Write an exponential of a symbolic multiple of i arctan as powers of the linears

BREAKING-CHANGES.md

@@ -607,6 +607,19 @@ partial fractions divide. It is taken out in front of the integral first
 | `"(b*d + 2*c*d*x)^9/(a + b*x + c*x^2)^3".ToEntity().Integrate("x")` | past a minute | 76,249 characters, the length the content-free one already had |
 | `"(b*d + 2*c*d*x)^8/(a + b*x + c*x^2)^3".ToEntity().Integrate("x")` | past a minute | 65,236 characters |
 
+### An exponential of a symbolic multiple of i arctan is written as powers of the linears
+
+**Answers where there were none.** `(c + a^2 c x^2)^p/e^(2 i p arctan(a x))` was left unevaluated. For a real `L`,
+`e^(k i arctan(L))` is `(1 + i L)^(k/2) (1 - i L)^(-k/2)` and `(K (1 + L^2))^q` is `K^q (1 + i L)^q (1 - i L)^q`, for any
+`k` and `q`, so the integrand is a product of powers of two linears, and with these exponents a single power
+([#718](https://github.com/asc-community/AngouriMath/issues/718)).
+
+| Input | Was (2.5.0) | Now |
+|---|---|---|
+| `"(c+a^2*c*x^2)^p/e^(2*i*p*atan(a*x))".ToEntity().Integrate("x")` | `integral(...)` | 84 characters |
+| `"e^(2*i*p*atan(a*x))*(c+a^2*c*x^2)^p".ToEntity().Integrate("x")` | `integral(...)` | 86 characters |
+| `"e^(i*n*atan(a*x))/(c+a^2*c*x^2)".ToEntity().Integrate("x")` | `integral(...)` | 123 characters |
+
 ### A whole power is taken out of a power of a product, in front of a constant
 
 **Answers where there were none.** `(c sin(a + b x)^3)^(1/3)/x` was declined: nothing read a function under a root of

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

@@ -9678,6 +9678,93 @@ node is Powf(var @base, Number.Rational power) && power is not Number.Integer &&
             return ratio;
         }
 
+        /// <summary>
+        /// An exponential of a symbolic multiple of <c>i arctan(L)</c>, <c>L</c> linear, beside
+        /// powers of multiples of <c>1 + L^2</c>, written as powers of the two linears:
+        /// <c>e^(k i arctan(L))</c> is <c>(1 + i L)^(k/2) (1 - i L)^(-k/2)</c> and
+        /// <c>(K (1 + L^2))^q</c> is <c>K^q (1 + i L)^q (1 - i L)^q</c>, each exactly for a real
+        /// <c>L</c> and any <c>k</c> and <c>q</c>: the two principal arguments are
+        /// <c>arctan(L)</c> and its negative, and <c>1 + L^2</c> is positive.
+        /// </summary>
+        /// <remarks>
+        /// Rubi's 5.3.6 <c>(c + a^2 c x^2)^p/e^(2 i p arctan(a x))</c> is then
+        /// <c>c^p (1 - i a x)^(2p)</c>, a power of a linear; it was left unevaluated.
+        /// <see cref="SolveByWritingAnExponentialOfAnInverseAlgebraically"/> takes a rational
+        /// multiple, and this the rest.
+        /// https://github.com/asc-community/AngouriMath/issues/718
+        /// </remarks>
+        internal static Entity? SolveAnExponentialOfASymbolicMultipleOfAnArctangentAsPowersOfTheLinears(Entity expr, Entity.Variable x, bool integrateByParts)
+        {
+            static IEnumerable<(Entity Factor, bool Below)> Factors(Entity e, bool below) => e switch
+            {
+                Mulf(var l, var r) => Factors(l, below).Concat(Factors(r, below)),
+                Divf(var l, var r) => Factors(l, below).Concat(Factors(r, !below)),
+                _ => new[] { (e, below) }
+            };
+            var factors = Factors(expr, false).ToList();
+            Entity? argument = null;
+            Entity? multiple = null;
+            foreach (var (factor, below) in factors)
+            {
+                if (factor is not Powf(var e, var exponent) || e != MathS.e || !exponent.ContainsNode(x))
+                    continue;
+                if (argument is not null)
+                    return null;
+                var arctangents = exponent.Nodes.OfType<Arctanf>().Distinct().ToList();
+                if (arctangents.Count != 1 || !TreeAnalyzer.TryGetPolyLinear(arctangents[0].Argument, x, out var slope, out _) || TreeAnalyzer.IsZero(slope))
+                    return null;
+                var placeholder = Variable.CreateUnique(exponent, "u_atan");
+                if (!TreeAnalyzer.TryGetPolyLinear(exponent.Replace(node => node == arctangents[0] ? placeholder : node), placeholder, out var coefficient, out var free)
+                    || coefficient is null || free is null || coefficient.ContainsNode(x) || !TreeAnalyzer.IsZero(free))
+                    return null;
+                // Simplified, since `2 i p/i` is not reduced to `2 p` short of it.
+                var k = (coefficient / MathS.i).Simplify();
+                // A rational multiple is the other rule's.
+                if (k.Evaled is Number.Rational)
+                    return null;
+                argument = arctangents[0].Argument;
+                multiple = below ? -k : k;
+            }
+            if (argument is null || multiple is null)
+                return null;
+            var quadratic = 1 + MathS.Sqr(argument);
+            Entity constant = Number.Integer.One;
+            Entity plusPower = multiple / 2, minusPower = -multiple / 2;
+            foreach (var (factor, below) in factors)
+            {
+                if (factor is Powf(var e, var exponent) && e == MathS.e && exponent.ContainsNode(x))
+                    continue;
+                if (!factor.ContainsNode(x))
+                {
+                    constant = below ? constant / factor : constant * factor;
+                    continue;
+                }
+                var (@base, power) = factor is Powf(var b, var q) ? (b, q) : (factor, (Entity)Number.Integer.One);
+                if (power.ContainsNode(x)
+                    || !TreeAnalyzer.TryGetPolyQuadratic(@base, x, out var a2, out var b2, out var c2)
+                    || !TreeAnalyzer.TryGetPolyQuadratic(quadratic, x, out var a3, out var b3, out var c3)
+                    || a2 is null || b2 is null || c2 is null || a3 is null || b3 is null || c3 is null || TreeAnalyzer.IsZero(a3))
+                    return null;
+                var ratio = (a2 / a3).InnerSimplified;
+                if (!Functions.PartialFractions.IsZeroAsAValue((b2 - ratio * b3).InnerSimplified)
+                    || !Functions.PartialFractions.IsZeroAsAValue((c2 - ratio * c3).InnerSimplified))
+                    return null;
+                var signed = below ? -power : power;
+                constant *= MathS.Pow(ratio, signed);
+                plusPower += signed;
+                minusPower += signed;
+            }
+            // Simplified, x-free and small: `-p + p` is a zero power only so.
+            plusPower = plusPower.Simplify();
+            minusPower = minusPower.Simplify();
+            Entity written = constant;
+            if (!TreeAnalyzer.IsZero(plusPower))
+                written *= MathS.Pow(1 + MathS.i * argument, plusPower);
+            if (!TreeAnalyzer.IsZero(minusPower))
+                written *= MathS.Pow(1 - MathS.i * argument, minusPower);
+            return Integration.ComputeAsAQuestionOfItsOwn(written, x, integrateByParts);
+        }
+
         /// <summary>
         /// An exponential of <c>i</c> times an inverse trigonometric function, written
         /// algebraically: <c>e^(i arctan(L))</c> is <c>(1 + i L)/sqrt(1 + L^2)</c>,

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

@@ -671,6 +671,8 @@ private static Entity WithItsPiecewiseSumsCombined(Entity expr)
             // question asked, before the rules that spent the budget: `e^(3 i atan(a + b x))/x^2` ran out of
             // time where, rewritten, it is answered in half a second. It is asked again in its place below.
             if (AnsweringTheQuestionAsked && (answer = IndefiniteIntegralSolver.SolveByWritingAnExponentialOfAnInverseAlgebraically(expr, x, integrateByParts)) is { }) return answer;
+            // The same for a symbolic multiple of the arctangent, beside powers of its quadratic.
+            if (AnsweringTheQuestionAsked && (answer = IndefiniteIntegralSolver.SolveAnExponentialOfASymbolicMultipleOfAnArctangentAsPowersOfTheLinears(expr, x, integrateByParts)) is { }) return answer;
             // A polynomial over a symbolic quartic that splits into two quadratics in a hidden variable,
             // palindromic or even after a shift: the rules below spent seconds declining both.
             if ((answer = IndefiniteIntegralSolver.SolveAPolynomialOverASymbolicQuarticThatSplitsIntoTwoQuadratics(expr, x, integrateByParts)) is { }) return answer;

Sources/Tests/UnitTests/Calculus/AnExponentialOfASymbolicMultipleOfAnArctangentIntegralTest.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 symbolic multiple of <c>i arctan(a x)</c> beside a power of its quadratic,
+    /// written as powers of the two linears: Rubi's 5.3.6
+    /// <c>(c + a^2 c x^2)^p/e^(2 i p arctan(a x))</c>, which was left unevaluated.
+    /// <a href="https://github.com/asc-community/AngouriMath/issues/718">#718</a>
+    /// </summary>
+    [Trait("Area", "Calculus")]
+    public sealed class AnExponentialOfASymbolicMultipleOfAnArctangentIntegralTest
+    {
+        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("(c+a^2*c*x^2)^p/e^(2*i*p*atan(a*x))")]
+        [InlineData("e^(2*i*p*atan(a*x))*(c+a^2*c*x^2)^p")]
+        [InlineData("e^(i*n*atan(a*x))/(c+a^2*c*x^2)")]
+        public void IsAnsweredAndDifferentiatesBack(string integrand)
+        {
+            DifferentiatesBack(integrand, ("a", 1.3), ("c", 0.7), ("p", 0.4), ("n", 0.6));
+            DifferentiatesBack(integrand, ("a", -0.5), ("c", -1.5), ("p", 1.3), ("n", -2.2));
+        }
+    }
+}