A difference is offered to the substitution search as a sum is

BREAKING-CHANGES.md

@@ -407,6 +407,19 @@ square and the cube keep the Hermite reduction's answers. Rubi's 1.1.3.8 `P(x) (
 | `"(c + d*x)/(a + b*x^3)^4".ToEntity().Integrate("x")` | `integral(...)` | the same at the roots of `a + b x^3` |
 | `"x^2/(a + b*x^5)^5".ToEntity().Integrate("x")` | `integral(...)` | four algebraic terms, and the integral of `x^2/(a + b x^5)` |
 
+### A difference is offered to the substitution search as a sum is
+
+**Answers where there were none.** `sin(1 + 2/(1 + x))` was answered under `u = 1 + x`, and
+`sin(1 + 2/(1 - x))` was declined: the substitution search offered every sum in the integrand as a
+candidate and no difference, so `u = 1 - x` was never tried. A difference in the integrand as asked
+is offered now as a sum is ([#718](https://github.com/asc-community/AngouriMath/issues/718)).
+
+| Input | Was (2.5.0) | Now |
+|---|---|---|
+| `"sin(1 + 2/(1 - x))".ToEntity().Integrate("x")` | `integral(...)` | in the sine and cosine integrals of `2/(1 - x)` |
+| `"cos(1 + 2/(x - 1))".ToEntity().Integrate("x")` | `integral(...)` | in the sine and cosine integrals of `2/(x - 1)` |
+| `"sin(p + k/(c - d*x))".ToEntity().Integrate("x")` | `integral(...)` | in the sine and cosine integrals of `k/(c - d x)` |
+
 ### A polynomial over a power of a trinomial past the square is integrated
 
 `P(x)/(a + b x^n + c x^(2n))^k`, a polynomial over a power of a trinomial in `x^n` with symbols in

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

@@ -26266,7 +26266,7 @@ private static bool IsAFunctionOfTheTangentAlone(Entity expr, Entity.Variable u)
                 // and not a subexpression anyone wrote: `1 + 2x^2` and `1 + 2x^2 + 4x^3` are
                 // the left-nested partial sums of Hearn's radicand `1 + 2x^2 + 4x^3 + x^4`,
                 // and each was half a second of rewriting and simplifying to be refused.
-                if (u is Sumf && IsOnlyAPartialSum(expr, u))
+                if (u is Sumf or Minusf && IsOnlyAPartialSum(expr, u))
                     continue;
 
                 // A candidate that does not vary with x is no substitution at all, and its
@@ -27147,8 +27147,14 @@ bool APowerCanBeExact(int k)
                     case var special when IsASpecialFunction(special) && TheDifferentialCanBeThere(special, expr, x):
                         candidates.Add(node);
                         break;
-                    case Sumf(var aug, var add) when !rational && !large && node.Complexity <= LargestSumOffered:
-                        if (aug.ContainsNode(x) || add.ContainsNode(x)) candidates.Add(node); // Linear expressions ax + b
+                    // A difference as a sum: `1 - x` below the bar of `sin(1 + 2/(1 - x))` is the
+                    // substitution that `1 + x` is for `sin(1 + 2/(1 + x))`, and was never offered.
+                    // In the question asked only: below it every substitution writes differences,
+                    // `x = (u^2 - c)/d`, and offering them was two fifths more search on Rubi's
+                    // `(c + d x)^(5/2)/(x (a + b x)^2)`, past the harness's patience.
+                    case Sumf or Minusf when !rational && !large && node.Complexity <= LargestSumOffered
+                        && (node is Sumf || Integration.AnsweringTheQuestionAsked):
+                        if (node.DirectChildren.Any(child => child.ContainsNode(x))) candidates.Add(node); // Linear expressions ax + b
                         break;
                 }
             // Sort by complexity - try simpler substitutions first

Sources/Tests/UnitTests/Calculus/DifferenceAsASubstitutionIntegralTest.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
+{
+    /// <summary>
+    /// A difference offered to the substitution search as a sum is: <c>sin(1 + 2/(1 - x))</c> is
+    /// <c>-sin(1 + 2/u)</c> under <c>u = 1 - x</c>, as <c>sin(1 + 2/(1 + x))</c> is
+    /// <c>sin(1 + 2/u)</c> under <c>u = 1 + x</c>, and only the second was answered.
+    /// <a href="https://github.com/asc-community/AngouriMath/issues/718">#718</a>
+    /// </summary>
+    /// <remarks>
+    /// Checked by differentiating back with <c>p = 0.3</c>, <c>k = 0.9</c>, <c>c = 1.1</c>,
+    /// <c>d = 0.7</c>, on both sides of the pole of every row.
+    /// </remarks>
+    [Trait("Area", "Calculus")]
+    public sealed class DifferenceAsASubstitutionIntegralTest
+    {
+        [Theory]
+        [InlineData("sin(1 + 2/(1 - x))")]
+        [InlineData("cos(1 + 2/(1 - x))")]
+        [InlineData("sin(1 + 2/(x - 1))")]
+        [InlineData("sin(1 + 2/(3 - 2*x))")]
+        [InlineData("sin(p + k/(c - d*x))")]
+        public void IsSubstitutedForAsASumIs(string integrand)
+        {
+            var integral = integrand.ToEntity().Integrate("x");
+            Assert.DoesNotContain("integral(", integral.Stringize());
+            Entity Pinned(Entity e) => e.Substitute("p", 0.3).Substitute("k", 0.9).Substitute("c", 1.1).Substitute("d", 0.7);
+            var derivative = Pinned(integral.Substitute("C", 0)).Differentiate("x");
+            var original = Pinned(integrand.ToEntity());
+            foreach (var at in new[] { -1.0, 0.3, 0.8, 2.2, 3.0 })
+            {
+                var want = original.Substitute("x", at).EvalNumerical();
+                var got = derivative.Substitute("x", at).EvalNumerical();
+                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}");
+            }
+        }
+    }
+}