Integrate x^m (d + e x^2)^p times an inverse trigonometric function by parts

BREAKING-CHANGES.md

@@ -6593,6 +6593,25 @@ leaves are read as quotients of the sine and the cosine
 | `"1/(a+b*asin(c*x))".Integrate("x")` | `integral(1 / (a + b * arcsin(c * x)), x)` | `Si` and `Ci` of `(a + b arcsin(c x))/b` |
 | `"x/((c+a^2*c*x^2)^2*atan(a*x))".Integrate("x")` | `integral(x / ((c + a ^ 2 * c * x ^ 2) ^ 2 * arctan(a * x)), x)` | `1/2 * Si(2 * arctan(a * x)) / (a ^ 2 * c ^ 2) + C` |
 
+### A power of x over a power of a quadratic times an inverse trigonometric function is integrated by parts against one term
+
+**Answers where there were none.** `x^3 (a + b F(c x))/(d + e x^2)^3` for `F` the inverse sine, tangent, secant and
+cosecant, Rubi's 5.1.4, 5.3.4, 5.5.1 and 5.6.1, was declined. By parts against the integrator's own antiderivative of
+`x^3/(d + e x^2)^3`, a piecewise on `e = 0`, spent the budget on every arm. Where `m + 2p + 3 = 0`,
+`x^m (d + e x^2)^p` has the one-term antiderivative `v = x^(m + 1) (d + e x^2)^(p + 1)/(d (m + 1))`, Rubi's, which
+vanishes at 0, and the integral is now `v (a + b F(c x)) - b integral v F(c x)'` with `F(c x)'` algebraic.
+`v' = x^m (d + e x^2)^p` is an identity of rational functions, so the parts hold wherever the integrand is defined,
+on any branch of `F`. Only
+for a whole `p` of `-3` or less, and not where `d + e x^2` is a multiple of the radicand `F` holds, which the
+substitution under `F` answers ([#233](https://github.com/asc-community/AngouriMath/issues/233)).
+
+| Input | Was (2.5.0) | Now |
+|---|---|---|
+| `"x^3*(a + b*arcsin(c*x))/(d + k*x^2)^3".ToEntity().Integrate("x")` | `integral(...)` | 6,013 characters |
+| `"x^3*(a + b*arctan(c*x))/(d + k*x^2)^3".ToEntity().Integrate("x")` | `integral(...)` | 4,066 characters |
+| `"x^3*(a + b*arcsec(c*x))/(d + k*x^2)^3".ToEntity().Integrate("x")` | `integral(...)` | 1,772 characters |
+| `"x^5*(a + b*arctan(c*x))/(d + k*x^2)^4".ToEntity().Integrate("x")` | `integral(...)` | 6,176 characters |
+
 ### The arcsecant beside a root of `x^2 - 1` is integrated right below `-1` too
 
 **Answers where there were none.** Under `x = sec(u)`, the substitution that undoes the arcsecant,

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

@@ -32123,6 +32123,134 @@ Piecewise piecewise when piecewise.Cases.Any() => piecewise.Cases.First().Expres
             return (alpha / (p0 * slope) * product).InnerSimplified;
         }
 
+        /// <summary>
+        /// <c>x^m (d + e x^2)^p (a + b F(c x))</c>, <c>F</c> an inverse trigonometric function and
+        /// <c>m + 2p + 3 = 0</c>, by parts against the antiderivative of <c>x^m (d + e x^2)^p</c> that is
+        /// one term, <c>v = x^(m + 1) (d + e x^2)^(p + 1)/(d (m + 1))</c>:
+        /// <c>integral = v (a + b F(c x)) - b integral v F(c x)'</c>, and <c>F(c x)'</c> is algebraic.
+        /// </summary>
+        /// <remarks>
+        /// <para>
+        /// <c>v' = x^m (d + e x^2)^p ((m + 1)(d + e x^2) + 2 (p + 1) e x^2)/(d (m + 1))</c>, and the bracket
+        /// is <c>(m + 1) d + (m + 2p + 3) e x^2</c>, which is <c>(m + 1) d</c> for every <c>e</c>: an
+        /// identity of rational functions, so the parts are exact wherever the integrand is defined, on
+        /// any branch of <c>F</c>. It is Rubi's choice of <c>v</c>, and the one that vanishes at
+        /// <c>x = 0</c>.
+        /// </para>
+        /// <para>
+        /// The integrator's own antiderivative of <c>x^3/(d + e x^2)^3</c> is a piecewise on <c>e = 0</c>,
+        /// and by parts against it spent the budget on every arm: Rubi's 5.1.4, 5.3.4, 5.5.1 and 5.6.1
+        /// <c>x^3 (a + b F(c x))/(d + e x^2)^3</c>, for <c>F</c> the inverse sine, tangent, secant and
+        /// cosecant, ran out of time, where the remainder against this <c>v</c> is answered in under a
+        /// second.
+        /// </para>
+        /// <para>
+        /// Narrow on purpose. <c>p</c> whole and at most <c>-3</c>, so that <c>v</c> is rational and
+        /// <c>m = -2p - 3</c> at least 3: <c>m = 1</c>, <c>x/(d + e x^2)^2</c>, is answered by parts as it
+        /// was, and a half-odd <c>p</c> is answered as it was beside the inverse sine and tangent and is
+        /// elliptic beside the secant. And not for <c>e = +-c^2 d</c>, the quadratic a multiple of
+        /// <c>F</c>'s own radicand, which the substitution under <c>F</c> answers.
+        /// https://github.com/asc-community/AngouriMath/issues/233
+        /// </para>
+        /// </remarks>
+        internal static Entity? SolveAPowerOfXOverAPowerOfAQuadraticTimesAnInverseTrigonometricFunctionByParts(Entity expr, Entity.Variable x, bool integrateByParts)
+        {
+            if (!Integration.AnsweringTheQuestionAsked || integratingAPowerOverAQuadraticBesideAnInverse)
+                return null;
+            // The one inverse trigonometric function, of c x.
+            Entity? function = null;
+            foreach (var node in expr.Nodes)
+                if ((node is Arcsinf or Arccosf or Arctanf or Arccotanf or Arcsecantf or Arccosecantf) && node.ContainsNode(x))
+                {
+                    if (function is not null && function != node)
+                        return null;
+                    function = node;
+                }
+            var argument = function switch
+            {
+                Arcsinf(var u) => u, Arccosf(var u) => u, Arctanf(var u) => u,
+                Arccotanf(var u) => u, Arcsecantf(var u) => u, Arccosecantf(var u) => u,
+                _ => null
+            };
+            if (function is null || argument is null || !TreeAnalyzer.TryGetPolyLinear(argument, x, out var rate, out var offset)
+                || rate.ContainsNode(x) || !Functions.PartialFractions.IsZeroAsAValue(offset) || Functions.PartialFractions.IsZeroAsAValue(rate))
+                return null;
+            // Above the bar one factor linear in it, a + b F, and the rest; below, nothing with it.
+            var (numerator, denominator) = Functions.SingleQuotient.Of(expr);
+            if (denominator.ContainsNode(function))
+                return null;
+            Entity? linear = null;
+            Entity rest = Number.Integer.One;
+            foreach (var factor in Mulf.LinearChildren(numerator))
+            {
+                if (!factor.ContainsNode(function))
+                    rest *= factor;
+                else if (linear is null)
+                    linear = factor;
+                else
+                    return null;
+            }
+            var t = Variable.CreateUnique(expr, "t_inv");
+            if (linear is null || !TreeAnalyzer.TryGetPolyLinear(linear.Substitute(function, t), t, out var slope, out var intercept)
+                || slope.ContainsNode(x) || intercept.ContainsNode(x) || Functions.PartialFractions.IsZeroAsAValue(slope))
+                return null;
+            // The rest over the denominator as K x^m Q^p, Q = d + e x^2.
+            Entity constant = Number.Integer.One;
+            Entity? quadratic = null;
+            var m = 0;
+            var p = 0;
+            foreach (var (side, sign) in new[] { (rest, 1), (denominator, -1) })
+                foreach (var factor in Mulf.LinearChildren(side))
+                {
+                    if (!factor.ContainsNode(x))
+                    {
+                        constant = sign > 0 ? constant * factor : constant / factor;
+                        continue;
+                    }
+                    var (@base, power) = factor is Powf(var b0, Number.Integer { EInteger: var e0 }) && e0.CanFitInInt32()
+                        ? (b0, e0.ToInt32Unchecked()) : (factor, 1);
+                    if (@base == x)
+                        m += sign * power;
+                    else if (quadratic is null || quadratic == @base)
+                    {
+                        quadratic = @base;
+                        p += sign * power;
+                    }
+                    else
+                        return null;
+                }
+            if (quadratic is null || p > -3 || m != -2 * p - 3
+                || !TreeAnalyzer.TryGetPolynomial(quadratic, x, out var read)
+                || read.Keys.Any(key => !key.IsZero && !key.Equals(EInteger.FromInt32(2)))
+                || !read.TryGetValue(EInteger.Zero, out var d) || !read.TryGetValue(EInteger.FromInt32(2), out var e)
+                || d.ContainsNode(x) || e.ContainsNode(x)
+                || Functions.PartialFractions.IsZeroAsAValue(d) || Functions.PartialFractions.IsZeroAsAValue(e)
+                || Functions.PartialFractions.IsZeroAsAValue((e - rate * rate * d).Expand())
+                || Functions.PartialFractions.IsZeroAsAValue((e + rate * rate * d).Expand()))
+                return null;
+            var v = constant * MathS.Pow(x, Number.Integer.Create(m + 1)) * MathS.Pow(quadratic, Number.Integer.Create(p + 1))
+                / (d * (m + 1));
+            // The remainder holds no inverse function, so it is not this integrand again; the flag is for
+            // whatever its own search writes.
+            var remainder = (slope * v * function.Differentiate(x)).InnerSimplified;
+            Entity? integral;
+            integratingAPowerOverAQuadraticBesideAnInverse = true;
+            try
+            {
+                integral = Integration.ComputeAsAQuestionOfItsOwn(remainder, x, integrateByParts);
+            }
+            finally
+            {
+                integratingAPowerOverAQuadraticBesideAnInverse = false;
+            }
+            if (integral is null)
+                return null;
+            var answer = v * linear - integral;
+            return Functions.PartialFractions.DerivativeDiffersWhereReal(answer, expr, x) ? null : answer;
+        }
+
+        [System.ThreadStatic] private static bool integratingAPowerOverAQuadraticBesideAnInverse;
+
         /// <summary>
         /// <c>x^m (a + b atan(c x))^p</c> over a power of <c>d (1 +- i c x)</c>, under <c>x = tan(u)/c</c>, where
         /// <c>1 + i tan(u)</c> is <c>e^(i u)/cos(u)</c>: the integrand in <c>u</c> is

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

@@ -707,6 +707,10 @@ private static Entity WithItsPiecewiseSumsCombined(Entity expr)
             // A rational function times the logarithm of a nested product of powers of linears, by parts,
             // its logarithm's derivative read off as written: `ln(e (f (a + b x)^p (c + d x)^q)^r)`.
             if ((answer = IndefiniteIntegralSolver.SolveARationalFunctionTimesALogarithmOfPowersOfLinears(expr, x, integrateByParts)) is { }) return answer;
+            // x^m (d + e x^2)^p (a + b F(c x)) with m + 2p + 3 = 0 and F an inverse trigonometric function, by
+            // parts against v = x^(m + 1) (d + e x^2)^(p + 1)/(d (m + 1)), before by parts against the
+            // integrator's piecewise v timed out on `x^3 (a + b asin(c x))/(d + e x^2)^3`.
+            if ((answer = IndefiniteIntegralSolver.SolveAPowerOfXOverAPowerOfAQuadraticTimesAnInverseTrigonometricFunctionByParts(expr, x, integrateByParts)) is { }) return answer;
             // A function of an inverse of something not linear times its derivative up to a factor
             // constant though written with x: `atan(c x/sqrt(a - c^2 x^2))^2/sqrt(d - c^2 d x^2/a)`. Early,
             // since it fires only where that factor holds x, and the rules before it spent 28 seconds on

Sources/Tests/UnitTests/Calculus/APowerOfXOverAPowerOfAQuadraticTimesAnInverseTrigonometricFunctionByPartsIntegralTest.cs

@@ -0,0 +1,66 @@
+//
+// 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 System.Globalization;
+using System.Linq;
+using AngouriMath.Extensions;
+using Xunit;
+
+namespace AngouriMath.Tests.Calculus
+{
+    /// <summary>
+    /// <c>x^m (d + k x^2)^p (a + b F(c x))</c> with <c>m + 2p + 3 = 0</c> and <c>F</c> an inverse
+    /// trigonometric function, by parts against <c>v = x^(m + 1) (d + k x^2)^(p + 1)/(d (m + 1))</c>.
+    /// By parts against the integrator's own antiderivative, a piecewise on <c>k = 0</c>, ran out of
+    /// time. Rubi's 5.1.4, 5.3.4, 5.5.1 and 5.6.1.
+    /// <a href="https://github.com/asc-community/AngouriMath/issues/233">#233</a>
+    /// </summary>
+    [Trait("Area", "Calculus")]
+    public sealed class APowerOfXOverAPowerOfAQuadraticTimesAnInverseTrigonometricFunctionByPartsIntegralTest
+    {
+        private const string Inside = "-1.3 -0.9 -0.4 0.3 0.8 1.2";
+        private const string Outside = "-4.1 -2.9 -2.1 2.1 2.9 4.1";
+
+        [Theory]
+        [InlineData("x^3*(a + b*arcsin(c*x))/(d + k*x^2)^3", Inside)]
+        [InlineData("x^3*(a + b*arctan(c*x))/(d + k*x^2)^3", Inside)]
+        [InlineData("x^3*(a + b*arcsec(c*x))/(d + k*x^2)^3", Outside)]
+        [InlineData("x^3*(a + b*arccsc(c*x))/(d + k*x^2)^3", Outside)]
+        [InlineData("x^3*(a + b*arccos(c*x))/(d + k*x^2)^3", Inside)]
+        [InlineData("x^3*(a + b*arccot(c*x))/(d + k*x^2)^3", Inside)]
+        [InlineData("x^3*arcsin(c*x)/(d + k*x^2)^3", Inside)]
+        [InlineData("x^5*(a + b*arctan(c*x))/(d + k*x^2)^4", Inside)]
+        public void ByPartsAgainstTheOneTermAntiderivative(string integrand, string points)
+        {
+            var integral = integrand.ToEntity().Integrate("x");
+            var text = integral.Stringize();
+            Assert.DoesNotContain("integral(", text);
+            Assert.True(text.Length < 20000, $"{text.Length} characters of answer for {integrand}");
+            foreach (var (d, k) in new[] { (2.1, 0.9), (-1.7, 1.3), (1.9, -0.8) })
+            {
+                Entity Pinned(Entity e) => e.Substitute("a", 1.3).Substitute("b", 0.7).Substitute("c", 0.6)
+                    .Substitute("d", d).Substitute("k", k);
+                var derivative = Pinned(integral.Substitute("C", 0)).Differentiate("x");
+                var original = Pinned(integrand.ToEntity());
+                var compared = 0;
+                foreach (var at in points.Split(' ').Select(point => double.Parse(point, CultureInfo.InvariantCulture)))
+                {
+                    var want = original.Substitute("x", at).EvalNumerical();
+                    var got = derivative.Substitute("x", at).EvalNumerical();
+                    if (want.IsNaN)
+                        continue;
+                    compared++;
+                    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}, d = {d}, k = {k}, where the integrand is {want}");
+                }
+                Assert.True(compared >= 5, $"only {compared} points could be compared for {integrand} at d = {d}, k = {k}");
+            }
+        }
+    }
+}