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Rafael-SOWNet merged 1 commit into from
Sep 26, 2026
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A whole power of a perfect square in a power of the variable is written as a power of its root #1494

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18 changes: 18 additions & 0 deletions BREAKING-CHANGES.md
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Expand Up @@ -1483,6 +1483,24 @@ Rubi's 1.2.1.4, all 632 problems that count: 250 to 409, no row lost, 61 timeout
new timeouts have a radicand of four symbols beside a power of `x`, and went from a quick decline
to one past the budget.

### A whole power of a perfect square in a power of the variable is written as a power of its root

`1/(a^2 + 2 a b x^2 + b^2 x^4)^2` was left unevaluated. The quartic is `b^2 (x^2 + a/b)^2`, so a
whole power of it is `(b^2)^n (x^2 + a/b)^(2n)`, exactly and with no sign, and the rational
integrator reads a power of `x^2 + a/b` where it declined the quartic. It is written so at the
question asked, for a square written out as a sum with a symbol in it and `x` below the bar. It
is not written so in a polynomial, or where the square is already written as a power. Answers
that came out in the expanded square before now come out in `x^2 + a/b`.

| Input | Was (2.5.0) | Now |
|---|---|---|
| `"1/(a^2+2*a*b*x^2+b^2*x^4)^2".Integrate("x")` | left unevaluated | the antiderivative, in `x^2 + a/b` |
| `"x/(a^2+2*a*b*x^2+b^2*x^4)^3".Integrate("x")` | a piecewise on `b^2 = 0`, in `2 b^2 x^2 + 2 a b` | `-(x^2 + a/b)^(-5)/(10 (b^2)^3)` |
| `"x^2*(a^2+2*a*b*x^3+b^2*x^6)^(-2)".Integrate("x")` | a piecewise on `b^2 = 0`, in `2 b^2 x^3 + 2 a b` | `-(b^2)^(-2) b^3/(9 (a + b x^3)^3)` |

Rubi's 1.2.2.2, 1.2.2.4, 1.2.2.7 and 1.2.3.2, the 574 problems that count with a perfect square
written with symbols: 388 to 409, no row lost.

### `binomial(n, k)` is a function

**Addition, not silent.** The binomial coefficient is a node, `Entity.Binomialf`, spelled
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47 changes: 33 additions & 14 deletions Sources/AngouriMath/Functions/Continuous/Integration/IndefiniteIntegralSolver.cs
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Expand Up @@ -10602,12 +10602,28 @@ private static bool IsAPerfectSquareDiscriminant(Entity a, Entity b, Entity c)
// `u = e^(2x)` cost twenty seconds that way. A bare `x^2` is the exception at any
// depth, nothing else reading `(x^2)^(3/2)`.
var atTheTop = Integration.AnsweringTheQuestionAsked;
// A whole power too, at the top, of a square with a symbol in it, and in an integrand
// with x below the bar: `(a^2 + 2 a b x^2 + b^2 x^4)^(-2)` is `(b^2)^(-2) (x^2 + a/b)^(-4)`,
// exact and with no sign, and the rational integrator reads a power of `x^2 + a/b`
// where it declined the quartic. Not in a polynomial, which is answered as one already;
// and read off the whole integrand, since `1/P^2` holds `P^2` with a positive exponent.
// And only of a square written out as a sum: `((a + b - b u^2)^2)^(-1)`, what the
// hyperbolic tangent makes of `coth(x)^2/(a + b sech(x)^2)^2`, is a square written as
// one, and taken apart into `(u^2 - a/b - 1)^(-2)` it went from under a second to past
// a minute.
bool? xBelowTheBar = null;
bool XIsBelowTheBar()
=> xBelowTheBar ??= Functions.SingleQuotient.Of(Functions.SingleQuotient.Combine(expr)).Denominator.ContainsNode(x);
bool MayBeARootOfASquare(Entity node, Entity.Variable x)
=> node is Powf(var radicand, Number.Rational r) && r.ERational.Denominator.Equals(EInteger.FromInt32(2))
&& (atTheTop || r.ERational.Equals(ERational.Create(1, 2))
|| radicand is Powf(var @base, var degree) && @base == x && degree == Number.Integer.Create(2))
&& radicand.ContainsNode(x) && radicand.Complexity <= (atTheTop ? 40 : 12)
&& radicand.Nodes.Count(inner => inner == x) <= 2;
=> node is Powf(var radicand, Number.Rational r)
&& (r.ERational.Denominator.Equals(EInteger.FromInt32(2))
&& (atTheTop || r.ERational.Equals(ERational.Create(1, 2))
|| radicand is Powf(var @base, var degree) && @base == x && degree == Number.Integer.Create(2))
&& radicand.Complexity <= (atTheTop ? 40 : 12)
|| atTheTop && r is Number.Integer && r != Number.Integer.Zero && r != Number.Integer.One
&& radicand is Sumf or Minusf
&& radicand.Complexity <= 40 && radicand.Vars.Any(symbol => symbol != x) && XIsBelowTheBar())
&& radicand.ContainsNode(x) && radicand.Nodes.Count(inner => inner == x) <= 2;
if (!expr.Nodes.Any(node => MayBeARootOfASquare(node, x)))
return null;
var changed = false;
Expand All @@ -10631,6 +10647,11 @@ bool MayBeARootOfASquare(Entity node, Entity.Variable x)
if (monomials.Keys.Any(k => !k.IsZero && !k.Equals(half) && !k.Equals(degree))
|| !half.Equals(EInteger.One) && !monomials.ContainsKey(EInteger.Zero))
return node;
// A whole power of a square in x itself is a rational function the splits answer
// already, in the form they give; in a power of x, `x^2 + a/b`, they did not.
var whole = r is Number.Integer;
if (whole && half.Equals(EInteger.One))
return node;
var a = monomials[degree];
var b = monomials.TryGetValue(half, out var b1) ? b1 : Number.Integer.Zero;
var c = monomials.TryGetValue(EInteger.Zero, out var c0) ? c0 : Number.Integer.Zero;
Expand All @@ -10640,16 +10661,14 @@ bool MayBeARootOfASquare(Entity node, Entity.Variable x)
// `a^2 + 2 a b x + b^2 x^2` is the square Rubi writes -- whose condition travels
// with the answer. The other two coefficients are real or symbolic, not a
// number off the line, for the same reason.
Entity leading;
if (a.Evaled is Number.Real { IsPositive: true })
leading = a;
else if (IsPositiveForARealParameter(a, x, out var leadingAssumed, assumed))
// A whole power needs no sign of anything: (a (w + h)^2)^n is a^n (w + h)^(2n).
if (!whole && a.Evaled is not Number.Real { IsPositive: true })
{
leading = a;
if (!IsPositiveForARealParameter(a, x, out var leadingAssumed, assumed))
return node;
assumed = leadingAssumed;
}
else
return node;
var leading = a;
if (b.Evaled is Number.Complex and not Number.Real || c.Evaled is Number.Complex and not Number.Real
|| !IsAPerfectSquareDiscriminant(a, b, c))
return node;
Expand All @@ -10660,14 +10679,14 @@ bool MayBeARootOfASquare(Entity node, Entity.Variable x)
if (h.Vars.Any())
h = Functions.PartialFractions.Bare(h.Simplify());
var linear = h == Number.Integer.Zero ? w : (w + h).InnerSimplified;
var twoR = Number.Integer.Create(r.ERational.Numerator);
var twoR = Number.Integer.Create(whole ? r.ERational.Numerator * EInteger.FromInt32(2) : r.ERational.Numerator);
Entity power = twoR == Number.Integer.One ? linear : MathS.Pow(linear, twoR);
// The sign in front of the integral rather than inside it: it is constant
// between the linear factor's zeros, and a rule below that differentiates the
// integrand cannot evaluate `derivative(sgn(...))` -- which is the exception
// `sqrt(a^2 + 2abx + b^2x^2) sqrt(c + ex + dx^2)` threw with it left in place.
// None where it is plainly one: an even power of x plus a positive number.
if (!(half.IsEven && h.Evaled is Number.Real { IsPositive: true }))
if (!whole && !(half.IsEven && h.Evaled is Number.Real { IsPositive: true }))
signs = signs * MathS.Signum(linear);
changed = true;
return leading == Number.Integer.One ? power : MathS.Pow(leading, r) * power;
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34 changes: 34 additions & 0 deletions Sources/Tests/UnitTests/Calculus/RootOfAPerfectSquareIntegralTest.cs
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Expand Up @@ -128,6 +128,40 @@ Entity Pin(Entity e) => e.Substitute("a", -0.9).Substitute("b", 1.7).Substitute(
Assert.True(compared >= 5, $"only {compared} points were comparable for {integrand}");
}

/// <summary>
/// A whole power of the same square: <c>(a^2 + 2 a b x^2 + b^2 x^4)^(-2)</c> is
/// <c>(b^2)^(-2) (x^2 + a/b)^(-4)</c>, exactly and with no sign, a power of a quadratic in
/// <c>x^2</c> the rational integrator reads. Pinned with <c>a/b</c> positive, so that the
/// square has no real zero among the points.
/// </summary>
[Theory]
[InlineData("1/(a^2 + 2*a*b*x^2 + b^2*x^4)^2")]
[InlineData("x/(a^2 + 2*a*b*x^2 + b^2*x^4)^3")]
[InlineData("x^2*(a^2 + 2*a*b*x^3 + b^2*x^6)^(-2)")]
public void AWholePowerOfASquareInAPowerOfTheVariable(string integrand)
{
var integral = integrand.ToEntity().Integrate("x").Substitute("C", 0);
Assert.DoesNotContain("integral(", integral.Stringize());
Assert.DoesNotContain("NaN", integral.Stringize());
Entity Pin(Entity e) => e.Substitute("a", 0.9).Substitute("b", 1.7);
var derivative = Pin(integral).Differentiate("x");
var original = Pin(integrand.ToEntity());
var compared = 0;
foreach (var at in Points)
{
var got = derivative.Substitute("x", at).EvalNumerical();
var want = original.Substitute("x", at).EvalNumerical();
if (got.IsNaN || want.IsNaN)
continue;
compared++;
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-9,
$"d/dx of the antiderivative of {integrand} is {got} at x = {at}, where the integrand is {want}");
}
Assert.True(compared >= 5, $"only {compared} points were comparable for {integrand}");
}

[Fact]
public void TheSignIsTheSignOfTheLinearFactor()
{
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