Answer a difference of two cubes where the sampled check cannot tell

BREAKING-CHANGES.md

@@ -396,6 +396,19 @@ in Rubi's 1.1.3.4 come out shorter as well, and those that took two seconds take
 | `"1/(x^8*(a + b*x^3)*(c + d*x^3))".ToEntity().Integrate("x")` | `integral(...)` | 1,217 characters |
 | `"1/(x^6*(a + b*x^4)*(c + d*x^4))".ToEntity().Integrate("x")` | `integral(...)` | 795 characters |
 
+### A difference of two cubes is split over the cube roots of unity where its answer cannot be sampled
+
+**Answers where there were none.** `1/(a - b cosh(x)^3)` was declined, which Rubi's 6.2.7 has. Under `u = e^x` it is
+`8 u^2/(8 a u^3 - b (u^2 + 1)^3)`, a difference of two cubes, which the rule for a sum of two cubes split correctly and
+then refused: with the second cube negative, the real cube root makes all three quadratics complex at the sampled
+values, and too few points could be compared. The answer is now refused only where it differs
+([#718](https://github.com/asc-community/AngouriMath/issues/718)).
+
+| Input | Was (2.5.0) | Now |
+|---|---|---|
+| `"1/(a-b*cosh(x)^3)".ToEntity().Integrate("x")` | `integral(...)` | 666 characters |
+| `"8*x^2/(8*a*x^3-b*(x^2+1)^3)".ToEntity().Integrate("x")` | `integral(...)` | 654 characters |
+
 ### A negative power of an inverse function of `c + d x^2` is integrated in sine and cosine integrals
 
 **Answers where there were none.** `1/(a + b acos(1 + d x^2))` was declined, and so were seventeen more of Rubi's

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

@@ -19676,7 +19676,11 @@ Piecewise piecewise when piecewise.Cases.Any() => piecewise.Cases.First().Expres
             foreach (var unity in unities)
                 answer = answer + unity * each.Substitute(namedB, unity * root).Substitute(namedS, shift);
             answer = (constant / (3 * d6 * MathS.Pow(root, Number.Integer.Create(2))) * answer).InnerSimplified;
-            return Functions.PartialFractions.DerivativeHoldsAtSampledPoints(answer, expr, x) ? answer : null;
+            // Refused where it differs, not where it cannot be told: with `q < 0` the real cube root
+            // makes all three quadratics' coefficients complex at the pinned values, the intervals
+            // settle too few points, and `8 x^2/(8 a x^3 - b (x^2 + 1)^3)` -- `1/(a - b cosh(x)^3)`
+            // under `u = e^x` -- was declined where its `+ b` twin was answered.
+            return Functions.PartialFractions.DerivativeDiffersWhereReal(answer, expr, x) ? null : answer;
         }
 
         /// <summary>

Sources/Tests/UnitTests/Calculus/ASumOfTwoCubesIntegralTest.cs

@@ -55,5 +55,18 @@ public void IsAnsweredAndDifferentiatesBack(string integrand)
             DifferentiatesBack(integrand, ("a", 2), ("b", 3), ("c", 5));
             DifferentiatesBack(integrand, ("a", -1), ("b", 2), ("c", -3));
         }
+
+        // A difference of two cubes, the second cube's coefficient negative: `1/(a - b cosh(x)^3)` is
+        // `8 u^2/(8 a u^3 - b (u^2 + 1)^3)` under `u = e^x`. The real cube root makes all three
+        // quadratics complex at sampled values, and the answer was refused for want of points.
+        [Theory]
+        [InlineData("8*x^2/(8*a*x^3-b*(x^2+1)^3)")]
+        [InlineData("1/(a-b*cosh(x)^3)")]
+        [InlineData("x/(27*a^3+27*a^2*b*x^2-27*a^2*c*x^3+9*a*b^2*x^4+b^3*x^6)")]
+        public void ADifferenceIsAnsweredAndDifferentiatesBack(string integrand)
+        {
+            DifferentiatesBack(integrand, ("a", 2), ("b", 3), ("c", 5));
+            DifferentiatesBack(integrand, ("a", -1), ("b", 2), ("c", -3));
+        }
     }
 }