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Rafael-SOWNet merged 1 commit into from
Oct 19, 2026
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Integrate an exponential of the reciprocal of a linear beside another linear #1950

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13 changes: 13 additions & 0 deletions BREAKING-CHANGES.md
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Expand Up @@ -55,6 +55,19 @@ numbers for every coefficient the solve is now done in exact rationals, to degre
| `"1/((3-2*x+x^2)^(11/2)*(1+x+2*x^2)^5)".ToEntity().Integrate("x")` | `integral(...)` | 1,853 characters |
| `"1/((3-2*x+x^2)^(21/2)*(1+x+2*x^2)^10)".ToEntity().Integrate("x")` | `integral(...)` | 3,475 characters |

### An exponential of something of the reciprocal of a linear is integrated beside another linear

**Answers where there were none.** `f^(c/(a + b x)) x^2` was declined, and so were 24 more of Rubi's 2.3:
`F^(a + b/(c + d x))/(e + f x)^m`, `e^(e/(c + d x)^2) (a + b x)^m` and `F^(e + f (a + b x)/(c + d x))/(g + h x)^m`.
Under `u = 1/(c + d x)` another linear `p + q x` is `((d p - q c) u + q)/d` times `c + d x`, so the question in `u`
is an exponential beside a rational function, which the exponential integral and the moments answer. Only
powers of `c + d x` itself were read before ([#233](https://github.com/asc-community/AngouriMath/issues/233)).

| Input | Was (2.5.0) | Now |
|---|---|---|
| `"e^(k/(c+d*x))/(a+b*x)".ToEntity().Integrate("x")` | `integral(...)` | two `Ei`, of `k/(c + d x)` and of `k/(c + d x) - k b/(a d - b c)` |
| `"f^(c/(a+b*x)^2)*x^4".ToEntity().Integrate("x")` | `integral(...)` | 1,262 characters, the error function and powers of `1/(a + b x)` |

### A polynomial over a symbolic quartic that is two quadratics in a hidden variable is split over them

**Answers where there were none.** `1/(a + b x + c x^2 + b x^3 + a x^4)` was declined, and so were Rubi's 1.3.1
Expand Down
88 changes: 71 additions & 17 deletions Sources/AngouriMath/Functions/Continuous/Integration/IndefiniteIntegralSolver.cs
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Expand Up @@ -21983,7 +21983,8 @@ private static bool TryReadAPowerTimesARadicalQuadratic(Entity expr, Entity.Vari
/// </remarks>
internal static Entity? SolveAGaussianInAReciprocal(Entity expr, Entity.Variable x)
{
if (ReadAnExponentialInTheReciprocalOfALinear(expr, x) is not var (constant, @base, exponentInU, u, linear, slope, power)
// Beside another linear, the moments alone do not answer it: that is the next rule's.
if (ReadAnExponentialInTheReciprocalOfALinear(expr, x) is not (var constant, var @base, var exponentInU, var u, var linear, var slope, var power, null)
|| !TreeAnalyzer.TryGetPolyQuadratic(exponentInU, u, out var square, out _, out _) || TreeAnalyzer.IsZero(square))
return null;
var powerInU = -power - EInteger.FromInt32(2);
Expand Down Expand Up @@ -22013,7 +22014,7 @@ private static bool TryReadAPowerTimesARadicalQuadratic(Entity expr, Entity.Vari
/// </remarks>
internal static Entity? SolveAnExponentialInTheReciprocalOfALinear(Entity expr, Entity.Variable x, bool integrateByParts)
{
if (ReadAnExponentialInTheReciprocalOfALinear(expr, x) is not var (constant, @base, exponentInU, u, linear, slope, power)
if (ReadAnExponentialInTheReciprocalOfALinear(expr, x) is not var (constant, @base, exponentInU, u, linear, slope, power, beside)
|| !TreeAnalyzer.TryGetPolynomial(exponentInU, u, out var monomials)
|| monomials.Keys.Any(k => k.Sign < 0 || k.CompareTo(EInteger.FromInt32(8)) > 0)
|| !monomials.Keys.Any(k => k.Sign > 0))
Expand All @@ -22023,10 +22024,23 @@ private static bool TryReadAPowerTimesARadicalQuadratic(Entity expr, Entity.Vari
return null;
var exponential = MathS.Pow(@base, exponentInU);
var inU = powerInU.IsZero ? exponential : MathS.Pow(u, Number.Integer.Create(powerInU)) * exponential;
// Another linear is a linear in u, and the rest one quotient, the way the rules read it.
var factor = -constant / slope;
if (beside is var (above, below, slopePower))
{
// -constant d^(slopePower - 1), one power of d.
var raised = slopePower - EInteger.One;
factor = raised.IsZero ? -constant
: raised.Sign > 0 ? -constant * (raised.Equals(EInteger.One) ? slope : MathS.Pow(slope, Number.Integer.Create(raised)))
: -constant / (raised.Equals(EInteger.FromInt32(-1)) ? slope : MathS.Pow(slope, Number.Integer.Create(-raised)));
var top = powerInU.Sign > 0 ? MathS.Pow(u, Number.Integer.Create(powerInU)) * above : above;
var bottom = powerInU.Sign < 0 ? MathS.Pow(u, Number.Integer.Create(-powerInU)) * below : below;
inU = bottom == Number.Integer.One ? top * exponential : top * exponential / bottom;
}
if (Integration.ComputeAsAQuestionOfItsOwn(inU, u, integrateByParts) is not { } answerInU
|| answerInU.Nodes.Any(node => node == MathS.NaN))
return null;
return (-constant / slope * answerInU.Substitute(u, 1 / linear)).InnerSimplified;
return (factor * answerInU.Substitute(u, 1 / linear)).InnerSimplified;
}

/// <summary>
Expand All @@ -22037,12 +22051,23 @@ private static bool TryReadAPowerTimesARadicalQuadratic(Entity expr, Entity.Vari
/// <c>b/L^k</c> is <c>b u^k</c> -- rather than by substituting <c>x = (1/u - c)/d</c> and
/// asking the simplifier to see that <c>1/(1/u)^2</c> is <c>u^2</c>.
/// </summary>
private static (Entity Constant, Entity Base, Entity ExponentInU, Variable U, Entity Linear, Entity Slope, EInteger Power)?
/// <remarks>
/// A whole power of another linear <c>M = p + q x</c> is read too, as <c>M/L</c> written in
/// <c>u</c> times <c>L</c>: with <c>L = c + d x</c>, <c>M/L = ((d p - q c) u + q)/d</c>, a
/// linear in <c>u</c>, so <c>M^k</c> adds <c>k</c> to the power of <c>L</c> and
/// <c>((d p - q c) u + q)^k</c> to <c>Beside</c>, and <c>-k</c> to its power of <c>d</c>, kept
/// apart from the constant so that the caller writes one power of <c>d</c>, not <c>d/d</c>.
/// <c>Beside</c> is <see langword="null"/> when there is no such factor. A quotient <c>N/L^k</c> in the exponent with <c>N</c> linear is
/// <c>(N/L) u^(k - 1)</c> the same way. Rubi's 2.3 <c>f^(c/(a + b x)) x^m</c>,
/// <c>F^(a + b/(c + d x))/(e + f x)^m</c> and <c>F^(e + f (a + b x)/(c + d x))/(g + h x)^m</c>
/// were declined, for want of the other linear.
/// https://github.com/asc-community/AngouriMath/issues/233
/// </remarks>
private static (Entity Constant, Entity Base, Entity ExponentInU, Variable U, Entity Linear, Entity Slope, EInteger Power, (Entity Above, Entity Below, EInteger SlopePower)? Beside)?
ReadAnExponentialInTheReciprocalOfALinear(Entity expr, Entity.Variable x)
{
Powf? exponential = null;
Entity? linear = null;
var power = EInteger.Zero;
var linears = new List<(Entity Linear, EInteger Power)>();
Entity constant = Number.Integer.One;
foreach (var (factor, underneath) in FactorsOfTheIntegrand(expr))
{
Expand All @@ -22056,30 +22081,59 @@ private static (Entity Constant, Entity Base, Entity ExponentInU, Variable U, En
exponential = (Powf)factor;
continue;
}
// The linear, or a whole power of it, above or below the bar.
// A linear, or a whole power of one, above or below the bar.
var (candidate, k) = factor is Powf(var b, Number.Integer n) && n.EInteger.CanFitInInt32() ? (b, n.EInteger) : (factor, EInteger.One);
if (linear is not null && linear != candidate
|| !TreeAnalyzer.TryGetPolyLinear(candidate, x, out var slopeOfIt, out _) || slopeOfIt is null || TreeAnalyzer.IsZero(slopeOfIt))
if (!TreeAnalyzer.TryGetPolyLinear(candidate, x, out var slopeOfIt, out _) || slopeOfIt is null || TreeAnalyzer.IsZero(slopeOfIt))
return null;
linear = candidate;
power = underneath ? power - k : power + k;
linears.Add((candidate, underneath ? -k : k));
}
if (exponential is null)
return null;
linear ??= ALinearBelowABar(exponential.Exponent, x);
if (linear is null || !TreeAnalyzer.TryGetPolyLinear(linear, x, out var slope, out _) || slope is null || TreeAnalyzer.IsZero(slope))
// L is the linear the exponent divides by. Wherever the exponent could be written in u
// before other linears were read, it is the linear beside it as well.
var linear = ALinearBelowABar(exponential.Exponent, x);
if (linear is null || !TreeAnalyzer.TryGetPolyLinear(linear, x, out var slope, out var intercept) || slope is null || TreeAnalyzer.IsZero(slope))
return null;
var u = Variable.CreateUnique(expr, "u");
// M/L in u, for a linear M: ((d p - q c) u + q)/d's numerator, the d left to the caller.
Entity? OverTheLinearInU(Entity other)
{
if (!TreeAnalyzer.TryGetPolyLinear(other, x, out var q, out var p) || q is null || TreeAnalyzer.IsZero(q))
return null;
return ((slope * p - q * intercept).InnerSimplified * u + q).InnerSimplified;
}
var power = EInteger.Zero;
var slopePower = EInteger.Zero;
Entity above = Number.Integer.One, below = Number.Integer.One;
var beside = false;
foreach (var (other, k) in linears)
{
power += k;
if (other == linear)
continue;
if (k.Abs().CompareTo(EInteger.FromInt32(32)) > 0 || OverTheLinearInU(other) is not { } inU)
return null;
beside = true;
var raised = k.Abs().Equals(EInteger.One) ? inU : MathS.Pow(inU, Number.Integer.Create(k.Abs()));
if (k.Sign > 0)
above *= raised;
else
below *= raised;
slopePower -= k;
}
var exponentInU = exponential.Exponent.Replace(node => node switch
{
Divf(var numerator, Powf(var below, Number.Integer k)) when below == linear => numerator * MathS.Pow(u, k),
Divf(var numerator, var below) when below == linear => numerator * u,
Powf(var below, Number.Integer { EInteger.Sign: < 0 } k) when below == linear => MathS.Pow(u, -k),
Divf(var numerator, Powf(var under, Number.Integer k)) when under == linear && !numerator.ContainsNode(x) => numerator * MathS.Pow(u, k),
Divf(var numerator, var under) when under == linear && !numerator.ContainsNode(x) => numerator * u,
Divf(var numerator, Powf(var under, Number.Integer k)) when under == linear && OverTheLinearInU(numerator) is { } inU
=> inU / slope * MathS.Pow(u, (k - 1).InnerSimplified),
Divf(var numerator, var under) when under == linear && OverTheLinearInU(numerator) is { } inU => inU / slope,
Powf(var under, Number.Integer { EInteger.Sign: < 0 } k) when under == linear => MathS.Pow(u, -k),
_ => node
});
if (exponentInU.ContainsNode(x))
return null;
return (constant, exponential.Base, exponentInU, u, linear, slope, power);
return (constant, exponential.Base, exponentInU, u, linear, slope, power, beside ? (above.InnerSimplified, below.InnerSimplified, slopePower) : null);
}

/// <summary>
Expand Down
61 changes: 61 additions & 0 deletions ...nitTests/Calculus/AnExponentialInTheReciprocalOfALinearBesideAnotherLinearIntegralTest.cs
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@@ -0,0 +1,61 @@
//
// 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>
/// An exponential of something of <c>1/L</c>, <c>L = c + d x</c>, beside a whole power of another
/// linear <c>M</c>. Under <c>u = 1/L</c>, <c>M</c> is <c>((d p - q c) u + q)/d</c> times <c>L</c>,
/// so the question in <c>u</c> is an exponential beside a rational function, which the
/// exponential integral and the moments answer. They were declined, since only powers of
/// <c>L</c> itself were read. Rubi's 2.3.
/// <a href="https://github.com/asc-community/AngouriMath/issues/233">#233</a>
/// </summary>
[Trait("Area", "Calculus")]
public sealed class AnExponentialInTheReciprocalOfALinearBesideAnotherLinearIntegralTest
{
[Theory]
[InlineData("f^(c/(a + b*x))*x^2", 1.1)]
[InlineData("f^(c/(a + b*x))/x", 1.1)]
[InlineData("f^(c/(a + b*x)^2)*x", 1.1)]
[InlineData("f^(m + k/(c + d*x))/(g + h*x)^2", 1.1)]
[InlineData("e^(k/(c + d*x))*(a + b*x)^2", 1.1)]
[InlineData("e^(k/(c + d*x))/(a + b*x)", 1.1)]
[InlineData("e^(k/(c + d*x))/(a + b*x)", -0.9)]
[InlineData("e^(k/(c + d*x)^2)*(a + b*x)", -0.9)]
[InlineData("f^(m + k*(a + b*x)/(c + d*x))/(g + h*x)", 1.1)]
[InlineData("e^(1/x)/(x + 1)", 1.1)]
public void WrittenInTheReciprocal(string integrand, double k)
{
var integral = integrand.ToEntity().Integrate("x");
var text = integral.Stringize();
Assert.DoesNotContain("integral(", text);
Assert.True(text.Length < 5000, $"{text.Length} characters of answer for {integrand}");
Entity Pinned(Entity e) => e.Substitute("a", 1.3).Substitute("b", 0.7).Substitute("c", 0.4).Substitute("d", 0.3)
.Substitute("f", 2.3).Substitute("g", 0.6).Substitute("h", 1.9).Substitute("m", 0.5).Substitute("k", k);
var derivative = Pinned(integral.Substitute("C", 0)).Differentiate("x");
var original = Pinned(integrand.ToEntity());
var compared = 0;
foreach (var at in new[] { -0.9, -0.4, 0.3, 0.8, 1.3, 2.9 })
{
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}, where the integrand is {want}");
}
Assert.True(compared >= 5, $"only {compared} points could be compared for {integrand}");
}
}
}
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