Integrate a function of an inverse function of a linear beside its powers under the linear

BREAKING-CHANGES.md

@@ -457,6 +457,19 @@ back after. Each answer is in three cases by the sign of `4 a/c - (b/c)^2`
 | `"(a + b*x^3)^(1/3)/(x^2*(a*d - b*d*x^3))".ToEntity().Integrate("x")` | `integral(...)` | 773 characters |
 | `"1/(x^2*(1 - x^3)^(2/3)*(1 + x^3))".ToEntity().Integrate("x")` | `integral(...)` | 471 characters |
 
+### A function of an inverse function of a linear beside powers of multiples of that linear is integrated under it
+
+**Answers where there were none.** `(c k + d k x)^3 (a + b asinh(c + d x))^2` ran out of time, and so did three more
+of Rubi's 7.1.5 and 7.2.5. Under `u = c + d x` it is `k^3 u^3 (a + b asinh(u))^2`, which is answered, divided by `d`
+([#718](https://github.com/asc-community/AngouriMath/issues/718)).
+
+| Input | Was (2.5.0) | Now |
+|---|---|---|
+| `"(c*k+d*k*x)^3*(a+b*asinh(c+d*x))^2".ToEntity().Integrate("x")` | `integral(...)` | 3,574 characters |
+| `"(c*k+d*k*x)^3*(a+b*asinh(c+d*x))^3".ToEntity().Integrate("x")` | `integral(...)` | 6,188 characters |
+| `"(c*k+d*k*x)^3*(a+b*asinh(c+d*x))^4".ToEntity().Integrate("x")` | `integral(...)` | 9,539 characters |
+| `"(c*k+d*k*x)*(a+b*acosh(c+d*x))^4".ToEntity().Integrate("x")` | `integral(...)` | 3,607 characters |
+
 ### `a + i a tan` beside the tangent's differential is lowered over its conjugate
 
 **Answers where there were none.** `sqrt(a + i a tan(x))/(c + d tan(x))^(3/2)` ran past thirty seconds:

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

@@ -26647,6 +26647,72 @@ static void Add(Dictionary<int, ERational> into, int k, ERational value)
             return answer.Nodes.Any(node => node == MathS.NaN) ? null : answer;
         }
 
+        /// <summary>
+        /// A function of an inverse function of a linear <c>L = c + d x</c> times powers of multiples of
+        /// <c>L</c>, <c>(c e + d e x)^3 (a + b arsinh(c + d x))^2</c>, under <c>u = L</c>: each power of
+        /// <c>k L</c> is <c>(k u)^m</c>, the inverse function's argument is <c>u</c>, and the integral is
+        /// the one in <c>u</c> over <c>d</c>.
+        /// </summary>
+        /// <remarks>
+        /// <c>x^3 (a + b arsinh(x))^2</c> is answered in half a second, and the rules timed out on
+        /// Rubi's 7.1.5 <c>(c e + d e x)^3 (a + b arsinh(c + d x))^2</c>, which is that in <c>u</c>.
+        /// https://github.com/asc-community/AngouriMath/issues/718
+        /// </remarks>
+        internal static Entity? SolveAFunctionOfAnInverseOfALinearBesideItsPowersUnderTheLinear(Entity expr, Entity.Variable x)
+        {
+            if (!Integration.AnsweringTheQuestionAsked)
+                return null;
+            // The linear: the argument of an inverse function, with a constant term, as in `arsinh(c + d x)`,
+            // which arrives as `ln(c + d x + sqrt((c + d x)^2 + 1))`.
+            Entity? linear = null;
+            foreach (var node in expr.Nodes)
+            {
+                Entity? argument = node switch
+                {
+                    Arcsinf(var y) => y, Arccosf(var y) => y, Arctanf(var y) => y,
+                    _ => null
+                };
+                // The logarithm's argument without its root: `c + d x` of `ln(c + d x + sqrt((c + d x)^2 + 1))`.
+                if (argument is null && node is Logf(var b0, var of) && b0 == MathS.e && of is Sumf)
+                {
+                    var terms = Sumf.LinearChildren(of).ToList();
+                    var roots = terms.Where(term => term is Powf(_, Number.Rational { ERational.Denominator: var den }) && den.Equals(EInteger.FromInt32(2))).ToList();
+                    if (roots.Count == 1)
+                        argument = terms.Where(term => term != roots[0]).Aggregate((Entity)Number.Integer.Zero, (sum, term) => sum == Number.Integer.Zero ? term : sum + term);
+                }
+                if (argument is null || !argument.ContainsNode(x))
+                    continue;
+                if (!TreeAnalyzer.TryGetPolyLinear(argument, x, out var slope0, out var intercept0)
+                    || Functions.PartialFractions.IsZeroAsAValue(slope0) || Functions.PartialFractions.IsZeroAsAValue(intercept0))
+                    continue;
+                linear = argument;
+                break;
+            }
+            if (linear is null)
+                return null;
+            if (!TreeAnalyzer.TryGetPolyLinear(linear, x, out var slope, out var intercept) || slope is null || intercept is null)
+                return null;
+            var u = Variable.CreateUnique(expr, "u_lin");
+            // Every node that is a constant multiple of L becomes that multiple of u.
+            var inU = expr.Replace(node =>
+            {
+                if (node == linear)
+                    return u;
+                if (node is not (Sumf or Minusf) || !node.ContainsNode(x)
+                    || !TreeAnalyzer.TryGetPolyLinear(node, x, out var s1, out var c1) || s1 is null || c1 is null
+                    || Functions.PartialFractions.IsZeroAsAValue(s1))
+                    return node;
+                var k = (s1 / slope).InnerSimplified;
+                return Functions.PartialFractions.IsZeroAsAValue((c1 - k * intercept).InnerSimplified) ? k * u : node;
+            });
+            if (inU.ContainsNode(x))
+                return null;
+            if (Integration.ComputeAsAQuestionOfItsOwn(inU.InnerSimplified, u, integrateByParts: true) is not { } integrated)
+                return null;
+            var answer = (integrated.Substitute(u, linear) / slope).InnerSimplified;
+            return answer.Nodes.Any(node => node == MathS.NaN) ? null : answer;
+        }
+
         /// <summary>
         /// A rational function of a sine, or of a cosine, over a quadratic in it with a symbol
         /// among the coefficients: <c>sin(x)/(a + b sin(x) + c sin(x)^2)</c>, and the same with

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

@@ -680,6 +680,9 @@ private static Entity WithItsPiecewiseSumsCombined(Entity expr)
             // A power of a sine, cosine or hyperbolic one over a binomial in a power of it, over the
             // binomial's roots, before the rules that spent the budget on `sinh(y)^4/(a + b sinh(y)^3)`.
             if ((answer = IndefiniteIntegralSolver.SolveAPowerOverABinomialInAPowerOfASineOverItsRoots(expr, x)) is { }) return answer;
+            // A function of an inverse function of c + d x beside powers of multiples of it, under
+            // u = c + d x, before the rules that timed out on `(c e + d e x)^3 (a + b arsinh(c + d x))^2`.
+            if ((answer = IndefiniteIntegralSolver.SolveAFunctionOfAnInverseOfALinearBesideItsPowersUnderTheLinear(expr, x)) is { }) return answer;
             // A power of x times an inverse tangent and the logarithm of its quadratic, by parts with the
             // two remainders that have no antiderivative apiece paired, before the rules that timed out.
             if ((answer = IndefiniteIntegralSolver.SolveAPowerOfXTimesAnArctangentAndTheLogarithmOfItsQuadratic(expr, x)) is { }) return answer;

Sources/Tests/UnitTests/Calculus/AnInverseOfALinearBesideItsPowersIntegralTest.cs

@@ -0,0 +1,58 @@
+//
+// 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 function of an inverse function of <c>c + d x</c> beside powers of multiples of it, under
+    /// <c>u = c + d x</c>: Rubi's 7.1.5 <c>(c e + d e x)^3 (a + b arsinh(c + d x))^2</c> and 7.2.5
+    /// <c>(c e + d e x)(a + b arcosh(c + d x))^4</c>, which timed out.
+    /// <a href="https://github.com/asc-community/AngouriMath/issues/718">#718</a>
+    /// </summary>
+    [Trait("Area", "Calculus")]
+    public sealed class AnInverseOfALinearBesideItsPowersIntegralTest
+    {
+        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}");
+            }
+        }
+
+        // c + d x above 1 at every sampled x, where the inverse hyperbolic cosine is real.
+        [Theory]
+        [InlineData("(c*k+d*k*x)^3*(a+b*asinh(c+d*x))^2")]
+        [InlineData("(c*k+d*k*x)^3*(a+b*asinh(c+d*x))^4")]
+        [InlineData("(c*k+d*k*x)*(a+b*acosh(c+d*x))^4")]
+        public void IsAnsweredAndDifferentiatesBack(string integrand)
+        {
+            DifferentiatesBack(integrand, ("a", 1.3), ("b", 0.7), ("c", 3), ("d", 0.5), ("k", 1.5));
+            DifferentiatesBack(integrand, ("a", -0.5), ("b", 2), ("c", 4), ("d", -0.6), ("k", -0.7));
+        }
+    }
+}