[WIP] Add symbolic integral

Project.toml

@@ -41,7 +41,6 @@ Reexport = "0.2, 1"
 ReferenceTests = "0.9"
 Requires = "1.1"
 RuntimeGeneratedFunctions = "0.4.3, 0.5"
-SafeTestsets = "0"
 SciMLBase = "1.8"
 Setfield = "0.7"
 SpecialFunctions = "0.7, 0.8, 0.9, 0.10, 1.0"

src/Symbolics.jl

@@ -6,6 +6,9 @@ using LinearAlgebra
 
 using Reexport
 
+using DomainSets
+
+import DomainSets: Domain
 @reexport using SymbolicUtils
 
 import SymbolicUtils: Term, Add, Mul, Pow, Sym, symtype,
@@ -52,6 +55,9 @@ export Differential, expand_derivatives
 
 include("diff.jl")
 
+export Integral
+include("integral.jl")
+
 include("linear_algebra.jl")
 
 import Libdl

src/diff.jl

@@ -113,6 +113,25 @@ function expand_derivatives(O::Symbolic, simplify=false; occurances=nothing)
                     return expand_derivatives(operation(arg)(inner), simplify)
                 end
             end
+        elseif isa(operation(arg) , Integral)
+            if isa(operation(arg).domain , AbstractInterval)
+                domain = operation(arg).domain
+                a , b = DomainSets.endpoints(domain)
+                c = 0
+                inner_function = expand_derivatives_(arguments(arg)[1])
+                if isa(value(a), Term)
+                    t1 = replaceSym(operation(arg).x ,value(a) , inner_function )
+                    t2 = D(a)
+                    c -= t1*t2
+                elseif isa(value(b) , Term)
+                    t1 = replaceSym(operation(arg).x ,value(b) , inner_function )
+                    t2 = D(b)
+                    c += t1*t2
+                end
+                inner = expand_derivatives_(D(arguments(arg)[1]))
+                c += operation(arg)(inner)
+                return c
+            end
         end
 
         l = length(arguments(arg))
@@ -151,6 +170,8 @@ function expand_derivatives(O::Symbolic, simplify=false; occurances=nothing)
             x = +((!_iszero(c) ? vcat(c, exprs) : exprs)...)
             return simplify ? SymbolicUtils.simplify(x) : x
         end
+    elseif isa(operation(O) , Integral )
+        return operation(O)(expand_derivatives_(arguments(O)[1]))  
     elseif !hasderiv(O)
         return O
     else

src/integral.jl

@@ -0,0 +1,34 @@
+# basic integral struct with upper bound and lower bound.
+struct Integral{X, T <: Domain} <: Function
+    x
+    domain::T
+    Integral(x,domain) = new{typeof(x),typeof(domain)}(Symbolics.value(x), domain)
+end
+
+(I::Integral)(x) = Term{SymbolicUtils.symtype(x)}(I, [x])
+(I::Integral)(x::Num) = Num(I(Symbolics.value(x)))
+SymbolicUtils.promote_symtype(::Integral, x) = x
+
+function Base.show(io::IO, I::Integral)
+    print(io, "Integral(", I.x, ", ", I.domain, ")")
+end
+
+Base.:(==)(I1::Integral, I2::Integral) = (isequal(I1.x, I2.x) && isequal(I1.domain, I2.domain))
+(D::Differential)(I::Integral{X,T}) where{X,T} = I∘D
+
+function replaceSym(a::Sym, b, O)
+    if isa(O , Sym)
+        if isequal(O , a)
+            return b
+        else
+            return O
+        end
+    else
+        args_replace = Vector{Symbolic{Real}}()
+        args_ = arguments(O)
+        for i in 1:length(args_)
+            push!(args_replace, replaceSym(a , b , args_[i]))
+        end
+        return operation(O)(args_replace...)
+    end
+end

test/integral.jl

@@ -0,0 +1,7 @@
+using Symbolics, Test
+using DomainSets
+@variables x y
+
+I1 = Integral( x , ClosedInterval(1, 5))
+I2 = Integral( x , ClosedInterval(1, 5))
+@test I1 == I2