Do not divide by a leading coefficient that is zero spelled otherwise

BREAKING-CHANGES.md

@@ -534,6 +534,19 @@ power, and the integrand is answered in a fraction of a second
 | `"csch(c+d*x)*(a+b*sinh(c+d*x)^4)^2".ToEntity().Integrate("x")` | past fifteen minutes | 426 characters |
 | `"csch(c+d*x)*(a+b*sinh(c+d*x)^4)^3".ToEntity().Integrate("x")` | past fifteen minutes | 646 characters |
 
+### A power of a polynomial whose leading coefficient is zero spelled otherwise is not divided by it
+
+**Answers where there were none.** `1/((a (1 + x^2))^2 - (a (1 - x^2))^2)^3` was left unevaluated. That is
+`1/(64 a^6 x^6)`: the polynomial's fourth power cancels, and its leading coefficient as read, `a^2 - a^2` spelled
+otherwise, was divided by to make it monic. The radical substitution writes this shape for an exponential of an
+inverse hyperbolic cotangent ([#718](https://github.com/asc-community/AngouriMath/issues/718)).
+
+| Input | Was (2.5.0) | Now |
+|---|---|---|
+| `"1/((a*(1+x^2))^2-(a*(1-x^2))^2)^3".ToEntity().Integrate("x")` | `integral(...)` | 133 characters |
+| `"x^2/((a*(1+x^2))^2-(a*(1-x^2))^2)^2".ToEntity().Integrate("x")` | `integral(...)` | 235 characters |
+| `"1/((-1/a^2+((1+x^2)/(a*(1-x^2)))^2)^2*(1-x^2)^2)".ToEntity().Integrate("x")` | `integral(...)` | 518 characters |
+
 ### An exponential of a whole multiple of an inverse hyperbolic tangent is written algebraically first
 
 **Answers where there were none.** `e^(n artanh(u))` with u linear is written as powers of `1 + u` and `1 - u`, and then

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

@@ -2510,6 +2510,10 @@ Entity InU(Dictionary<EInteger, Entity> polynomial)
                 else if (factor is Powf(var polynomial, Number.Integer power) && power.EInteger.CompareTo(EInteger.FromInt32(2)) >= 0
                          && TreeAnalyzer.TryGetPolynomial(polynomial, x, out var read) && read.Count > 1
                          && read.Keys.Max() is { } top && read[top] is var leading && !leading.ContainsNode(x) && leading.Evaled is not Number
+                         // A leading coefficient that is zero spelled otherwise, `-a^2 + a^2` in
+                         // `(a (1 + u^2))^2 - (a (1 - u^2))^2`, is not one: divided by, every other
+                         // coefficient was over zero, and the answer `NaN + C`.
+                         && Functions.PartialFractions.InLowestTermsOverTheSymbols(leading) is not Number.Complex { IsZero: true }
                          && !numerator.Nodes.Concat(denominator.Nodes).Any(node =>
                                 node is Powf(var radicand, Number.Rational r) && r is not Number.Integer && radicand.Nodes.Contains(polynomial)))
                 {

Sources/Tests/UnitTests/Calculus/ALeadingCoefficientThatVanishesIntegralTest.cs

@@ -0,0 +1,47 @@
+//
+// 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 power of a polynomial whose leading coefficient is zero spelled otherwise is not made monic by
+    /// dividing by it: <c>(a (1 + x^2))^2 - (a (1 - x^2))^2</c> is <c>4 a^2 x^2</c>, its fourth power
+    /// cancels, and over its cube the answer was <c>NaN + C</c>. The radical substitution writes it for
+    /// Rubi's 7.4.2 <c>1/(e^(arccoth(a x)) (c - a^2 c x^2)^2)</c>.
+    /// <a href="https://github.com/asc-community/AngouriMath/issues/718">#718</a>
+    /// </summary>
+    [Trait("Area", "Calculus")]
+    public sealed class ALeadingCoefficientThatVanishesIntegralTest
+    {
+        [Theory]
+        [InlineData("1/((a*(1+x^2))^2-(a*(1-x^2))^2)^3")]
+        [InlineData("x^2/((a*(1+x^2))^2-(a*(1-x^2))^2)^2")]
+        [InlineData("1/((-1/a^2+((1+x^2)/(a*(1-x^2)))^2)^2*(1-x^2)^2)")]
+        public void IsNotANaN(string integrand)
+        {
+            var integral = integrand.ToEntity().Integrate("x");
+            Assert.DoesNotContain("NaN", integral.Stringize());
+            Assert.DoesNotContain("integral(", integral.Stringize());
+            var pinned = integral.Substitute("C", 0).Substitute("a", 1.3);
+            Entity original = integrand.ToEntity().Substitute("a", 1.3);
+            var derivative = pinned.Differentiate("x");
+            foreach (var at in new[] { -2.3, 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}");
+            }
+        }
+    }
+}