@@ -294,6 +294,20 @@ angle, each of which the integrator answers
 | `"x^3*(1-c^2*x^2)^(5/2)/(a+b*asin(c*x))".ToEntity().Integrate("x")` | `integral(...)` | 508 characters |
 | `"x^3*(1-c^2*x^2)^(5/2)/(a+b*asin(c*x))^2".ToEntity().Integrate("x")` | `integral(...)` | 1,470 characters |
 
+### A constant common to a half-angle denominator is divided out
+
+**Answers where there were none.** `x/(k + k sin(x))^2` was declined, and so were `x/(d c + c d sin(x))^2` and Rubi's
+5.1.4 `(a + b asin(c x))/((d + c d x)^(5/2) sqrt(f - c f x))`. After the half-angle substitution, such a denominator
+is a constant times a polynomial with number coefficients, `k^2 (t + 1)^4`. The constant is now divided out before the
+partial fractions
+([#718](https://github.com/asc-community/AngouriMath/issues/718)).
+
+| Input | Was (2.5.0) | Now |
+|---|---|---|
+| `"x/(k+k*sin(x))^2".ToEntity().Integrate("x")` | `integral(...)` | 285 characters |
+| `"x/(d*c+c*d*sin(x))^2".ToEntity().Integrate("x")` | `integral(...)` | 329 characters |
+| `"(a+b*asin(c*x))/((d+c*d*x)^(5/2)*sqrt(f-c*f*x))".ToEntity().Integrate("x")` | `integral(...)` | 927 characters |
+
 ### A piecewise is defined where the case it takes is
 
 **Domain conditions that were false.** The domain condition of a piecewise asked every case's predicate
 
@@ -32678,6 +32678,33 @@ bool IsAPolynomialInBoth(Entity polynomial)
                 ?.Pipe(answer => underneath ? answer / constant : answer * constant);
         }
 
+        /// <summary>
+        /// A rational function of <paramref name="t"/> as a polynomial over a polynomial, both divided by
+        /// the denominator's leading coefficient with each coefficient in lowest terms over the symbols,
+        /// so that a constant common to the denominator's coefficients cancels; null where either side is
+        /// not a polynomial in <paramref name="t"/> with coefficients free of it.
+        /// </summary>
+        private static Entity? WithTheLeadingCoefficientDividedOut(Entity rational, Entity.Variable t)
+        {
+            var (above, below) = Functions.SingleQuotient.Of(rational);
+            if (!TreeAnalyzer.TryGetPolynomial(above.Expand(), t, out var top) || !TreeAnalyzer.TryGetPolynomial(below.Expand(), t, out var bottom)
+                || bottom.Count == 0 || top.Keys.Concat(bottom.Keys).Any(key => key.Sign < 0 || !key.CanFitInInt32())
+                || top.Values.Concat(bottom.Values).Any(coefficient => coefficient.ContainsNode(t)))
+                return null;
+            var lead = bottom[bottom.Keys.Max()!];
+            // Only where the symbols in the denominator are one factor common to all of it: there the
+            // rules below formed `k^2/k^2`, and anywhere else this respelling sent fifteen of Rubi's
+            // 4.x problems the half-angle answered in a second to a timeout.
+            if (!lead.Vars.Any() || bottom.Values.Any(coefficient => Functions.PartialFractions.InLowestTermsOverTheSymbols(coefficient / lead) is not Number))
+                return null;
+            Entity Rebuilt(Dictionary<EInteger, Entity> monomials)
+                => monomials.Aggregate((Entity)Number.Integer.Zero, (sum, pair)
+                    => sum + Functions.PartialFractions.InLowestTermsOverTheSymbols(pair.Value / lead)
+                        // t^0 written as 1: as a power it brings `provided not t = 0` with it.
+                        * (pair.Key.IsZero ? Number.Integer.One : MathS.Pow(t, Number.Integer.Create(pair.Key))));
+            return (Rebuilt(top) / Rebuilt(bottom)).InnerSimplified;
+        }
+
         /// <summary>
         /// A rational function of <c>sin(x)</c> and <c>cos(x)</c>, turned into a rational function
         /// of one variable by the half-angle substitution <c>t = tan(x/2)</c> and handed to the
@@ -32869,6 +32896,14 @@ static bool NoRemainder(Entity rest)
             if (divided)
                 integrand = (numerator / denominator).InnerSimplified;
 
+            // Over the denominator's leading coefficient, every coefficient in lowest terms over the
+            // symbols: `1/(k + k sin(x))^2` is `2 (1 + t^2)/(k^2 (t + 1)^4)`, and with `k^2` left in the
+            // coefficients the rules below formed `k^2/k^2`, which the simplification does not cancel,
+            // so the quadratic's discriminant was not decidably zero and `x/(k + k sin(x))^2` was
+            // answered `NaN` through a branch of the piecewise that can never hold.
+            if (integrand.Vars.Any(symbol => symbol != t) && WithTheLeadingCoefficientDividedOut(integrand, t) is { } normalized)
+                integrand = normalized;
+
             return Integration.ComputeIndefiniteIntegral(integrand, t, integrateByParts) is { } result
                 ? result.Substitute(t, MathS.Tan(argument / 2))
                 : null;
 
@@ -0,0 +1,59 @@
+//
+// 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 constant common to the whole denominator after the half-angle substitution, divided out:
+    /// <c>x/(k + k sin(x))^2</c> was <c>NaN</c>, and so was Rubi's 5.1.4
+    /// <c>(a + b asin(c x))/((d + c d x)^(5/2) sqrt(f - c f x))</c>, which comes to it.
+    /// <a href="https://github.com/asc-community/AngouriMath/issues/718">#718</a>
+    /// </summary>
+    [Trait("Area", "Calculus")]
+    public sealed class AConstantCommonToAHalfAngleDenominatorIntegralTest
+    {
+        private static void DifferentiatesBack(string integrand, params (string, double)[] pins)
+        {
+            var integral = integrand.ToEntity().Integrate("x");
+            Assert.DoesNotContain("integral(", integral.Stringize());
+            Assert.DoesNotContain("NaN", 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("x/(k+k*sin(x))^2")]
+        [InlineData("1/(d*q+q*d*sin(x))^2")]
+        [InlineData("(a+b*asin(q*x))/((d+q*d*x)^(5/2)*sqrt(f-q*f*x))")]
+        public void IsAnsweredAndDifferentiatesBack(string integrand)
+        {
+            // |q x| below 1 at every sampled x, where the inverse sine and the roots are real.
+            DifferentiatesBack(integrand, ("k", 1.3), ("a", 0.7), ("b", -0.4), ("d", 1.1), ("f", 1.5), ("q", 0.3));
+            DifferentiatesBack(integrand, ("k", -0.7), ("a", -2), ("b", 1.5), ("d", 0.6), ("f", 2), ("q", -0.2));
+        }
+    }
+}
@@ -0,0 +1,63 @@
+//
+// 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 half-angle substitution with a constant common to the denominator's coefficients: it is
+    /// divided out in lowest terms over the symbols, which the simplification does not do for
+    /// <c>k^2/k^2</c>. Kept, it left a quadratic whose discriminant was not decidably zero, and
+    /// <c>x/(k + k sin(x))^2</c> -- and Rubi's 5.1.4 <c>(a + b asin(c x))/((d + c d x)^(5/2) sqrt(f - c f x))</c>,
+    /// which comes to it -- were answered <c>NaN</c>.
+    /// <a href="https://github.com/asc-community/AngouriMath/issues/718">#718</a>
+    /// </summary>
+    [Trait("Area", "Calculus")]
+    public sealed class AHalfAngleSubstitutionWithACommonConstantIntegralTest
+    {
+        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}");
+            }
+        }
+
+        // d with the sign that keeps the argument real at every sampled x, and |d| below 2/2.3^2
+        // so that the inverse sine and cosine stay inside [-1, 1]; off it the exact checks hold too.
+        [Theory]
+        [InlineData("x/(k+k*sin(x))^2")]
+        [InlineData("1/(d*q+q*d*sin(x))^2")]
+        [InlineData("x/(k-k*cos(x))^2")]
+        [InlineData("(a+b*asin(q*x))/((d+q*d*x)^(5/2)*sqrt(f-q*f*x))")]
+        public void IsAnsweredAndDifferentiatesBack(string integrand)
+        {
+            Assert.DoesNotContain("NaN", integrand.ToEntity().Integrate("x").Stringize());
+            DifferentiatesBack(integrand, ("a", 0.6), ("b", 1.25), ("q", 0.3), ("d", 0.75), ("f", 1.4), ("k", 1.5));
+            DifferentiatesBack(integrand, ("a", -0.75), ("b", 0.4), ("q", -0.4), ("d", -2), ("f", 0.5), ("k", -0.3));
+        }
+    }
+}