Integrate half-odd powers of a ± a sin beside a fractional power of the cosine

BREAKING-CHANGES.md

@@ -3708,6 +3708,27 @@ Rubi's 1.1.4.2, `(c x)^m (a x^j + b x^n)^p` ([#718](https://github.com/asc-commu
 | `"x/sqrt(a*x + b*x^4)".Integrate("x")` | left unevaluated | `K` times `2 ln(2 b x^(3/2) + 2 sqrt(b) sqrt(a + b x^3))/(3 sqrt(b))` for `b > 0`, an arcsine for `b < 0`; real where `x < -(a/b)^(1/3)` as well, where `K` is `-1` |
 | `"x^(3/2*(n - 1))/(a*x^(n - 1) + b*x^n + c*x^(n + 1))^(3/2)".Integrate("x")` | left unevaluated | `K (2 b + 4 c x)/((4 a c - b^2) sqrt(a + b x + c x^2))`; `K` is 1 for a positive `x`, and a sign for a negative one, `-1` for `n = 2.3` |
 
+### Half-odd powers of `a ± a sin` beside a fractional power of the cosine are integrated through the sine
+
+**Answers where there were none.** `(e cos(c + d x))^(5/2) (a + a sin(c + d x))^(3/2)`,
+`sqrt(e cos(c + d x))/sqrt(a + a sin(c + d x))` and the 22 others of Rubi's 4.1.1.2 with a half-odd power of each were
+declined. Under `u = sin(y)` they are `(1 + u)^A (1 - u)^B` with `A` and `B` quarter-odd, which by Chebyshev's theorem
+is elementary only where `A + B` is whole. The rule below now takes numeric powers where `A`, `B` or `A + B` is whole
+and the rest is a polynomial in `u`, also one level below the question, and declines
+`(a + a sin(y)) sqrt(e cos(y))`, which is elliptic, before asking anything. Three of the file the unreleased master
+answered in six to eleven seconds, under a condition, now come through it in a twentieth of a second and without one,
+and `sqrt(e sin(c + d x))/sqrt(a - a cos(c + d x))`, which it answered by the half angle only for positive `a` and `e`, is
+answered for either sign ([#718](https://github.com/asc-community/AngouriMath/issues/718)).
+
+| Input | Was (2.5.0) | Now |
+|---|---|---|
+| `"(k*cos(c + d*x))^(5/2)*(a + a*sin(c + d*x))^(3/2)".ToEntity().Integrate("x")` | `integral(...)` | 1,156 characters |
+| `"sqrt(k*cos(c + d*x))/sqrt(a + a*sin(c + d*x))".ToEntity().Integrate("x")` | `integral(...)` | 859 characters |
+| `"(a + a*sin(c + d*x))^(5/2)/(k*cos(c + d*x))^(13/2)".ToEntity().Integrate("x")` | `integral(...)` | 468 characters |
+| `"(k*cos(c + d*x))^(9/2)/(a + a*sin(c + d*x))^(5/2)".ToEntity().Integrate("x")` | `integral(...)` | 1,073 characters |
+| `"(a + a*sin(c + d*x))^(3/2)/(k*cos(c + d*x))^(9/2)".ToEntity().Integrate("x")` | `integral(...)`; 456 characters under a condition, in 10.6 s, on the unreleased master | 403 characters, in 61 ms |
+| `"sqrt(k*sin(c + d*x))/sqrt(a - a*cos(c + d*x))".ToEntity().Integrate("x")` | `integral(...)`; on the unreleased master, an answer only where `a > 0` and `k > 0` | 825 characters, for either sign |
+
 ### Symbolic powers of `a ± a sin` beside a power of the cosine are integrated through the sine
 
 **Answers where there were none.** `(a + a sin(h + f x))^m sqrt(c - c sin(h + f x))` and its kin,

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

@@ -5469,8 +5469,8 @@ Entity Integral(ERational power)
         }
 
         /// <summary>
-        /// Powers of <c>a ± a sin(y)</c> with a symbolic exponent, beside a power of
-        /// <c>g cos(y)</c> and anything else in the sine alone, under <c>u = sin(y)</c>:
+        /// Powers of <c>a ± a sin(y)</c> with a symbolic exponent, or beside a fractional power of
+        /// <c>g cos(y)</c>, and anything else in the sine alone, under <c>u = sin(y)</c>:
         /// <c>(1 + sin(y))(1 - sin(y))</c> is <c>cos(y)^2</c>, so each factor is a power of
         /// <c>1 + u</c> or of <c>1 - u</c> up to a factor constant on each interval where it is
         /// defined, and <c>dy = du/cos(y)</c> is one more pair of half powers. The cosine's the
@@ -5495,12 +5495,24 @@ Entity Integral(ERational power)
         /// <c>(a (1 + u))^m</c> is not <c>a^m (1 + u)^m</c> for every <c>a</c>, and the quotient
         /// of the two is constant on each interval either way. At the question asked only, and
         /// only where an exponent is symbolic: numeric half powers are the half angle's.
+        /// </para>
+        /// <para>
+        /// But for a fractional power of the cosine, which the half angle does not read: Rubi's
+        /// <c>(e cos(c + d x))^p (a + a sin(c + d x))^m</c> with half-odd <c>p</c> and <c>m</c>
+        /// were past the budget, the substitution search spending it on the two radicals. In
+        /// <c>u</c> they are <c>(1 + u)^A (1 - u)^B</c> with <c>A</c> and <c>B</c> quarter-odd, which
+        /// by Chebyshev's theorem is elementary only where <c>A + B = m + p - 1</c> is whole, and
+        /// with every power a number the rule asks only where <c>A</c>, <c>B</c> or <c>A + B</c> is
+        /// whole and the rest is a polynomial in <c>u</c>: <c>(a + a sin(y)) sqrt(e cos(y))</c>, which
+        /// is elliptic, is declined before anything is asked. Numeric powers are read one level
+        /// below the question too, where the polynomial term hands <c>1/(A B)</c> on as
+        /// <c>(A B)^(-1)</c>, and a whole power of a product is read as the powers of its factors.
         /// https://github.com/asc-community/AngouriMath/issues/718
         /// </para>
         /// </remarks>
         internal static Entity? SolveSymbolicPowersOfOnePlusMinusASineThroughTheSine(Entity expr, Entity.Variable x, bool integrateByParts)
         {
-            if (!Integration.AnsweringTheQuestionAsked)
+            if (!Integration.AnsweringTheQuestionAskedOrOneBelow)
                 return null;
             Entity? argument = null;
             foreach (var node in expr.Nodes)
@@ -5519,8 +5531,12 @@ Entity Integral(ERational power)
             var cosine = MathS.Cos(argument);
             // Each factor as a base and an exponent, a power of a power read as one power: the
             // quotient of the two spellings is constant on each interval, and K carries it.
+            // A whole power of a product is the product of the powers: `1/(A B)` is handed on as
+            // `(A B)^(-1)` by the polynomial term.
             var factors = new List<(Entity Factor, Entity Base, Entity Exponent)>();
-            foreach (var factor in Mulf.LinearChildren(expr))
+            foreach (var factor in Mulf.LinearChildren(expr).SelectMany(factor => factor is Powf(var product, Number.Integer power) && product is Mulf or Divf
+                         ? Mulf.LinearChildren(product).Select(part => (Entity)MathS.Pow(part, power))
+                         : new[] { factor }))
             {
                 Entity @base = factor, exponent = Number.Integer.One;
                 while (@base is Powf(var inner, var power))
@@ -5580,12 +5596,22 @@ Entity Integral(ERational power)
                     return null;
                 rest *= inU;
             }
-            if (!symbolic)
+            // Symbolic powers at the question asked only. Numeric powers are the half angle's, but
+            // for a fractional power of the complement, which it does not read; and those one level
+            // down too, where the polynomial term hands on `1/(A B)` as `(A B)^(-1)`.
+            if (symbolic ? !Integration.AnsweringTheQuestionAsked
+                    : !(ofTheComplement.InnerSimplified is Number.Rational complementPower && !IsWhole(complementPower.ERational)))
                 return null;
             // dy = du/complement takes one more half power of each.
             var half = Number.Rational.Create(1, 2);
             var a = (plus + ofTheComplement * half - half).Simplify();
             var b = (minus + ofTheComplement * half - half).Simplify();
+            // With every power a number, only where Chebyshev's theorem says the question in u is
+            // elementary, and beside a polynomial in u: otherwise it is a search past the budget.
+            if (!symbolic && !(a is Number.Rational ra && b is Number.Rational rb
+                    && (IsWhole(ra.ERational) || IsWhole(rb.ERational) || IsWhole(ra.ERational.Add(rb.ERational)))
+                    && IsAPolynomialInU(rest.InnerSimplified)))
+                return null;
             // A zero exponent leaves its factor out: `(1 - u)^0` is one provided `u` is not one, and
             // the condition would stand between the question and the rules that answer it.
             Entity PowerOf(Entity @base, Entity exponent) => exponent == Number.Integer.Zero ? Number.Integer.One : MathS.Pow(@base, exponent);
@@ -5619,6 +5645,14 @@ Entity Integral(ERational power)
                 }
                 return found;
             }
+
+            static bool IsWhole(ERational value) => value.IsInteger();
+
+            // Free of u, or a polynomial in it with coefficients free of it.
+            bool IsAPolynomialInU(Entity part)
+                => !part.ContainsNode(u)
+                   || TreeAnalyzer.TryGetPolynomial(part, u, out var terms)
+                      && terms.Keys.All(power => power.Sign >= 0) && terms.Values.All(coefficient => !coefficient.ContainsNode(u));
         }
 
         /// <summary>

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

@@ -1045,7 +1045,8 @@ private static Entity WithItsPiecewiseSumsCombined(Entity expr)
             if ((answer = IndefiniteIntegralSolver.SolveAPowerOfACosineAndASinePlusTheirAmplitude(expr, x)) is { }) return answer;
             // And symbolic powers of `a ± a sin(y)` beside a power of `g cos(y)`, by u = sin(y), where
             // `(1 + u)(1 - u)` is the cosine's square: the half angle wants numeric powers, and the
-            // substitution search spent the budget on these.
+            // substitution search spent the budget on these. Numeric ones too beside a fractional
+            // power of `g cos(y)`, which the half angle does not read.
             if ((answer = IndefiniteIntegralSolver.SolveSymbolicPowersOfOnePlusMinusASineThroughTheSine(expr, x, integrateByParts)) is { }) return answer;
             // And a half-odd power of a +- a sec(y), which is that square over cos(y): by the half-angle
             // tangent, in which the whole is rational beside one root.

Sources/Tests/UnitTests/Calculus/FractionalPowersOfACosineBesideOnePlusASineIntegralTest.cs

@@ -0,0 +1,61 @@
+//
+// 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>
+    /// Numeric powers of <c>a ± a sin(y)</c> beside a fractional power of <c>k cos(y)</c>, under
+    /// <c>u = sin(y)</c>: <c>(1 + u)^A (1 - u)^B</c> with <c>A + B</c> whole, which the half angle
+    /// does not read and the substitution search spent the budget on. Checked where the cosine is
+    /// positive, and with <c>k</c> negative where it is negative, both real. Rubi's 4.1.1.2.
+    /// <a href="https://github.com/asc-community/AngouriMath/issues/718">#718</a>
+    /// </summary>
+    [Trait("Area", "Calculus")]
+    public sealed class FractionalPowersOfACosineBesideOnePlusASineIntegralTest
+    {
+        [Theory]
+        [InlineData("(k*cos(c + d*x))^(5/2)*(a + a*sin(c + d*x))^(3/2)")]
+        [InlineData("sqrt(k*cos(c + d*x))/sqrt(a + a*sin(c + d*x))")]
+        [InlineData("(a + a*sin(c + d*x))^(5/2)/(k*cos(c + d*x))^(13/2)")]
+        [InlineData("(k*cos(c + d*x))^(9/2)/(a + a*sin(c + d*x))^(5/2)")]
+        [InlineData("1/((k*cos(c + d*x))^(7/2)*(a + a*sin(c + d*x))^(3/2))")]
+        [InlineData("(k*cos(c + d*x))^(3/2)*sqrt(a - a*sin(c + d*x))")]
+        [InlineData("sin(c + d*x)*sqrt(k*cos(c + d*x))*(a + a*sin(c + d*x))^(3/2)")]
+        [InlineData("(k*sin(c + d*x))^(3/2)*(a + a*cos(c + d*x))^(3/2)")]
+        public void ThroughTheSine(string integrand)
+        {
+            var integral = integrand.ToEntity().Integrate("x");
+            var text = integral.Stringize();
+            Assert.DoesNotContain("integral(", text);
+            Assert.True(text.Length < 3000, $"{text.Length} characters of answer for {integrand}");
+            // y = c + d x stays inside (0, pi/2) at the first pins and inside (pi/2, pi) at the second.
+            foreach (var (k, c) in new[] { (0.7, 0.4), (-0.7, 2.2) })
+            {
+                Entity Pinned(Entity e) => e.Substitute("a", 1.3).Substitute("k", k).Substitute("c", c).Substitute("d", 0.3);
+                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}, k = {k}, where the integrand is {want}");
+                }
+                Assert.True(compared >= 5, $"only {compared} points could be compared for {integrand}");
+            }
+        }
+    }
+}