Add more special-function rules

src/methods/rule_based/general.jl

@@ -244,4 +244,7 @@ all_rules_paths = [
 "7 Inverse hyperbolic functions/7.3 Inverse hyperbolic tangent/7.3.3 (d+e x)^m (a+b arctanh(c x^n))^p.jl"
 
 "8 Special functions/8.1 Bessel functions.jl"
+"8 Special functions/8.2 Airy functions.jl"
+"8 Special functions/8.3 Error and exponential integral functions.jl"
+"8 Special functions/8.4 Trigonometric integral functions.jl"
 ]

src/methods/rule_based/rules2/8 Special functions/8.1 Bessel functions.jl

@@ -90,4 +90,48 @@ file_rules = [
         ),
     )
 
+    (
+        "8_1_9",
+        :((~x)^(~!nu) * hankelh1((~!mu), (~!a) * (~x))) => :(
+            !contains_var((~a), (~mu), (~nu), (~x)) &&
+                eq((~mu), (~nu) - 1) &&
+                !eq((~a), 0) ?
+                (~x)^(~nu) * SpecialFunctions.hankelh1((~nu), (~a) * (~x)) ⨸ (~a) : nothing
+        ),
+    )
+
+    (
+        "8_1_10",
+        :((~x)^(~!nu) * hankelh2((~!mu), (~!a) * (~x))) => :(
+            !contains_var((~a), (~mu), (~nu), (~x)) &&
+                eq((~mu), (~nu) - 1) &&
+                !eq((~a), 0) ?
+                (~x)^(~nu) * SpecialFunctions.hankelh2((~nu), (~a) * (~x)) ⨸ (~a) : nothing
+        ),
+    )
+
+    (
+        "8_1_11",
+        :((~x)^(~!m) * hankelh1((~!nu), (~!a) * (~x))) => :(
+            !contains_var((~a), (~m), (~nu), (~x)) &&
+                igt(simplify((~m) - (~nu) - 1), 0) &&
+                ext_iseven(simplify((~m) - (~nu) - 1)) &&
+                !eq((~a), 0) ?
+                (~x)^(~m) * SpecialFunctions.hankelh1((~nu) + 1, (~a) * (~x)) ⨸ (~a) -
+                simplify((~m) - (~nu) - 1) ⨸ (~a) * ∫((~x)^((~m) - 1) * SpecialFunctions.hankelh1((~nu) + 1, (~a) * (~x)), (~x)) : nothing
+        ),
+    )
+
+    (
+        "8_1_12",
+        :((~x)^(~!m) * hankelh2((~!nu), (~!a) * (~x))) => :(
+            !contains_var((~a), (~m), (~nu), (~x)) &&
+                igt(simplify((~m) - (~nu) - 1), 0) &&
+                ext_iseven(simplify((~m) - (~nu) - 1)) &&
+                !eq((~a), 0) ?
+                (~x)^(~m) * SpecialFunctions.hankelh2((~nu) + 1, (~a) * (~x)) ⨸ (~a) -
+                simplify((~m) - (~nu) - 1) ⨸ (~a) * ∫((~x)^((~m) - 1) * SpecialFunctions.hankelh2((~nu) + 1, (~a) * (~x)), (~x)) : nothing
+        ),
+    )
+
 ]

src/methods/rule_based/rules2/8 Special functions/8.2 Airy functions.jl

@@ -0,0 +1,37 @@
+file_rules = [
+    #(* ::Subsection::Closed:: *)
+    #(* 8.2 Airy functions *)
+
+    (
+        "8_2_1",
+        :(airyaiprime((~!a) + (~!b) * (~x))) => :(
+            !contains_var((~a), (~b), (~x)) &&
+                !eq((~b), 0) ?
+                SpecialFunctions.airyai((~a) + (~b) * (~x)) ⨸ (~b) : nothing
+        ),
+    )
+
+    (
+        "8_2_2",
+        :(airybiprime((~!a) + (~!b) * (~x))) => :(
+            !contains_var((~a), (~b), (~x)) &&
+                !eq((~b), 0) ?
+                SpecialFunctions.airybi((~a) + (~b) * (~x)) ⨸ (~b) : nothing
+        ),
+    )
+
+    (
+        "8_2_3",
+        :((~x) * airyai(~x)) => :(
+            SpecialFunctions.airyaiprime(~x)
+        ),
+    )
+
+    (
+        "8_2_4",
+        :((~x) * airybi(~x)) => :(
+            SpecialFunctions.airybiprime(~x)
+        ),
+    )
+
+]

src/methods/rule_based/rules2/8 Special functions/8.3 Error and exponential integral functions.jl

@@ -0,0 +1,35 @@
+file_rules = [
+    #(* ::Subsection::Closed:: *)
+    #(* 8.3 Error and exponential integral functions *)
+
+    (
+        "8_3_1",
+        :(erf((~!a) + (~!b) * (~x))) => :(
+            !contains_var((~a), (~b), (~x)) &&
+                !eq((~b), 0) ?
+                (((~a) + (~b) * (~x)) * SymbolicUtils.erf((~a) + (~b) * (~x)) +
+                    exp(-((~a) + (~b) * (~x))^2) ⨸ sqrt(π)) ⨸ (~b) : nothing
+        ),
+    )
+
+    (
+        "8_3_2",
+        :(erfi((~!a) + (~!b) * (~x))) => :(
+            !contains_var((~a), (~b), (~x)) &&
+                !eq((~b), 0) ?
+                (((~a) + (~b) * (~x)) * SymbolicUtils.erfi((~a) + (~b) * (~x)) -
+                    exp(((~a) + (~b) * (~x))^2) ⨸ sqrt(π)) ⨸ (~b) : nothing
+        ),
+    )
+
+    (
+        "8_3_3",
+        :(expinti((~!a) + (~!b) * (~x))) => :(
+            !contains_var((~a), (~b), (~x)) &&
+                !eq((~b), 0) ?
+                (((~a) + (~b) * (~x)) * SymbolicUtils.expinti((~a) + (~b) * (~x)) -
+                    exp((~a) + (~b) * (~x))) ⨸ (~b) : nothing
+        ),
+    )
+
+]

src/methods/rule_based/rules2/8 Special functions/8.4 Trigonometric integral functions.jl

@@ -0,0 +1,25 @@
+file_rules = [
+    #(* ::Subsection::Closed:: *)
+    #(* 8.4 Trigonometric integral functions *)
+
+    (
+        "8_4_1",
+        :(sinint((~!a) + (~!b) * (~x))) => :(
+            !contains_var((~a), (~b), (~x)) &&
+                !eq((~b), 0) ?
+                (((~a) + (~b) * (~x)) * SymbolicUtils.sinint((~a) + (~b) * (~x)) +
+                    cos((~a) + (~b) * (~x))) ⨸ (~b) : nothing
+        ),
+    )
+
+    (
+        "8_4_2",
+        :(cosint((~!a) + (~!b) * (~x))) => :(
+            !contains_var((~a), (~b), (~x)) &&
+                !eq((~b), 0) ?
+                (((~a) + (~b) * (~x)) * SymbolicUtils.cosint((~a) + (~b) * (~x)) -
+                    sin((~a) + (~b) * (~x))) ⨸ (~b) : nothing
+        ),
+    )
+
+]

test/methods/rule_based/test_special_functions.jl

@@ -47,8 +47,8 @@ using Symbolics
     )
     @test isequal(
         integrate((1 + x) * SpecialFunctions.airyai(x), x, method),
-        SymbolicIntegration.∫(SpecialFunctions.airyai(x), x) +
-            SymbolicIntegration.∫(x * SpecialFunctions.airyai(x), x),
+        SpecialFunctions.airyaiprime(x) +
+            SymbolicIntegration.∫(SpecialFunctions.airyai(x), x),
     )
     @test isequal(
         integrate(x^3 * SpecialFunctions.besseli(0, a * x), x, method),
@@ -60,4 +60,50 @@ using Symbolics
         -x^3 * SpecialFunctions.besselk(1, a * x) / a -
             2 * x^2 * SpecialFunctions.besselk(2, a * x) / a^2,
     )
+
+    @test isequal(
+        integrate(x * SpecialFunctions.hankelh1(0, a * x), x, method),
+        x * SpecialFunctions.hankelh1(1, a * x) / a,
+    )
+    @test isequal(
+        integrate(x^3 * SpecialFunctions.hankelh2(0, a * x), x, method),
+        x^3 * SpecialFunctions.hankelh2(1, a * x) / a -
+            2 * x^2 * SpecialFunctions.hankelh2(2, a * x) / a^2,
+    )
+    @test isequal(
+        integrate(SpecialFunctions.airyaiprime(a * x), x, method),
+        SpecialFunctions.airyai(a * x) / a,
+    )
+    @test isequal(
+        integrate(SpecialFunctions.airybiprime(a * x), x, method),
+        SpecialFunctions.airybi(a * x) / a,
+    )
+    @test isequal(
+        integrate(x * SpecialFunctions.airyai(x), x, method),
+        SpecialFunctions.airyaiprime(x),
+    )
+    @test isequal(
+        integrate(x * SpecialFunctions.airybi(x), x, method),
+        SpecialFunctions.airybiprime(x),
+    )
+    @test isequal(
+        integrate(SpecialFunctions.erf(a * x), x, method),
+        (exp(-(a^2) * (x^2)) / sqrt(π) + a * x * SpecialFunctions.erf(a * x)) / a,
+    )
+    @test isequal(
+        integrate(SpecialFunctions.erfi(a * x), x, method),
+        (a * x * SpecialFunctions.erfi(a * x) - exp((a^2) * (x^2)) / sqrt(π)) / a,
+    )
+    @test isequal(
+        integrate(SpecialFunctions.expinti(a * x), x, method),
+        (-exp(a * x) + a * x * SpecialFunctions.expinti(a * x)) / a,
+    )
+    @test isequal(
+        integrate(SpecialFunctions.sinint(a * x), x, method),
+        (cos(a * x) + a * x * SpecialFunctions.sinint(a * x)) / a,
+    )
+    @test isequal(
+        integrate(SpecialFunctions.cosint(a * x), x, method),
+        (-sin(a * x) + a * x * SpecialFunctions.cosint(a * x)) / a,
+    )
 end