Integrate an exponential of a quadratic over the linear through its vertex

BREAKING-CHANGES.md

@@ -84,6 +84,19 @@ coefficients are the roots of that quadratic, and the polynomial over it is spli
 | `"1/(4*a*c+4*c^2*x^2+4*c*d*x^3+d^2*x^4)".ToEntity().Integrate("x")` | `integral(...)` | 3,920 characters |
 | `"x/(4*a*c+4*c^2*x^2+4*c*d*x^3+d^2*x^4)".ToEntity().Integrate("x")` | `integral(...)` | 2,315 characters |
 
+### An exponential of a quadratic over a power of the linear through its vertex is a moment about it
+
+**Answers where there were none.** `f^(a + b x + c x^2)/(b + 2 c x)` was declined, and so were five more of Rubi's 2.3,
+`f^(a + b x + c x^2)/(b + 2 c x)^n` and `f^(b x + c x^2)/(b + 2 c x)^n` for `n` up to 3. Under `u = b + 2 c x` the exponent is
+`u^2/(4 c) + a - b^2/(4 c)`, with no linear part, and the integrand is a moment about 0 of the Gaussian: the
+exponential integral of the square for an odd `n`, the error function for an even one
+([#233](https://github.com/asc-community/AngouriMath/issues/233)).
+
+| Input | Was (2.5.0) | Now |
+|---|---|---|
+| `"f^(a+b*x+c*x^2)/(b+2*c*x)".ToEntity().Integrate("x")` | `integral(...)` | `f^(a - b^2/(4 c)) Ei(ln(f) (b + 2 c x)^2/(4 c))/(4 c)` |
+| `"e^(x^2+2*x)/(x+2)".ToEntity().Integrate("x")` | `integral(...)` | `integral(...)`, not through the vertex |
+
 ### `|` is divisibility, and was disjunction
 
 **Read this one first.** It is the only entry here that changes what existing input *means* rather

Sources/AngouriMath/Functions/Continuous/Integration/IndefiniteIntegralSolver.cs

@@ -6995,6 +6995,65 @@ private static bool TryReadAMonomial(Entity term, Entity.Variable x, out ERation
             return Functions.PartialFractions.Bare((constant * shifted / d * (sum + ei * MathS.Ei(rate * u)).Substitute(u, linear)).InnerSimplified);
         }
 
+        /// <summary>
+        /// An exponential of a quadratic over a whole power of the linear through its vertex:
+        /// <c>F^(A x^2 + B x + C)/(p + q x)^n</c> with <c>2 A p = q B</c>, which is how
+        /// <c>f^(a + b x + c x^2)/(b + 2 c x)^n</c> arrives. Under <c>u = p + q x</c> the exponent is
+        /// <c>A/q^2 u^2 + C - B^2/(4 A)</c>, with no linear part, and the integrand
+        /// <c>F^(A/q^2 u^2 + C - B^2/(4 A))/(q u^n)</c> is a moment about 0 of the Gaussian: the
+        /// exponential integral of the square for an odd <c>n</c>, the error function for an even
+        /// one. Rubi's 2.3. https://github.com/asc-community/AngouriMath/issues/233
+        /// </summary>
+        /// <remarks>
+        /// Only through the vertex: anywhere else the shift leaves a linear part in the exponent,
+        /// and the moments about 0 do not answer it. The exponent in <c>u</c> is written from the
+        /// coefficients, not by substituting <c>x = (u - p)/q</c> and asking the simplifier to see
+        /// that the linear part cancels.
+        /// </remarks>
+        internal static Entity? SolveAnExponentialOfAQuadraticOverTheLinearThroughItsVertex(Entity expr, Entity.Variable x, bool integrateByParts)
+        {
+            if (expr is not Divf)
+                return null;
+            Powf? exponential = null;
+            Entity? linear = null;
+            var power = EInteger.Zero;
+            Entity constant = Number.Integer.One;
+            foreach (var (factor, underneath) in FactorsOfTheIntegrand(expr))
+            {
+                if (!factor.ContainsNode(x))
+                {
+                    constant = underneath ? constant / factor : constant * factor;
+                    continue;
+                }
+                if (!underneath && exponential is null && factor is Powf(var @base, var exponent) && !@base.ContainsNode(x) && exponent.ContainsNode(x))
+                {
+                    exponential = (Powf)factor;
+                    continue;
+                }
+                // The linear, or a whole power of it, below the bar; nothing else.
+                if (!underneath)
+                    return null;
+                var (candidate, k) = factor is Powf(var below, Number.Integer n) && n.EInteger.Sign > 0 && n.EInteger.CanFitInInt32()
+                    ? (below, n.EInteger) : (factor, EInteger.One);
+                if (linear is not null && linear != candidate)
+                    return null;
+                linear = candidate;
+                power += k;
+            }
+            if (exponential is null || linear is null || power.CompareTo(EInteger.FromInt32(32)) > 0
+                || !TreeAnalyzer.TryGetPolyLinear(linear, x, out var q, out var p) || TreeAnalyzer.IsZero(q)
+                || !TreeAnalyzer.TryGetPolyQuadratic(exponential.Exponent, x, out var a, out var b, out var c) || TreeAnalyzer.IsZero(a)
+                || !VanishesIdentically(2 * a * p - q * b))
+                return null;
+            var u = Variable.CreateUnique(expr, "u");
+            var exponentInU = (a / MathS.Sqr(q)).InnerSimplified * MathS.Sqr(u) + (c - MathS.Sqr(b) / (4 * a)).InnerSimplified;
+            var question = MathS.Pow(exponential.Base, exponentInU) / (power.Equals(EInteger.One) ? u : MathS.Pow(u, Number.Integer.Create(power)));
+            if (Integration.ComputeAsAQuestionOfItsOwn(question, u, integrateByParts) is not { } answer
+                || answer.Nodes.Any(node => node is Integralf || node == MathS.NaN))
+                return null;
+            return (constant / q * answer.Substitute(u, linear)).InnerSimplified;
+        }
+
         /// <summary>
         /// <c>int u^m e^(k u) du</c>, where <paramref name="exponential"/> is <c>e^(k u)</c>, as its
         /// elementary part and the coefficient of <c>Ei(k u)</c> in it:

Sources/AngouriMath/Functions/Continuous/Integration/Integration.Definition.cs

@@ -827,6 +827,8 @@ private static Entity WithItsPiecewiseSumsCombined(Entity expr)
             if ((answer = IndefiniteIntegralSolver.SolveAProductOfExponentialsOfQuadratics(expr, x)) is { }) return answer;
             // An exponential of a linear over a power of a linear, onto the exponential integral.
             if ((answer = IndefiniteIntegralSolver.SolveAnExponentialOfALinearOverAPowerOfALinear(expr, x)) is { }) return answer;
+            // And of a quadratic over a power of the linear through its vertex, onto the moments about 0.
+            if ((answer = IndefiniteIntegralSolver.SolveAnExponentialOfAQuadraticOverTheLinearThroughItsVertex(expr, x, integrateByParts)) is { }) return answer;
             // And a sum of exponentials, which is how sinh and cosh arrive, onto Shi and Chi.
             if ((answer = IndefiniteIntegralSolver.SolveAHyperbolicOfALinearOverAPowerOfALinear(expr, x)) is { }) return answer;
             // And either over several linears, split into partial fractions over them first, as the

Sources/Tests/UnitTests/Calculus/AnExponentialOfAQuadraticOverTheLinearThroughItsVertexIntegralTest.cs

@@ -0,0 +1,57 @@
+//
+// Copyright (c) 2019-2026 Angouri.
+// AngouriMath is licensed under MIT.
+// Details: https://github.com/asc-community/AngouriMath/blob/master/LICENSE.md.
+// Website: https://am.angouri.org.
+//
+
+using System;
+using AngouriMath.Extensions;
+using Xunit;
+
+namespace AngouriMath.Tests.Calculus
+{
+    /// <summary>
+    /// An exponential of a quadratic over a power of the linear through its vertex,
+    /// <c>f^(a + b x + c x^2)/(b + 2 c x)^n</c>. Under <c>u = b + 2 c x</c> the exponent has no linear
+    /// part, and the integrand is a moment about 0 of the Gaussian, which the integrator answers.
+    /// They were declined. Rubi's 2.3.
+    /// <a href="https://github.com/asc-community/AngouriMath/issues/233">#233</a>
+    /// </summary>
+    [Trait("Area", "Calculus")]
+    public sealed class AnExponentialOfAQuadraticOverTheLinearThroughItsVertexIntegralTest
+    {
+        [Theory]
+        [InlineData("f^(a + b*x + c*x^2)/(b + 2*c*x)")]
+        [InlineData("f^(a + b*x + c*x^2)/(b + 2*c*x)^2")]
+        [InlineData("f^(a + b*x + c*x^2)/(b + 2*c*x)^3")]
+        [InlineData("f^(b*x + c*x^2)/(b + 2*c*x)^3")]
+        [InlineData("e^(x^2 + 2*x)/(x + 1)")]
+        public void AMomentAboutTheVertex(string integrand)
+        {
+            var integral = integrand.ToEntity().Integrate("x");
+            var text = integral.Stringize();
+            Assert.DoesNotContain("integral(", text);
+            Entity Pinned(Entity e) => e.Substitute("a", 1.3).Substitute("b", 0.7).Substitute("c", 0.4).Substitute("f", 2.3);
+            var derivative = Pinned(integral.Substitute("C", 0)).Differentiate("x");
+            var original = Pinned(integrand.ToEntity());
+            var compared = 0;
+            foreach (var at in new[] { -2.9, -0.4, 0.3, 0.8, 1.3, 2.1 })
+            {
+                var want = original.Substitute("x", at).EvalNumerical();
+                var got = derivative.Substitute("x", at).EvalNumerical();
+                if (want.IsNaN)
+                    continue;
+                compared++;
+                Assert.True(Math.Abs((double)(got - want).RealPart) + Math.Abs((double)(got - want).ImaginaryPart)
+                            < 1e-9 * Math.Max(1, Math.Abs((double)want.RealPart) + Math.Abs((double)want.ImaginaryPart)),
+                    $"d/dx of the antiderivative of {integrand} is {got} at x = {at}, where the integrand is {want}");
+            }
+            Assert.True(compared >= 5, $"only {compared} points could be compared for {integrand}");
+        }
+
+        [Fact]
+        public void NotThroughTheVertexIsDeclined()
+            => Assert.Contains("integral(", "e^(x^2 + 2*x)/(x + 2)".ToEntity().Integrate("x").Stringize());
+    }
+}