Read an exponential of an inverse below the bar as a negative multiple

BREAKING-CHANGES.md

@@ -595,6 +595,21 @@ no numeric root finder applies, and its discriminant vanishes identically. Such
 | `"1/(x^3-3*b*x+2*b^(3/2))".ToEntity().Integrate("x")` | `integral(...)` | 122 characters |
 | `"1/(-9*b*x+9*x^3+2*b^(3/2)*sqrt(3))".ToEntity().Integrate("x")` | `integral(...)` | 176 characters |
 
+### An exponential of i arctan below the bar is read as a negative multiple
+
+**Answers where there were none.** `x^2/(e^(i arctan(a x)) (c + a^2 c x^2)^(3/2))` was left unevaluated: the exponential
+below the bar was read as `n = 1`. It is taken above the bar with its exponent negated, and for a negative whole `n`,
+`e^(n i arctan(L))` is written over the conjugate, `(1 - i L)^(-n) (1 + L^2)^(n/2)`, which puts the linear with `i` in
+it above the bar where the rational rules read it
+([#718](https://github.com/asc-community/AngouriMath/issues/718)).
+
+| Input | Was (2.5.0) | Now |
+|---|---|---|
+| `"x^2/(e^(i*atan(a*x))*(c+a^2*c*x^2)^(3/2))".ToEntity().Integrate("x")` | `integral(...)` | 211 characters |
+| `"x/(e^(i*atan(a*x))*(c+a^2*c*x^2)^(3/2))".ToEntity().Integrate("x")` | `integral(...)` | 109 characters |
+| `"1/(e^(i*atan(a*x))*(c+a^2*c*x^2)^(3/2))".ToEntity().Integrate("x")` | `integral(...)` | 132 characters |
+| `"x^2/(e^(5*i*atan(a*x))*(c+a^2*c*x^2)^(27/2))".ToEntity().Integrate("x")` | `integral(...)` | 189 characters |
+
 ### A symbol shared by a numerator sum's coefficients is taken out before the partial fractions
 
 **Answers where there were none.** `(b d + 2c d x)^9/(a + b x + c x^2)^3` ran past a minute, where

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

@@ -9778,6 +9778,21 @@ node is Powf(var @base, Number.Rational power) && power is not Number.Integer &&
         /// </summary>
         internal static Entity? SolveByWritingAnExponentialOfAnInverseAlgebraically(Entity expr, Entity.Variable x, bool integrateByParts)
         {
+            // An exponential below the bar taken above it with its exponent negated, so that a
+            // negative multiple is read as one: `x^2/(e^(i arctan(a x)) (c + a^2 c x^2)^(3/2))` is
+            // `x^2 e^(-i arctan(a x))/(c + a^2 c x^2)^(3/2)`, which is written over the conjugate.
+            if (expr is Divf(var overIt, var underIt) && Mulf.LinearChildren(underIt).Any(IsAnExponentialOfAnInverse))
+            {
+                Entity kept = Number.Integer.One, raised = Number.Integer.One;
+                foreach (var factor in Mulf.LinearChildren(underIt))
+                    if (IsAnExponentialOfAnInverse(factor) && factor is Powf(_, var power))
+                        raised *= MathS.Pow(MathS.e, -power);
+                    else
+                        kept *= factor;
+                expr = overIt * raised / kept;
+            }
+            bool IsAnExponentialOfAnInverse(Entity node) => node is Powf(var e0, var power0) && e0 == MathS.e && power0.ContainsNode(x)
+                && power0.Nodes.Any(inner => inner is Arctanf or Arcsinf or Arccosf);
             var rewrote = false;
             // A power of a quadratic in x outside the exponentials, which `(1 + L^2)^(-n/2)` meets.
             var quadraticElsewhere = expr.Nodes.Any(node => node is Powf(var q, _) && q.ContainsNode(x) && !q.Nodes.Any(inner => inner is Arctanf or Arcsinf or Arccosf)
@@ -9846,6 +9861,16 @@ node is Powf(var @base, Number.Rational power) && power is not Number.Integer &&
                     var halfPower = Number.Rational.Create(n.ERational.Negate().Divide(2));
                     if (n is not Number.Integer)
                         return MathS.Pow((1 + MathS.i * argument) / (1 - MathS.i * argument), Number.Rational.Create(n.ERational.Divide(2)));
+                    // A negative n over the conjugate, `(1 + i L)^(-1)` being `(1 - i L)/(1 + L^2)`:
+                    // the linear with `i` in it is then above the bar, where the rational rules read
+                    // it, and `x^2/((1 + i a x)(1 + a^2 x^2))` beside it was left unevaluated where
+                    // `x^2 (1 - i a x)/(1 + a^2 x^2)^2` is answered.
+                    if (n is Number.Integer { IsNegative: true })
+                    {
+                        var conjugatePower = Number.Integer.Create(((Number.Integer)n).EInteger.Negate());
+                        return (conjugatePower == Number.Integer.One ? 1 - MathS.i * argument : MathS.Pow(1 - MathS.i * argument, conjugatePower))
+                            * MathS.Pow(1 + MathS.Sqr(argument), Number.Rational.Create(n.ERational.Divide(2)));
+                    }
                     return (n == Number.Integer.One ? 1 + MathS.i * argument : MathS.Pow(1 + MathS.i * argument, n)) * MathS.Pow(1 + MathS.Sqr(argument), halfPower);
                 }
                 Entity unit = inverse is Arcsinf

Sources/Tests/UnitTests/Calculus/AnExponentialOfAnArctangentBelowTheBarIntegralTest.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 <c>i arctan(a x)</c> below the bar, taken above it as a negative multiple and
+    /// written over the conjugate: Rubi's 5.3.6 <c>x^2/(e^(i arctan(a x)) (c + a^2 c x^2)^(3/2))</c>,
+    /// which was left unevaluated.
+    /// <a href="https://github.com/asc-community/AngouriMath/issues/718">#718</a>
+    /// </summary>
+    [Trait("Area", "Calculus")]
+    public sealed class AnExponentialOfAnArctangentBelowTheBarIntegralTest
+    {
+        private static void DifferentiatesBack(string integrand, params (string, double)[] pins)
+        {
+            var integral = integrand.ToEntity().Integrate("x");
+            Assert.DoesNotContain("integral(", integral.Stringize());
+
+            var pinned = integral.Substitute("C", 0);
+            Entity original = integrand.ToEntity();
+            foreach (var (name, value) in pins)
+            {
+                pinned = pinned.Substitute(name, value);
+                original = original.Substitute(name, value);
+            }
+            var derivative = pinned.Differentiate("x");
+            foreach (var at in new[] { -2.3, -0.7, 0.4, 1.9 })
+            {
+                var got = derivative.Substitute("x", at).EvalNumerical();
+                var want = original.Substitute("x", at).EvalNumerical();
+                var difference = Math.Abs((double)(got - want).RealPart) + Math.Abs((double)(got - want).ImaginaryPart);
+                var scale = Math.Max(1.0, Math.Abs((double)want.RealPart) + Math.Abs((double)want.ImaginaryPart));
+                Assert.True(difference / scale < 1e-8,
+                    $"d/dx of the antiderivative of {integrand} is {got} at x = {at}, where the integrand is {want}");
+            }
+        }
+
+        [Theory]
+        [InlineData("x^2/(e^(i*atan(a*x))*(c+a^2*c*x^2)^(3/2))")]
+        [InlineData("x/(e^(i*atan(a*x))*(c+a^2*c*x^2)^(3/2))")]
+        [InlineData("1/(e^(i*atan(a*x))*(c+a^2*c*x^2)^(3/2))")]
+        public void IsAnsweredAndDifferentiatesBack(string integrand)
+        {
+            DifferentiatesBack(integrand, ("a", 1.3), ("c", 0.7));
+            DifferentiatesBack(integrand, ("a", -0.5), ("c", 2.5));
+        }
+    }
+}