diff --git a/BREAKING-CHANGES.md b/BREAKING-CHANGES.md
index db2c34985..409cf8bd6 100644
--- a/BREAKING-CHANGES.md
+++ b/BREAKING-CHANGES.md
@@ -208,6 +208,19 @@ after it
|---|---|---|
| `"(a*c + b*c*x)^(-3-2*p)*(f + g*x)*(a^2 + 2*a*b*x + b^2*x^2)^p".ToEntity().Integrate("x")` | `integral(...)`; an answer with no value on the unreleased master | the antiderivative |
+### A quadratic with rational roots counts as two linears before the substitution search
+
+**Answers where there were none.** `sec(x)^2 sin(x)/(a + b sin(x))^3` ran past thirty seconds: under the
+half-angle tangent it is a rational function over `(1 - u^2)^2 (a u^2 + 2 b u + a)^3`, and the substitution
+search spent eighty seconds on it before the partial fractions, which split it at once, were asked. The search
+already declines a rational function over symbolic linear factors and a quadratic; `1 - u^2` is two linear
+factors written together, and is counted so ([#718](https://github.com/asc-community/AngouriMath/issues/718)).
+
+| Input | Was (2.5.0) | Now |
+|---|---|---|
+| `"sec(x)^2*sin(x)/(a + b*sin(x))^3".ToEntity().Integrate("x")` | `integral(...)`; past thirty seconds on the unreleased master | 5,017 characters |
+| `"tan(x)^2/(a + b*sin(x))^3".ToEntity().Integrate("x")` | `integral(...)`; past thirty seconds on the unreleased master | 4,816 characters |
+
### A sine of a double argument beside the argument is written as a product
**Answers where there were none.** `csc(a + b x)^3 sin(2a + 2b x)^7` was declined: the rule that writes multiples
diff --git a/Sources/AngouriMath/Functions/Continuous/Integration/IndefiniteIntegralSolver.cs b/Sources/AngouriMath/Functions/Continuous/Integration/IndefiniteIntegralSolver.cs
index aca16ef3c..fcbd8c337 100644
--- a/Sources/AngouriMath/Functions/Continuous/Integration/IndefiniteIntegralSolver.cs
+++ b/Sources/AngouriMath/Functions/Continuous/Integration/IndefiniteIntegralSolver.cs
@@ -21277,6 +21277,12 @@ private static bool IsAProductOfSymbolicLinearFactors(Entity denominator, Entity
return false;
if (read.Keys.Max()!.Equals(EInteger.One))
linears++;
+ // A quadratic with rational coefficients and rational roots is two linear factors
+ // written together: the half-angle tangent's `1 - u^2` beside `a u^2 + 2 b u + a` is
+ // `(1 - u)(1 + u)` to the split, and the search spent eighty seconds declining Rubi's
+ // `sec(x)^2 sin(x)/(a + b sin(x))^3` before it was asked.
+ else if (read.Keys.Max()!.Equals(EInteger.FromInt32(2)) && HasRationalRoots(read))
+ linears += 2;
else if (read.Keys.Max()!.Equals(EInteger.FromInt32(2)))
quadratics++;
else
@@ -21286,6 +21292,23 @@ private static bool IsAProductOfSymbolicLinearFactors(Entity denominator, Entity
return (linears >= 1 || !aLinearAmongThem) && linears + quadratics >= 2 && symbolic;
}
+ ///
+ /// Whether a quadratic read as its coefficients has rational coefficients and a
+ /// discriminant that is the square of a rational.
+ ///
+ private static bool HasRationalRoots(Dictionary quadratic)
+ {
+ Number.Rational? Coefficient(int power)
+ => quadratic.TryGetValue(EInteger.FromInt32(power), out var c) ? c.Evaled as Number.Rational : Number.Integer.Zero;
+ if (Coefficient(2) is not { } a || Coefficient(1) is not { } b || Coefficient(0) is not { } c)
+ return false;
+ var discriminant = (b * b - 4 * a * c).Evaled;
+ if (discriminant is not Number.Rational d || d.ERational.IsNegative)
+ return false;
+ var root = MathS.Sqrt(d).Evaled;
+ return root is Number.Rational;
+ }
+
///
/// Whether a written factor of with a symbol in it stands
/// to a whole power of two or more: where the Hermite reduction's one solve takes those
diff --git a/Sources/Tests/UnitTests/Calculus/AHalfAngleRationalFunctionOverOneMinusTheSquareIntegralTest.cs b/Sources/Tests/UnitTests/Calculus/AHalfAngleRationalFunctionOverOneMinusTheSquareIntegralTest.cs
new file mode 100644
index 000000000..4e3f9a19d
--- /dev/null
+++ b/Sources/Tests/UnitTests/Calculus/AHalfAngleRationalFunctionOverOneMinusTheSquareIntegralTest.cs
@@ -0,0 +1,50 @@
+//
+// 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
+{
+ ///
+ /// A rational function of the sine over a power of a + b sin(x) with the cosine below the bar: under
+ /// the half-angle tangent, (1 - u^2) beside a u^2 + 2 b u + a, split by its partial fractions
+ /// rather than searched for a substitution. Rubi's 4.1.1.2, 4.1.1.3 and 4.1.2.2.
+ /// #718
+ ///
+ [Trait("Area", "Calculus")]
+ public sealed class AHalfAngleRationalFunctionOverOneMinusTheSquareIntegralTest
+ {
+ [Theory]
+ [InlineData("sec(x)^2*sin(x)/(a + b*sin(x))^3")]
+ [InlineData("tan(x)^2/(a + b*sin(x))^3")]
+ public void ByItsPartialFractions(string integrand)
+ {
+ var integral = integrand.ToEntity().Integrate("x");
+ var text = integral.Stringize();
+ Assert.DoesNotContain("integral(", text);
+ Assert.True(text.Length < 20000, $"{text.Length} characters of answer for {integrand}");
+ Entity Pinned(Entity e) => e.Substitute("a", 2.3).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}");
+ }
+ }
+}