A quotient by the square of a base is integrated by parts against the base's reciprocal

BREAKING-CHANGES.md

@@ -838,6 +838,21 @@ but `sqrt(q)^2 = q` ([#1598](https://github.com/asc-community/AngouriMath/issues
 | `"1/sqrt(-x^2 + i)".Integrate("x")` | `-arcsin(-2x/sqrt(4i)) + C` | `ln(-2x + 2i sqrt(-x^2 + i))/i + C`, an antiderivative as before, in the logarithm |
 | `"1/(sqrt(x)*sqrt(1 + i*x))".Integrate("x")`, and Rubi 4.3.2.1's `sqrt(tan(c + d x))/(a + i a tan(c + d x))^(5/2)` | `integral(...)` | the antiderivative |
 
+### A quotient by the square of a base is integrated by parts against the base's reciprocal
+
+**Answers where there were none.** `x^2/(a x cos(a x) - sin(a x))^2` was declined, with the rest of
+Rubi's 4.7.7 over the square of `a x cos(a x) - sin(a x)` or `cos(a x) + a x sin(a x)`. A quotient
+`N/(M v^2)` by the square of a base with a function of `x` in it is `g v'/v^2` with `g = N/(M v')`, and
+`v'/v^2` is the derivative of `-1/v`: its integral is `-g/v` plus the integral of `g'/v`, taken where
+the base's derivative is one term and `v` cancels from `g'/v`
+([#718](https://github.com/asc-community/AngouriMath/issues/718)).
+
+| Input | Was (2.5.0) | Now |
+|---|---|---|
+| `"x^2/(a*x*cos(a*x) - sin(a*x))^2".ToEntity().Integrate("x")` | `integral(...)` | `x/(a^2 sin(a x) (a x cos(a x) - sin(a x))) - cot(a x)/a^3` |
+| `"sin(a*x)^2/(a*x*cos(a*x) - sin(a*x))^2".ToEntity().Integrate("x")` | `integral(...)` | a quotient by `a x cos(a x) - sin(a x)` and `1/(a^2 x)` |
+| `"sin(a*x)^3/(x*(a*x*cos(a*x) - sin(a*x))^2)".ToEntity().Integrate("x")` | `integral(...)` | the same with the sine integral `Si(a x)` |
+
 ### Trigonometric functions of multiples of one linear argument are written in it
 
 `csc(a + b x) csc(2a + 2b x)^2` was left unevaluated, and `csc(1 + x) csc(2 + 2x)^2` ran twenty

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

@@ -367,6 +367,81 @@ private static bool IsADerivativeOfAFactorBelowTheBar(Entity expr, Entity.Variab
             return Integration.ComputeAsTheSameQuestion(written, x, integrateByParts);
         }
 
+        /// <summary>
+        /// A quotient by the square of a base, by parts against the base's reciprocal, where the
+        /// base's derivative makes the rest exact: <c>N/(M v^2)</c> is <c>g v'/v^2</c> with
+        /// <c>g = N/(M v')</c>, and <c>v'/v^2</c> is the derivative of <c>-1/v</c>, so the integral
+        /// is <c>-g/v</c> plus the integral of <c>g'/v</c>; taken only where <c>v</c> cancels from
+        /// that, with the functions of <paramref name="x"/> in it taken for indeterminates.
+        /// </summary>
+        /// <remarks>
+        /// <c>x^2/(a x cos(a x) - sin(a x))^2</c> was declined, with the rest of Rubi's 4.7.7 by the
+        /// square of <c>a x cos(a x) - sin(a x)</c> or <c>cos(a x) + a x sin(a x)</c>: the base's
+        /// derivative is <c>-a^2 x sin(a x)</c>, so <c>g</c> is <c>-x csc(a x)/a^2</c>, and
+        /// <c>g'</c> is <c>a^2 v/(a^2 sin(a x))^2</c>, whose integral is <c>-cot(a x)/a^3</c>. Only
+        /// for a base with a function of <paramref name="x"/> in it that is not a polynomial, since
+        /// a quotient of polynomials is the partial fractions', and only where <c>v</c> cancels,
+        /// since otherwise the remainder is the harder question.
+        /// https://github.com/asc-community/AngouriMath/issues/718
+        /// </remarks>
+        internal static Entity? SolveByPartsAgainstTheReciprocalOfASquaredBase(Entity expr, Entity.Variable x)
+        {
+            if (!TryReadAsQuotient(expr, out var numerator, out var denominator) || !denominator.ContainsNode(x))
+                return null;
+            Entity? squared = null;
+            Entity rest = Number.Integer.One;
+            foreach (var factor in Mulf.LinearChildren(denominator))
+                if (squared is null && factor is Powf(var @base, Number.Integer { EInteger: var power }) && power.Equals(EInteger.FromInt32(2))
+                    && @base is Sumf or Minusf && @base.ContainsNode(x)
+                    && @base.Nodes.Any(node => node.ContainsNode(x) && IsAFunctionOfTheVariable(node))
+                    // The variable outside the functions as well, as in `a x cos(a x) - sin(a x)`: a
+                    // base that is a function of `sin(x)` alone, `(a + b sin(x))^2`, is the
+                    // substitutions' and the reductions', and was answered in a second where this
+                    // ran past the budget.
+                    && @base.Replace(node => IsAFunctionOfTheVariable(node) ? Number.Integer.One : node).ContainsNode(x))
+                    squared = @base;
+                else
+                    rest = rest * factor;
+            if (squared is null)
+                return null;
+            // A power of x every term of the base has goes to the rest: `x^2 v^2` arrives as
+            // `(a x^2 cos(a x) - x sin(a x))^2`, the square of `x v`.
+            var lowest = Sumf.LinearChildren(squared).Select(term => Mulf.LinearChildren(term)
+                .Select(factor => factor == x ? 1 : factor is Powf(var b, Number.Integer { EInteger: var n }) && b == x && n.CanFitInInt32() ? n.ToInt32Unchecked() : 0)
+                .Sum()).Min();
+            if (lowest > 0)
+            {
+                squared = (squared / MathS.Pow(x, lowest)).Expand().InnerSimplified;
+                rest = rest * MathS.Pow(x, 2 * lowest);
+            }
+            // Expanded, since the derivative of `a x cos(a x)` and of `sin(a x)` each have an
+            // `a cos(a x)`, which cancel only once multiplied out.
+            var derivative = squared.Differentiate(x).Expand().InnerSimplified;
+            // A derivative of one term, so that what it leaves below the bar of the remainder is
+            // no new square of a sum to take apart the same way.
+            if (TreeAnalyzer.IsZero(derivative) || derivative is Sumf or Minusf)
+                return null;
+            // The ring's answer first, where it has one: `x^2/(x cos(x) - sin(x))^2` is
+            // `(x sin(x) + cos(x))/(x cos(x) - sin(x))` there, continuous across the zeros of the
+            // base's derivative, where `-g/v` and the remainder each have a pole that cancels.
+            if (SolveByTrigonometricTowerAnsatz(expr, x) is { } fromTheRing)
+                return fromTheRing;
+            var (gTop, gBottom) = CancelledWithFunctionsAsIndeterminates(numerator, rest * derivative, x, expr);
+            // g' over v, cancelled: (A' B - A B')/(B^2 v) for g = A/B.
+            var above = (gTop.Differentiate(x) * gBottom - gTop * gBottom.Differentiate(x)).InnerSimplified;
+            var (rTop, rBottom) = CancelledWithFunctionsAsIndeterminates(above, MathS.Pow(gBottom, 2) * squared, x, expr);
+            if (rBottom.Nodes.Any(node => node == squared) || !rTop.ContainsNode(x) && !rBottom.ContainsNode(x) && TreeAnalyzer.IsZero(rTop))
+                return null;
+            var remainder = Functions.PartialFractions.Bare(rTop / rBottom);
+            if (Integration.ComputeIndefiniteIntegral(remainder, x) is not { } integral || integral.Nodes.Any(node => node is Integralf))
+                return null;
+            return -gTop / (gBottom * squared) + integral;
+        }
+
+        /// <summary>Whether <paramref name="node"/> is a trigonometric function, a logarithm or an exponential.</summary>
+        private static bool IsAFunctionOfTheVariable(Entity node)
+            => node is Sinf or Cosf or Tanf or Cotanf or Secantf or Cosecantf or Logf or Powf(Number, _);
+
         /// <summary>Whether <paramref name="expr"/> holds a fractional power of something in <paramref name="x"/>.</summary>
         private static bool HasARadicalOf(Entity expr, Entity.Variable x)
             => expr.Nodes.Any(node => node is Powf(var @base, Number.Rational power) && power is not Number.Integer && @base.ContainsNode(x));

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

@@ -784,6 +784,9 @@ private static Entity Normalized(Entity expr, Entity.Variable x) =>
             // is `x^2/(L^2 sqrt(q))` cancelled, and was five terms over `L^5 (1 + L sqrt(q))`
             // each searched for seconds when the reduction reached it first.
             if ((answer = IndefiniteIntegralSolver.SolveByCancellingWithFunctionsAsIndeterminates(expr, x, integrateByParts)) is { }) return answer;
+            // A quotient by the square of a base with a function in it, by parts against the base's
+            // reciprocal where the base's derivative makes the rest exact.
+            if ((answer = IndefiniteIntegralSolver.SolveByPartsAgainstTheReciprocalOfASquaredBase(expr, x)) is { }) return answer;
             // A polynomial over a power of a linear and something that is not a polynomial,
             // the polynomial written in powers of the linear at its root: the part the power
             // divides goes over the rest alone. Before the split, so that the whole polynomial

Sources/Tests/UnitTests/Calculus/SquaredBaseByPartsIntegralTest.cs

@@ -0,0 +1,52 @@
+//
+// 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 quotient by the square of a base with a function in it, by parts against the base's
+    /// reciprocal: <c>N/(M v^2)</c> is <c>g v'/v^2</c> with <c>g = N/(M v')</c>, whose integral is
+    /// <c>-g/v</c> plus that of <c>g'/v</c>, which the base cancels from where its derivative is
+    /// one term. <c>x^2/(a x cos(a x) - sin(a x))^2</c> was declined; Rubi's 4.7.7.
+    /// <a href="https://github.com/asc-community/AngouriMath/issues/718">#718</a>
+    /// </summary>
+    /// <remarks>
+    /// Checked by differentiating back with <c>a = 1.3</c> on both sides of zero.
+    /// </remarks>
+    [Trait("Area", "Calculus")]
+    public sealed class SquaredBaseByPartsIntegralTest
+    {
+        [Theory]
+        [InlineData("x^2/(a*x*cos(a*x) - sin(a*x))^2")]
+        [InlineData("sin(a*x)^2/(a*x*cos(a*x) - sin(a*x))^2")]
+        [InlineData("sin(a*x)^3/(x*(a*x*cos(a*x) - sin(a*x))^2)")]
+        [InlineData("sin(a*x)^4/(x^2*(a*x*cos(a*x) - sin(a*x))^2)")]
+        [InlineData("x^2/(cos(a*x) + a*x*sin(a*x))^2")]
+        [InlineData("cos(a*x)^2/(cos(a*x) + a*x*sin(a*x))^2")]
+        [InlineData("cos(a*x)^3/(x*(cos(a*x) + a*x*sin(a*x))^2)")]
+        public void IsIntegratedAgainstTheReciprocalOfTheBase(string integrand)
+        {
+            var integral = integrand.ToEntity().Integrate("x");
+            Assert.DoesNotContain("integral(", integral.Stringize());
+            Entity Pinned(Entity e) => e.Substitute("a", 1.3);
+            var derivative = Pinned(integral.Substitute("C", 0)).Differentiate("x");
+            var original = Pinned(integrand.ToEntity());
+            foreach (var at in new[] { -2.3, -1.1, -0.4, 0.4, 1.1, 2.3 })
+            {
+                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)),
+                    $"d/dx of the antiderivative of {integrand} is {got} at x = {at}, where the integrand is {want}");
+            }
+        }
+    }
+}