Integrate fractional trigonometric powers two apart by the reduction, and a sine raised from nothing as a derivative

BREAKING-CHANGES.md

@@ -3679,6 +3679,38 @@ of the six functions read inside a power as it is
 | `"(csc(x) - sin(x))^(5/2)".ToEntity().Integrate("x")` | `integral(...)` | powers of `sin(x)` times `(cos(x)^2/sin(x))^(5/2)` |
 | `"1/(sec(x) - cos(x))^(3/2)".ToEntity().Integrate("x")` | `integral(...)` | a root of `cos(x)` with a logarithm and an arctangent, times `(sin(x)^2/cos(x))^(-3/2)` |
 
+### Fractional trigonometric powers two apart, and a sine raised from nothing, are integrated as derivatives
+
+**Answers where there were none.** Rubi's trigonometric integrands of these shapes were declined or ran out of time
+([#233](https://github.com/asc-community/AngouriMath/issues/233)):
+- A sum of powers of one sine, cosine, secant or cosecant of a linear, the powers not whole and two apart, by the
+  reduction formula from the highest down: `(cos sin^(n - 1))' = (n - 1) sin^(n - 2) - n sin^n`, and its kind for the
+  others. Each power alone is elliptic, so split, `(3 + 3 sec(c + d x)^2)/sqrt(sec(c + d x))` was two declined
+  integrals. Beside x, by parts once: `x/csc(c + d x)^(3/2) - x sqrt(csc(c + d x))/3`. The written function is kept.
+- The derivative of a sine raised from nothing times powers of the cosine and sums in it, or of a cosine beside the
+  sine: `(a + a cos(x))^(3/2)/cos(x)^(5/4)` is `(4 a^2 sin(x)/(cos(x)^(1/4) sqrt(a + a cos(x))))'`. Beside x, by parts
+  once: `x (b + a sin(x))/(a + b sin(x))^2` is `x G - integral G` for `G = -cos(x)/(a + b sin(x))`.
+- A cosine beside a function of the sine with a power of it that is not rational, in `s = sin(x)`:
+  `cos(x)/(sin(x) + sin(x)^sqrt(2))` is `1/(s + s^sqrt(2))`.
+- `(c sin(a + b x)^n)^(1/n)` is `K sin(a + b x)` with `K = (c sin(a + b x)^n)^(1/n)/sin(a + b x)`, constant between the
+  zeros of the sine, and so for the cosine.
+- Under the homogeneous substitution, factors written alike on both sides are cancelled where the quotient was
+  simplified: `sin(c + d x)^6/(a + b sin(c + d x)^2)^2`. Under the tangent, a whole power of a product is distributed
+  over its factors: `cot(c + d x)^2/(a + b sec(c + d x)^2)^2`. And beside a root of `1 - cos(x)`, the root of
+  `1 - cos(x)^2` is the product of the roots of `1 - cos(x)` and `1 + cos(x)`: `sqrt(1 - cos(x))/sqrt(a - cos(x))`.
+
+| Input | Was (2.5.0) | Now |
+|---|---|---|
+| `"(3+3*sec(c+d*x)^2)/sec(c+d*x)^(1/2)".ToEntity().Integrate("x")` | `integral(...)` | 60 characters |
+| `"x/csc(c+d*x)^(3/2)-1/3*x*sqrt(csc(c+d*x))".ToEntity().Integrate("x")` | `integral(...)` | 131 characters |
+| `"(a+a*cos(c+d*x))^(3/2)/cos(c+d*x)^(5/4)".ToEntity().Integrate("x")` | `integral(...)` | 96 characters |
+| `"x*(b+a*sin(x))/(a+b*sin(x))^2".ToEntity().Integrate("x")` | `integral(...)` | 64 characters |
+| `"cos(x)/(sin(x)+sin(x)^sqrt(2))".ToEntity().Integrate("x")` | `integral(...)` | 86 characters |
+| `"(c*sin(a+b*x)^n)^(1/n)".ToEntity().Integrate("x")` | `integral(...)` | 77 characters |
+| `"sin(c+d*x)^6/(a+b*sin(c+d*x)^2)^2".ToEntity().Integrate("x")` | `integral(...)` | 2,169 characters |
+| `"cot(c+d*x)^2/(a+b*sec(c+d*x)^2)^2".ToEntity().Integrate("x")` | `integral(...)` | 1,963 characters |
+| `"sqrt(1-cos(x))/sqrt(a-cos(x))".ToEntity().Integrate("x")` | `integral(...)` | 70 characters |
+
 ### A product of powers of the trigonometric functions, some not whole, is integrated as sine and cosine powers
 
 **Answers where there were none.** `(a sin(x))^(5/2) sqrt(b sec(x))` was declined, and so were

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

@@ -684,6 +684,17 @@ private static Entity WithItsPiecewiseSumsCombined(Entity expr)
             // A symbolic power of a sine, cosine, secant or cosecant times `alpha + beta f^2`, where it is
             // the derivative of `cos sin^(m + 1)`, `sin cos^(m + 1)`, `tan sec^m` or `cot csc^m`.
             if ((answer = IndefiniteIntegralSolver.SolveASymbolicPowerOfATrigonometricFunctionAsTheDerivativeOfAProduct(expr, x)) is { }) return answer;
+            // And numbers not whole for the exponents, two apart, by the reduction from the highest
+            // down: split, `1/csc(u)^(3/2) - sqrt(csc(u))/3` is two elliptic integrals.
+            if ((answer = IndefiniteIntegralSolver.SolveFractionalPowersOfATrigonometricFunctionTwoApartByTheReduction(expr, x)) is { }) return answer;
+            // And the derivative of a sine raised from nothing times powers of the cosine and sums in it,
+            // or of a cosine beside the sine, alone or beside x by parts: the half angle declined
+            // `(a + a cos(u))^(3/2)/cos(u)^(5/4)`, and parts integrated `x (b + a sin(x))/(a + b sin(x))^2`'s
+            // derivative into a function of tan(x/2) and ran out of time.
+            if ((answer = IndefiniteIntegralSolver.SolveAsTheDerivativeOfASineRaisedFromNothingBesidePowersOfTheCosine(expr, x, integrateByParts)) is { }) return answer;
+            // A cosine beside a function of the sine with a power off the rationals in it, in the sine:
+            // the substitution search declined `cos(x)/(sin(x) + sin(x)^sqrt(2))`.
+            if ((answer = IndefiniteIntegralSolver.SolveACosineBesideAFunctionOfTheSineWithAnIrrationalPowerInIt(expr, x, integrateByParts)) is { }) return answer;
             // A power of a sine, cosine or hyperbolic one over a binomial in a power of it, over the
             // binomial's roots, before the rules that spent the budget on `sinh(y)^4/(a + b sinh(y)^3)`.
             if ((answer = IndefiniteIntegralSolver.SolveAPowerOverABinomialInAPowerOfASineOverItsRoots(expr, x)) is { }) return answer;
@@ -1096,6 +1107,9 @@ private static Entity WithItsPiecewiseSumsCombined(Entity expr)
             // A fractional power of a constant times an even power of a function, the function
             // taken out with its sign: `(a sin(x)^2)^(5/2)` is `a^(5/2) sgn(sin(x)) sin(x)^5`.
             if ((answer = IndefiniteIntegralSolver.SolveByTakingAFunctionOutOfAPowerOfItsEvenPower(expr, x, integrateByParts)) is { }) return answer;
+            // And a symbolic root of a power, `(c sin(u)^n)^(1/n)`, the function out and the rest a
+            // constant written back.
+            if ((answer = IndefiniteIntegralSolver.SolveByTakingAFunctionOutOfASymbolicRootOfItsPower(expr, x, integrateByParts)) is { }) return answer;
             // A sum of two roots below the bar, multiplied above and below by its conjugate:
             // `1/(sqrt(A) - sqrt(B))` is `(sqrt(A) + sqrt(B))/(A - B)`.
             if ((answer = IndefiniteIntegralSolver.SolveByRationalisingASumOfRoots(expr, x, integrateByParts)) is { }) return answer;

Sources/Tests/UnitTests/Calculus/ACosineBesideAFunctionOfTheSineWithAnIrrationalPowerIntegralTest.cs

@@ -0,0 +1,49 @@
+//
+// 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>
+    /// A cosine beside a function of the sine with a power of it off the rationals, integrated in the
+    /// sine, and a sine beside the cosine the same: Rubi's 4.7.7 <c>cos(x)/(sin(x) + sin(x)^sqrt(2))</c>.
+    /// <a href="https://github.com/asc-community/AngouriMath/issues/233">#233</a>
+    /// </summary>
+    [Trait("Area", "Calculus")]
+    public sealed class ACosineBesideAFunctionOfTheSineWithAnIrrationalPowerIntegralTest
+    {
+        [Theory]
+        [InlineData("cos(x)/(sin(x)+sin(x)^sqrt(2))")]
+        [InlineData("cos(c+d*x)/(sin(c+d*x)+sin(c+d*x)^p)")]
+        [InlineData("sin(c+d*x)*cos(c+d*x)^pi*(1+cos(c+d*x))")]
+        public void IsAnsweredAndDifferentiatesBack(string integrand)
+        {
+            var integral = integrand.ToEntity().Integrate("x");
+            var text = integral.Stringize();
+            Assert.DoesNotContain("integral(", text);
+            Entity Pinned(Entity e) => e.Substitute("c", 0.3).Substitute("d", 0.75).Substitute("p", 2.5);
+            var derivative = Pinned(integral.Substitute("C", 0)).Differentiate("x");
+            var original = Pinned(integrand.ToEntity());
+            var compared = 0;
+            foreach (var at in new[] { -2.3, -1.1, -0.7, 0.4, 1.9, 2.8 })
+            {
+                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}");
+        }
+    }
+}

Sources/Tests/UnitTests/Calculus/AFunctionOutOfASymbolicRootOfItsPowerIntegralTest.cs

@@ -0,0 +1,60 @@
+//
+// 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>
+    /// A symbolic root of a constant times a power of a function, <c>(c sin(u)^n)^(1/n)</c>, as the
+    /// function times a factor constant between its zeros, written back: Rubi's 4.1.7 and 4.2.0.
+    /// <a href="https://github.com/asc-community/AngouriMath/issues/233">#233</a>
+    /// </summary>
+    [Trait("Area", "Calculus")]
+    public sealed class AFunctionOutOfASymbolicRootOfItsPowerIntegralTest
+    {
+        [Theory]
+        [InlineData("(c*sin(a+b*x)^n)^(1/n)")]
+        [InlineData("(c*cos(a+b*x)^m)^(1/m)")]
+        [InlineData("x*(c*cos(x)^m)^(1/m)")]
+        [InlineData("(c*sin(x)^n)^(2/n)")]
+        public void IsAnsweredAndDifferentiatesBack(string integrand)
+        {
+            var integral = integrand.ToEntity().Integrate("x");
+            var text = integral.Stringize();
+            Assert.DoesNotContain("integral(", text);
+            Entity Pinned(Entity e) => e.Substitute("a", 0.3).Substitute("b", 0.75).Substitute("c", 1.3).Substitute("n", 2.5).Substitute("m", 0.4);
+            var derivative = Pinned(integral.Substitute("C", 0)).Differentiate("x");
+            var original = Pinned(integrand.ToEntity());
+            var compared = 0;
+            foreach (var at in new[] { -2.3, -1.1, -0.7, 0.4, 1.9, 2.8 })
+            {
+                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 LeavesAPowerOfXToTheRulesForBinomials()
+        {
+            // (c x^n)^(1/n) is the rule's for a symbolic power of a monomial: `x (c x^n)^(1/n)/2`,
+            // whose derivative is `(c x^n)^(1/n)/2 + x (c x^n)^(1/n)/(2 x)`, the integrand on every
+            // branch, since d/dx (c x^n)^(1/n) is (c x^n)^(1/n)/x on every one.
+            var integral = "(c*x^n)^(1/n)".ToEntity().Integrate("x").Stringize();
+            Assert.Equal("x * (c * x ^ n) ^ (1 / n) / 2 + C", integral);
+        }
+    }
+}

Sources/Tests/UnitTests/Calculus/AQuotientInTheTangentWithItsSymbolsCancelledIntegralTest.cs

@@ -0,0 +1,51 @@
+//
+// 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>
+    /// A quotient of sines and cosines with symbols in it, written in the tangent with the factors alike
+    /// on both sides of the bar cancelled rather than simplified: Rubi's 4.1.7 and 4.5.7, whose
+    /// simplification ran past the budget.
+    /// <a href="https://github.com/asc-community/AngouriMath/issues/233">#233</a>
+    /// </summary>
+    [Trait("Area", "Calculus")]
+    public sealed class AQuotientInTheTangentWithItsSymbolsCancelledIntegralTest
+    {
+        [Theory]
+        [InlineData("sin(c+d*x)^6/(a+b*sin(c+d*x)^2)^2")]
+        [InlineData("csc(c+d*x)^2/(a-b*sin(c+d*x)^4)^2")]
+        [InlineData("cot(c+d*x)^2/(a+b*sec(c+d*x)^2)^2")]
+        [InlineData("sin(x)^4/(a+b*sin(x)^2)^2")]
+        public void IsAnsweredAndDifferentiatesBack(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", 0.3).Substitute("d", 0.75);
+            var derivative = Pinned(integral.Substitute("C", 0)).Differentiate("x");
+            var original = Pinned(integrand.ToEntity());
+            var compared = 0;
+            foreach (var at in new[] { -2.3, -1.1, -0.7, 0.4, 1.9, 2.8 })
+            {
+                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}");
+        }
+    }
+}

Sources/Tests/UnitTests/Calculus/ASineRaisedFromNothingBesidePowersOfTheCosineIntegralTest.cs

@@ -0,0 +1,65 @@
+//
+// 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>
+    /// The derivative of a sine raised from nothing times powers of the cosine, the secant and sums
+    /// in them, or of a cosine beside the sine, alone or beside x by parts: Rubi's 4.2.2.1, 4.5.1.2,
+    /// 4.5.4.2 and 4.7.7.
+    /// <a href="https://github.com/asc-community/AngouriMath/issues/233">#233</a>
+    /// </summary>
+    [Trait("Area", "Calculus")]
+    public sealed class ASineRaisedFromNothingBesidePowersOfTheCosineIntegralTest
+    {
+        // Points where the cosine is negative too, where the integrand is complex and the answer's
+        // principal powers still differentiate to it.
+        [Theory]
+        [InlineData("(a+a*cos(c+d*x))^(3/2)/cos(c+d*x)^(5/4)")]
+        [InlineData("(a+a*cos(c+d*x))^(3/2)*sec(c+d*x)^(5/4)")]
+        [InlineData("(a+a*sec(c+d*x))^(3/2)/sec(c+d*x)^(1/4)")]
+        [InlineData("(a+a*sec(c+d*x))^n*(-a*A*n-a*Q*(1+n)*sec(c+d*x))/(a*(1+n)*sec(c+d*x)^n)+sec(c+d*x)^(-1-n)*(a+a*sec(c+d*x))^n*(A+Q*sec(c+d*x)^2)")]
+        [InlineData("x*(b+a*sin(x))/(a+b*sin(x))^2")]
+        [InlineData("x*(b+a*cos(c+d*x))/(a+b*cos(c+d*x))^2")]
+        public void IsAnsweredAndDifferentiatesBack(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", 0.3).Substitute("d", 0.75)
+                .Substitute("A", 0.6).Substitute("Q", 1.25).Substitute("n", 0.4);
+            var derivative = Pinned(integral.Substitute("C", 0)).Differentiate("x");
+            var original = Pinned(integrand.ToEntity());
+            var compared = 0;
+            foreach (var at in new[] { -2.3, -1.1, -0.7, 0.4, 1.9, 2.8 })
+            {
+                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 IsTheRaisedProductInClosedForm()
+        {
+            // 4 a^2 sin/(cos^(1/4) sqrt(a + a cos)), Rubi's, with no half angle in it.
+            var integral = "(a+a*cos(x))^(3/2)/cos(x)^(5/4)".ToEntity().Integrate("x").Stringize();
+            Assert.DoesNotContain("x / 2", integral);
+            Assert.Contains("cos(x) ^ (-1/4)", integral);
+        }
+    }
+}