e^(i arctan(L)) beside a function of x is written as two powers of linears

BREAKING-CHANGES.md

@@ -240,6 +240,21 @@ integrals with `0` ([#1769](https://github.com/asc-community/AngouriMath/issues/
 | `"e^(-160)".ToEntity().InnerSimplified` | `0` | `e ^ (-160)` |
 | `"x^n*((1 - d^2)/x - x)^3*(1 + x^2 - d*x)/(x - d)/pi^2".ToEntity().Simplify()` | `0 provided not x - d = 0 and ...` | `x ^ n * ((1 - d ^ 2) / x - x) ^ 3 * (1 - d * x + x ^ 2) / (pi ^ 2 * (x - d))` |
 
+### `e^(i arctan(L))` beside a function of `x` is written as two powers of linears
+
+**Answers where there were none, and shorter ones.** `1/(e^(i arctan(a + b x)) x^2)` ran past thirty
+seconds: `e^(i arctan(L))` was written `(1 + i L)/sqrt(1 + L^2)`, a root of a quadratic beside the pole.
+It is `(1 + i L)^(1/2) (1 - i L)^(-1/2)` as well, for a real `L` -- the two principal powers' arguments are
+`arctan(L)/2` each and their moduli cancel -- powers of two linears, and beside a function of `x` it is
+written so. Of Rubi's 5.3.6, four integrals are answered that were not, and the 28 answers that change are
+a ninth of the length together ([#718](https://github.com/asc-community/AngouriMath/issues/718)).
+
+| Input | Was (2.5.0) | Now |
+|---|---|---|
+| `"1/(e^(i*atan(a + b*x))*x^2)".ToEntity().Integrate("x")` | `integral(...)`; past thirty seconds on the unreleased master | 340 characters |
+| `"x^3/e^(i*atan(a + b*x))".ToEntity().Integrate("x")` | `integral(...)`; past thirty seconds on the unreleased master | 373 characters |
+| `"x^3/e^(i*atan(a*x))".ToEntity().Integrate("x")` | `integral(...)`; 25,944 characters on the unreleased master | 218 characters |
+
 ### The floor and the ceiling of a rational are exact whatever its size
 
 **Wrong answers fixed.** The floor and the ceiling of an exact rational went through a decimal at the

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

@@ -8549,6 +8549,11 @@ node is Powf(var @base, Number.Rational power) && power is not Number.Integer &&
         internal static Entity? SolveByWritingAnExponentialOfAnInverseAlgebraically(Entity expr, Entity.Variable x, bool integrateByParts)
         {
             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)
+                && TreeAnalyzer.TryGetPolyQuadratic(q, x, out var leading, out _, out _) && !TreeAnalyzer.IsZero(leading));
+            // The variable outside the exponentials.
+            var xElsewhere = expr.Replace(node => node is Powf(var e, var power) && e == MathS.e && power.ContainsNode(x) ? Number.Integer.One : node).ContainsNode(x);
             var rewritten = expr.Replace(node =>
             {
                 if (node is not Powf(var @base, var exponent) || @base != MathS.e || !exponent.ContainsNode(x))
@@ -8583,6 +8588,20 @@ node is Powf(var @base, Number.Rational power) && power is not Number.Integer &&
                 // of different orders that nothing reads.
                 if (inverse is Arctanf)
                 {
+                    // `(1 + i L)^(n/2) (1 - i L)^(-n/2)`: the two principal powers' arguments are
+                    // `(n/2) arctan(L)` each and their moduli cancel, for a real L, so it is
+                    // `e^(n i arctan(L))` exactly, powers of two linears, which the rules for a
+                    // product of linear powers read. For n = 1 or -1 beside a function of x and no
+                    // power of a quadratic in x for the other form to meet: `e^(i arctan(a + b x))/x^2`
+                    // was `(1 + i L)/sqrt(1 + L^2)` over `x^2`, a root of a quadratic beside a pole, and
+                    // ran past thirty seconds, and `x^3 e^(-i arctan(a x))` came out in 26 thousand
+                    // characters where so it is 218. Alone the exponential is answered shorter as the
+                    // root, and for a power not whole the quotient's power is.
+                    if ((n == Number.Integer.One || n == Number.Integer.MinusOne) && !quadraticElsewhere && xElsewhere)
+                    {
+                        var half = Number.Rational.Create(n.ERational.Divide(2));
+                        return MathS.Pow(1 + MathS.i * argument, half) * MathS.Pow(1 - MathS.i * argument, Number.Rational.Create(half.ERational.Negate()));
+                    }
                     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)));

Sources/Tests/UnitTests/Calculus/AnExponentialOfAnArctangentBesideAPowerIntegralTest.cs

@@ -0,0 +1,52 @@
+//
+// 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>
+    /// <c>e^(+-i arctan(L))</c> beside a function of <c>x</c>, written as <c>(1 + i L)^(+-1/2) (1 - i L)^(-+1/2)</c>:
+    /// the two principal powers' arguments are <c>arctan(L)/2</c> each and their moduli cancel, for a real
+    /// <c>L</c>. Rubi's 5.3.6. The integrands are complex for a real <c>x</c> and are compared as complex numbers.
+    /// <a href="https://github.com/asc-community/AngouriMath/issues/718">#718</a>
+    /// </summary>
+    [Trait("Area", "Calculus")]
+    public sealed class AnExponentialOfAnArctangentBesideAPowerIntegralTest
+    {
+        [Theory]
+        [InlineData("1/(e^(i*atan(a + b*x))*x^2)")]
+        [InlineData("x^3/e^(i*atan(a + b*x))")]
+        [InlineData("x^3/e^(i*atan(a*x))")]
+        [InlineData("e^(i*atan(a*x))*x^2")]
+        public void AsTwoPowersOfLinears(string integrand)
+        {
+            var integral = integrand.ToEntity().Integrate("x");
+            var text = integral.Stringize();
+            Assert.DoesNotContain("integral(", text);
+            Assert.True(text.Length < 5000, $"{text.Length} characters of answer for {integrand}");
+            Entity Pinned(Entity e) => e.Substitute("a", 1.3).Substitute("b", 0.7);
+            var derivative = Pinned(integral.Substitute("C", 0)).Differentiate("x");
+            var original = Pinned(integrand.ToEntity());
+            var compared = 0;
+            foreach (var at in new[] { -1.2, -0.7, 0.3, 0.8, 1.3, 2.9 })
+            {
+                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}");
+        }
+    }
+}