ = 1.0
const β₃ = 1.0
const r₁ = 1.0
const r₂ = 1.0
const _DD = 100.0
const γ₁ = 0.1
const γ₂ = 0.1
const γ₃ = 0.1
const N = 32
const X = reshape([i for i in 1:N for j in 1:N], N, N)
const Y = reshape([j for i in 1:N for j in 1:N], N, N)
const α₁ = 1.0 .* (X .>= 4*N/5)

const Mx = Array(Tridiagonal([1.0 for i in 1:N-1], [-2.0 for i in 1:N], [1.0 for i in 1:N-1]))
const My = copy(Mx)
Mx[2, 1] = 2.0
Mx[end-1,end] = 2.0
My[1, 2] = 2.0
My[end,end-1] = 2.0

# Define the discretized PDE as an ODE function
function f(u, p, t)
    A = u[:,:,1]
    B = u[:,:,2]
    C = u[:,:,3]
    MyA = My*A
    AMx = A*Mx
    DA = @. _DD*(MyA + AMx)
    dA = @. DA + α₁ - β₁*A - r₁*A*B + r₂*C
    dB = @. α₂ - β₂*B - r₁*A*B + r₂*C
    dC = @. α₃ - β₃*C + r₁*A*B - r₂*C
    cat(dA, dB, dC, dims=3)
end
```

--------------------------------

### Create Matrix of Symbolic Variables

Source: https://github.com/juliasymbolics/symbolics.jl/blob/master/docs/src/manual/arrays.md

Creates a Julia matrix filled with symbolic variables using a shorthand notation.

```julia
u2 = Symbolics.variables(:x, 1:3, 3:6)
```

--------------------------------

### Evaluate Perturbative Solution Orders

Source: https://github.com/juliasymbolics/symbolics.jl/blob/master/docs/src/tutorials/perturbation.md

Print the numerical solutions for E at different orders of the perturbative series expansion in M, using Earth's eccentricity and a specific mean anomaly.

```julia
for n in 0:5
    println("$n-th order solution: E = ", substitute(taylor(E_pert, M, 0:n), vals_earth))
end
```

--------------------------------

### Verify Equivalence of Perturbative Series

Source: https://github.com/juliasymbolics/symbolics.jl/blob/master/docs/src/tutorials/perturbation.md

Assert that the two different perturbative series expansions (in M and in e) are equivalent by checking if their difference, when expanded as a Taylor series in both variables, results in zero.

```julia
@assert taylor(taylor(E_pert′ - E_pert, e, 0:4), M, 0:4) == 0 # use this as a test # hide
taylor(taylor(E_pert′ - E_pert, e, 0:4), M, 0:4)
```

--------------------------------

### Rule-Based Method for Non-Integer Powers in SymbolicIntegration.jl

Source: https://github.com/juliasymbolics/symbolics.jl/blob/master/docs/src/manual/integration.md

Illustrates using the rule-based method for integrals with non-integer powers. Ensure `Symbolics` and `SymbolicIntegration` are imported.

```julia
using Symbolics
using SymbolicIntegration

@variables x

# Rule-based method for cases not handled by Risch
integrate(x^(1/2), x)       # Uses rule-based method for non-integer powers
```

--------------------------------

### Expand Quintic Equation in Taylor Series

Source: https://github.com/juliasymbolics/symbolics.jl/blob/master/docs/src/tutorials/perturbation.md

Substitutes the Taylor series expansion of 'x' into the quintic equation and expands the result to a specified order in 'ϵ'.

```julia
quintic_taylor = substitute(quintic, x => x_taylor)
quintic_taylor = taylor(quintic_taylor, ϵ, 0:7)
```

--------------------------------

### Solve Equation with Mathematica

Source: https://github.com/juliasymbolics/symbolics.jl/blob/master/docs/src/manual/external.md

Use Mathematica's solving capabilities via SymbolicsMathLink.jl to solve algebraic equations. The `wcall` function executes Mathematica commands.

```julia
using SymbolicsMathLink

@variables x;
expr = x^2 + x - 1;

result = wcall("Solve", expr~0)
```

--------------------------------

### Creating a symbolic variable using interpolation

Source: https://github.com/juliasymbolics/symbolics.jl/blob/master/docs/src/manual/faq.md

Demonstrates how to use string interpolation with the `@variables` macro to dynamically set the name of a symbolic variable based on a Julia variable.

```julia
a = :c
b = only(@variables($a))
```

--------------------------------

### Create Array of Symbolic Expressions

Source: https://github.com/juliasymbolics/symbolics.jl/blob/master/docs/src/getting_started.md

Arrays of symbolic expressions can be created directly. For LaTeX rendering, use Latexify.jl.

```julia
A = [
x^2 + y 0 2x
     0       0 2y
     y^2 + x 0 0]

```

```julia
using Latexify
latexify(A)
```

--------------------------------

### Apply and Expand Derivatives

Source: https://github.com/juliasymbolics/symbolics.jl/blob/master/docs/src/getting_started.md

Apply the differential operator to an expression. Use expand_derivatives to compute the actual derivative.

```julia
z = t + t^2
D(z)
```

```julia
expand_derivatives(D(z))
```

--------------------------------

### Display Package Status

Source: https://github.com/juliasymbolics/symbolics.jl/blob/master/docs/src/index.md

Shows the current status of all direct dependencies in the package environment. This is useful for understanding the immediate dependencies of the project.

```julia
using Pkg # hide
Pkg.status() # hide
```

--------------------------------

### Load Function from File

Source: https://github.com/juliasymbolics/symbolics.jl/blob/master/docs/src/manual/io.md

Loads a Julia function previously saved to a .jl file using the include function.

```julia
g = include("function.jl")
```

--------------------------------

### Solve System of Polynomial Equations Symbolically

Source: https://github.com/juliasymbolics/symbolics.jl/blob/master/docs/src/manual/solver.md

Solve a system of polynomial equations using `symbolic_solve` with a list of equations and variables. Requires the `Groebner` package.

```julia
using Groebner;
@variables x y z;

eqs = [x^2 + y + z - 1, x + y^2 + z - 1, x + y + z^2 - 1]
Symbolics.symbolic_solve(eqs, [x,y,z])
```

--------------------------------

### Evaluate Perturbation Solution at ϵ=1

Source: https://github.com/juliasymbolics/symbolics.jl/blob/master/docs/src/tutorials/perturbation.md

Calculates the approximate solution to the original quintic equation by substituting ϵ=1 into the Taylor series expansion of x, up to various orders.

```julia
for n in 0:7
    x_pert_sol = substitute(taylor(x_pert, ϵ, 0:n), ϵ => big(1))
    println("$n-th order solution: x = $x_pert_sol = $(x_pert_sol * 1.0)")
end
```

--------------------------------

### Newton's Method for Quintic Equation

Source: https://github.com/juliasymbolics/symbolics.jl/blob/master/docs/src/tutorials/perturbation.md

Implements Newton's method to find a numerical solution for the quintic equation. It requires symbolic expressions for the function and its derivative.

```julia
function solve_newton(eq, x, x₀; abstol=1e-8, maxiters=50)
    # symbolic expressions for f(x) and f′(x)
    f = eq.lhs - eq.rhs # want to find root of f(x)
    f′ = Symbolics.derivative(f, x)

    xₙ = x₀ # numerical value of the initial guess
    for i = 1:maxiters
        # calculate new guess by numerically evaluating symbolic expression at previous guess
        xₙ₊₁ = Symbolics.value(substitute(x - f / f′, x => xₙ; fold = Val(true)))
        if abs(xₙ₊₁ - xₙ) < abstol
            return xₙ₊₁ # converged
        else
            xₙ = xₙ₊₁
        end
    end
    error("Newton's method failed to converge")
end

x_newton = solve_newton(quintic, x, 1.0)
println("Newton's method solution: x = ", x_newton)
```

--------------------------------

### Creating a symbolic variable with a specific name

Source: https://github.com/juliasymbolics/symbolics.jl/blob/master/docs/src/manual/faq.md

Shows how to create a symbolic variable and assign it to a Julia variable, where the Julia variable name differs from the symbolic variable name.

```julia
b = only(@variables(a))
```

--------------------------------

### Symbolic Matrix-Vector Multiplication

Source: https://github.com/juliasymbolics/symbolics.jl/blob/master/docs/src/manual/arrays.md

Demonstrates symbolic matrix-vector multiplication using symbolic array representations. This is more efficient than multiplying arrays of symbolic expressions.

```julia
@variables B[1:3, 1:3]
A * B
```

--------------------------------

### SymbolicLinearODE Documentation

Source: https://github.com/juliasymbolics/symbolics.jl/blob/master/docs/src/manual/ode.md

This snippet refers to the documentation for `Symbolics.SymbolicLinearODE`. It is used for representing and manipulating linear ODE systems symbolically.

```julia
Symbolics.SymbolicLinearODE
```

--------------------------------

### Save Symbolics Function to File

Source: https://github.com/juliasymbolics/symbolics.jl/blob/master/docs/src/manual/io.md

Generates an in-place function and saves its string representation to a .jl file using Julia's standard I/O.

```julia
using Symbolics
@variables u[1:3]
function f(u)
  [u[1]-u[3],u[1]^2-u[2],u[3]+u[2]]
end
ex1, ex2 = build_function(f(u),u)
write("function.jl", string(ex2))
```

--------------------------------

### Generate LaTeX Output

Source: https://github.com/juliasymbolics/symbolics.jl/blob/master/docs/src/manual/io.md

Uses Latexify.jl to generate LaTeX output from Symbolics models and expressions.

```julia
using Latexify
latexify(ex)
```

--------------------------------

### Create Symbolic Array Variables

Source: https://github.com/juliasymbolics/symbolics.jl/blob/master/docs/src/manual/arrays.md

Creates a symbolic matrix `A` and a symbolic vector `b` using the `@variables` macro. This establishes their dimensions and symbolic nature.

```julia
using Symbolics
@variables A[1:5, 1:3] b[1:3]
```

--------------------------------

### Make a struct symbolic

Source: https://github.com/juliasymbolics/symbolics.jl/blob/master/docs/src/manual/structs.md

Use the `@symstruct` macro to enable symbolic behavior for a struct. Ensure all type parameters are provided.

```julia
@symstruct Point2D{T}
```

--------------------------------

### Build Function from Symbolic Expressions

Source: https://github.com/juliasymbolics/symbolics.jl/blob/master/docs/src/getting_started.md

Converts symbolic expressions into Julia functions. Use this to generate efficient Julia code from symbolic representations. The output includes both a standard function and an in-place mutating version.

```julia
to_compute = [x^2 + y, y^2 + x]
f_expr = build_function(to_compute, [x, y])
Base.remove_linenums!.(f_expr)
```

--------------------------------

### Testing symbol membership using `any` and `isequal`

Source: https://github.com/juliasymbolics/symbolics.jl/blob/master/docs/src/manual/faq.md

Provides an alternative method for testing symbol membership in a collection using `any` with `isequal`.

```julia
any(isequal(x), [x])
```

--------------------------------

### Matrix-Vector Product using @arrayop

Source: https://github.com/juliasymbolics/symbolics.jl/blob/master/docs/src/manual/arrays.md

Defines and computes a matrix-vector product using the `@arrayop` macro for tensor notation. Operations are computed lazily.

```julia
d = Symbolics.@arrayop (i,) A[i,j]*b[j]
```

--------------------------------

### Create Symbolic Array of Variables

Source: https://github.com/juliasymbolics/symbolics.jl/blob/master/docs/src/getting_started.md

Use Symbolics.variables to create a Julia array of symbolic variables, useful for tracing functions.

```julia
u = Symbolics.variables(:u, 1:5)
f(u)
```

--------------------------------

### Solve Differential Equation with Mathematica

Source: https://github.com/juliasymbolics/symbolics.jl/blob/master/docs/src/manual/external.md

Employ Mathematica for solving differential equations through SymbolicsMathLink.jl. The `wcall` function is used to invoke Mathematica's DSolveValue.

```julia
using SymbolicsMathLink

@variables vars(x)[1:2];
expr = Differential(x)(vars[1]) + 2

result = wcall("DSolveValue", expr~0, vars[1], x)
```

--------------------------------

### SymPy Integration for Limits

Source: https://github.com/juliasymbolics/symbolics.jl/blob/master/docs/src/manual/limits.md

Enables the conversion of Symbolics expressions for use with SymPy's limit solvers.

```julia
using Symbolics
using Symbolics.SymPy

@variables x

sympy_limit(sin(x)/x, x=>0)
```

--------------------------------

### Type Wrapper API

Source: https://github.com/juliasymbolics/symbolics.jl/blob/master/docs/src/manual/types.md

Provides documentation for the core type wrapper functions used in Symbolics.jl to manage the integration of symbolic expressions with Julia's type system.

```APIDOC
## `Symbolics.wrap`

### Description
Wraps a symbolic expression tree in a type that conforms to Julia's type hierarchy, allowing it to be used where specific Julia types are expected.

### Method
`wrap(expr, T)`

### Parameters
#### Path Parameters
- `expr`: The symbolic expression tree to wrap.
- `T`: The target Julia type (e.g., `Real`, `Complex{Real}`, `Arr{Num}`).

### Response
- Returns a wrapped object that is a subtype of `T`.

## `Symbolics.unwrap`

### Description
Unwraps a symbolic expression tree from its wrapper type, returning the underlying expression tree for internal manipulation.

### Method
`unwrap(wrapped_expr)`

### Parameters
#### Path Parameters
- `wrapped_expr`: The wrapped symbolic object.

### Response
- Returns the underlying symbolic expression tree.
```

--------------------------------

### Compute Groebner Basis

Source: https://github.com/juliasymbolics/symbolics.jl/blob/master/docs/src/manual/external.md

Utilize the efficient Groebner.jl package for computing Groebner bases. Symbolics.jl provides an extension to automatically handle conversions.

```julia
using Symbolics

# Example usage (assuming Symbolics.jl is loaded)
# Symbolics.groebner_basis(polynomials)
# Requires polynomials to be defined in Symbolics.jl format
```

--------------------------------

### Symbolic Series Solution from Wikipedia

Source: https://github.com/juliasymbolics/symbolics.jl/blob/master/docs/src/tutorials/perturbation.md

The symbolic series solution for E in terms of e and M, as found on Wikipedia.

```julia
E_wiki = 1/(1-e)*M - e/(1-e)^4*M^3/factorial(3) + (9e^2+e)/(1-e)^7*M^5/factorial(5)
```

--------------------------------

### Verify C Function Computation

Source: https://github.com/juliasymbolics/symbolics.jl/blob/master/docs/src/tutorials/converting_to_C.md

This snippet verifies that the compiled C function `f` produces the same results as the original Julia function `lotka_volterra!`. It compares outputs with random inputs.

```julia
du = rand(2); du2 = rand(2)
u = rand(2)
p = rand(4)
t = rand()
f(du, u, p, t)
lotka_volterra!(du2, u, p, t)
du == du2 # true!
```

--------------------------------

### Attempting to symbolically trace a non-symbolic function

Source: https://github.com/juliasymbolics/symbolics.jl/blob/master/docs/src/manual/faq.md

Demonstrates the error that occurs when trying to symbolically represent an algorithm whose iteration count depends on symbolic input.

```julia
using Symbolics
@variables x
try
    factorial(x)
catch e
    e
end
```

--------------------------------

### Structure Detection Functions

Source: https://github.com/juliasymbolics/symbolics.jl/blob/master/docs/src/manual/sparsity_detection.md

Utilize `islinear` and `isaffine` to determine if a symbolic expression is linear or affine. These functions help in understanding the structural properties of the expressions.

```julia
Symbolics.islinear
```

```julia
Symbolics.isaffine
```

--------------------------------

### Simplify Symbolic Expressions

Source: https://github.com/juliasymbolics/symbolics.jl/blob/master/docs/src/getting_started.md

Use the `simplify` command to simplify symbolic expressions. This can be applied to arrays using Julia's broadcast mechanism.

```julia
simplify(2x + 2y)
```

```julia
B = simplify.([t + t^2 + t + t^2  2t + 4t
               x + y + y + 2t     x^2 - x^2 + y^2])
```

--------------------------------

### Define and Register Custom Metadata Type

Source: https://github.com/juliasymbolics/symbolics.jl/blob/master/docs/src/manual/metadata.md

Extend the metadata system by defining a new metadata type (e.g., MyCustomMetadata) and registering it using Symbolics.option_to_metadata_type. This allows for custom annotations in @variables.

```julia
using Symbolics

# Define a custom metadata type
struct MyCustomMetadata <: Symbolics.AbstractVariableMetadata end

# Register it for use in @variables
Symbolics.option_to_metadata_type(::Val{:my_custom}) = MyCustomMetadata

# Now you can use it
@variables x [my_custom = "some value"]
```

--------------------------------

### Solve for Coefficients and Substitute

Source: https://github.com/juliasymbolics/symbolics.jl/blob/master/docs/src/tutorials/perturbation.md

Solves the order-separated quintic equations for the Taylor series coefficients and substitutes these solutions back into the full series expansion of x.

```julia
x_coeffs_sol = solve_cascade(quintic_eqs, x_coeffs, 1, ϵ)
x_pert = substitute(x_taylor, x_coeffs_sol)
```

--------------------------------

### Create Vector of Symbolic Variables

Source: https://github.com/juliasymbolics/symbolics.jl/blob/master/docs/src/manual/arrays.md

Creates a Julia vector containing symbolic variables.

```julia
using Symbolics
@variables x y
u = [x,y]
```

--------------------------------

### Sparse Jacobian and Function Building

Source: https://github.com/juliasymbolics/symbolics.jl/blob/master/docs/src/getting_started.md

Calculates the sparse Jacobian of a function involving sparse matrix operations and then builds an efficient Julia function from it. Pre-allocate the output sparse matrix with the correct structure before calling the generated function.

```julia
using Symbolics, SparseArrays, LinearAlgebra

N = 10
_S = sprand(N, N, 0.1)
_Q = sprand(N, N, 0.1)

F(z) = [
    collect(_S * z)
    collect(_Q * z.^2)
]

Symbolics.@variables z[1:N]

sj = Symbolics.sparsejacobian(F(z), z)

f_expr = build_function(sj, z)

rows, cols, _ = findnz(sj)
out = sparse(rows, cols, zeros(length(cols)), size(sj)...)
myf = eval(last(f_expr))
myf(out, rand(N))
out
```

--------------------------------

### Linear ODE System Solver

Source: https://github.com/juliasymbolics/symbolics.jl/blob/master/docs/src/manual/ode.md

This snippet refers to the documentation for `Symbolics.solve_linear_ode_system`. This function is designed to solve systems of linear ODEs symbolically.

```julia
Symbolics.solve_linear_ode_system
```

--------------------------------

### Sparsity Detection Functions

Source: https://github.com/juliasymbolics/symbolics.jl/blob/master/docs/src/manual/sparsity_detection.md

Use `jacobian_sparsity` and `hessian_sparsity` to automatically detect the sparsity patterns for Jacobian and Hessian matrices, respectively. These functions leverage the tracing system of Symbolics.jl.

```julia
Symbolics.jacobian_sparsity
```

```julia
Symbolics.hessian_sparsity
```

--------------------------------

### Fast Substitution

Source: https://github.com/juliasymbolics/symbolics.jl/blob/master/docs/src/manual/expression_manipulation.md

Use `Symbolics.fast_substitute` for efficient substitution of multiple variables within a symbolic expression. This is optimized for performance when many replacements are needed.

```julia
Symbolics.fast_substitute
```

--------------------------------

### Convert Mathematica to Julia Expression

Source: https://github.com/juliasymbolics/symbolics.jl/blob/master/docs/src/manual/external.md

Convert Mathematica expressions back into Symbolics.jl expressions using SymbolicsMathLink.jl. This enables round-trip conversion between the two systems.

```julia
mathematica_to_expr(W`Some Mathematica expression`)
```

--------------------------------

### In-place Mutating Function Usage

Source: https://github.com/juliasymbolics/symbolics.jl/blob/master/docs/src/getting_started.md

Shows how to use the in-place mutating function generated by `build_function`. This function modifies its first argument directly, avoiding allocations. Initialize an output array of the correct size and type before calling.

```julia
myf! = eval(f_expr[2])
out = zeros(2)
myf!(out, [2.0, 3.0])
out
```

--------------------------------

### Solve Exponential Equation Symbolically

Source: https://github.com/juliasymbolics/symbolics.jl/blob/master/docs/src/manual/solver.md

Use `symbolic_solve` to find the exact symbolic solution for an exponential equation. Ensure `Symbolics` and `Nemo` are imported.

```julia
using Symbolics, Nemo;
@variables x;
Symbolics.symbolic_solve(9^x + 3^x ~ 8, x)
```

--------------------------------

### ADTypes.jl Sparsity Detector

Source: https://github.com/juliasymbolics/symbolics.jl/blob/master/docs/src/manual/sparsity_detection.md

The `SymbolicsSparsityDetector` is an interface provided by Symbolics.jl for integrating with ADTypes.jl, enabling automatic differentiation systems to leverage Symbolics.jl's sparsity detection capabilities.

```julia
Symbolics.SymbolicsSparsityDetector
```

--------------------------------

### C Target Function Compilation

Source: https://github.com/juliasymbolics/symbolics.jl/blob/master/docs/src/manual/build_function.md

Generates a compilable C function from Symbolics IR. Useful for integrating with C-based numerical solvers or applications.

```julia
Symbolics._build_function(target::Symbolics.CTarget,eqs::Array{<:Equation},args...;kwargs...)
```

--------------------------------

### Proving Multi-Variable Arithmetic Expressions

Source: https://github.com/juliasymbolics/symbolics.jl/blob/master/docs/src/manual/constraint_satisfaction.md

Use `isprovable` to determine if an expression is always true given a set of constraints on multiple integer variables.

```Julia
# Is sum always >= 2?
isprovable(m + n >= 2, multi_constraints)  # true - always satisfied when m,n >= 1
```

--------------------------------

### Integrate Rational Function with SymbolicIntegration.jl

Source: https://github.com/juliasymbolics/symbolics.jl/blob/master/docs/src/manual/integration.md

Solves a rational function integral, typically using the Risch method. Ensure `Symbolics` and `SymbolicIntegration` are imported.

```julia
using Symbolics
using SymbolicIntegration

@variables x

# Rational function with complex roots (typically uses Risch)
f = (x^3 + x^2 + x + 2)/(x^4 + 3*x^2 + 2)
integrate(f, x)  # Returns (1//2)*log(2 + x^2) + atan(x)
```

--------------------------------

### Testing symbol membership in a Set

Source: https://github.com/juliasymbolics/symbolics.jl/blob/master/docs/src/manual/faq.md

Demonstrates the correct way to test if a symbol is part of a collection by converting the collection to a `Set` and using the `in` operator.

```julia
x in Set([x])
```

--------------------------------

### Expand Variable as Taylor Series

Source: https://github.com/juliasymbolics/symbolics.jl/blob/master/docs/src/tutorials/perturbation.md

Expands the variable 'x' as a power series in 'ϵ' using Taylor series coefficients. This is a key step in perturbation theory.

```julia
x_coeffs, = @variables a[0:7] # create Taylor series coefficients
x_taylor = series(x_coeffs, ϵ) # expand x in a power series in ϵ
```

--------------------------------

### Define a basic 2D Point struct

Source: https://github.com/juliasymbolics/symbolics.jl/blob/master/docs/src/manual/structs.md

This snippet shows the standard Julia struct definition for a 2D point.

```julia
using Symbolics, SymbolicUtils

struct Point2D{T}
    x::T
    y::T
end
```

--------------------------------

### Separate Taylor Series into Order Equations

Source: https://github.com/juliasymbolics/symbolics.jl/blob/master/docs/src/tutorials/perturbation.md

Separates the expanded quintic Taylor series equation into individual equations for each order of 'ϵ'. The first five equations are shown for readability.

```julia
quintic_eqs = taylor_coeff(quintic_taylor, ϵ, 0:7)
quintic_eqs[1:5] # for readability, show only 5 shortest equations
```

--------------------------------

### SymPy ODE Solver Integration

Source: https://github.com/juliasymbolics/symbolics.jl/blob/master/docs/src/manual/ode.md

This snippet refers to the documentation for `Symbolics.sympy_ode_solve`. It indicates integration with SymPy for solving ODEs, likely leveraging SymPy's capabilities.

```julia
Symbolics.sympy_ode_solve
```

--------------------------------

### Create Symbolic Expressions

Source: https://github.com/juliasymbolics/symbolics.jl/blob/master/docs/src/getting_started.md

Symbolic expressions can be generated using Julia expressions with defined symbolic variables.

```julia
z = x^2 + y
```

--------------------------------

### Translate SymPy to Symbolics.jl

Source: https://github.com/juliasymbolics/symbolics.jl/blob/master/docs/src/manual/external.md

Use Symbolics.jl's functions to translate SymPy expressions back into Symbolics.jl format. This allows seamless integration of SymPy computations into a Julia workflow.

```julia
using SymPy

# Example usage (assuming Symbolics.jl is also loaded)
# sympy_to_symbolics(sympy_expression)
sympy_expr = sympy.symbols("x") + sympy.symbols("y")
sympy_to_symbolics(sympy_expr)
```

--------------------------------

### Define Variables with Default Values

Source: https://github.com/juliasymbolics/symbolics.jl/blob/master/docs/src/manual/metadata.md

Use the @variables macro to create symbolic variables and assign them default numerical values. This is useful for setting initial conditions or known constants.

```julia
using Symbolics

# Variable with default value
@variables x=1.0 y=2.0
```

--------------------------------

### Using a Registered Symbolic Array Function

Source: https://github.com/juliasymbolics/symbolics.jl/blob/master/docs/src/manual/functions.md

Utilize a previously registered symbolic array function with symbolic variables. The computation remains lazy.

```julia
@variables A[1:3, 1:3] b[1:3]
svdsolve(A,b)
```