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Rafael-SOWNet merged 1 commit into from
Sep 24, 2026
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A perfect square in a power of the variable is read as one #1491

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19 changes: 19 additions & 0 deletions BREAKING-CHANGES.md
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Expand Up @@ -1444,6 +1444,25 @@ multiple of it, `(d - c^2 d x^2)^(3/2)`, is written over it the same way, with
Rubi's 5.1.4, 5.1.5, 5.2.4 and 5.2.5, all 504 problems that count: 405 to 463, no row lost, 77
timeouts to 19.

### A perfect square in a power of the variable is read as one

`sqrt(a^2 + 2 a b x^2 + b^2 x^4) sqrt(c + e x + d x^2)` was left unevaluated. The radicand is
`(a + b x^2)^2`, so its root is the modulus `|a + b x^2|`, but the rule that reads a root of a
perfect square read only a quadratic in `x`. It now reads the same quadratic in a power of `x`,
`a x^(2k) + b x^k + c`, and puts `sgn(x^k + h)` in front of the integral, as it does for `k = 1`.
There is no sign where `k` is even and `h` is a positive number, since `x^k + h` is then
positive. The shift `h` is simplified where it holds a symbol: `2 a b/(2 b^2)` is `a/b`, for
`k = 1` as well.

| Input | Was (2.5.0) | Now |
|---|---|---|
| `"sqrt(c+pe*x+d*x^2)*sqrt(a^2+2*a*b*x^2+b^2*x^4)".Integrate("x")` | left unevaluated | the antiderivative, with `sgn(x^2 + a/b) sqrt(b^2)` in front |
| `"x/sqrt(a^2+2*a*b*x^3+b^2*x^6)".Integrate("x")` | left unevaluated | logarithms and an arctangent in `(a/b)^(1/3)`, with `sgn(x^3 + a/b)` in front |
| `"(1+2*x^2+x^4)^(3/2)".Integrate("x")` | left unevaluated | `x + 3 x^3/3 + 3 x^5/5 + x^7/7`, with no sign |

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: 182 to 388, no row lost, 19 timeouts to 9.

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

**Addition, not silent.** The binomial coefficient is a node, `Entity.Binomialf`, spelled
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35 changes: 26 additions & 9 deletions Sources/AngouriMath/Functions/Continuous/Integration/IndefiniteIntegralSolver.cs
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Expand Up @@ -10444,7 +10444,10 @@ private static bool IsAPerfectSquareDiscriminant(Entity a, Entity b, Entity c)
/// A square root of a perfect square in <paramref name="x"/> is the modulus:
/// <c>sqrt(x^2)</c> is <c>|x|</c>, which for a real <c>x</c> is <c>sgn(x) x</c>, and
/// <c>sqrt(a (x + h)^2)</c> is <c>sqrt(a) sgn(x + h) (x + h)</c> for a positive number
/// <c>a</c>. Every rule that reads a root of a quadratic assumes it is not one:
/// <c>a</c>, and a square in a power of <c>x</c> the same way:
/// <c>sqrt(a^2 + 2 a b x^2 + b^2 x^4)</c> is <c>sqrt(b^2) sgn(x^2 + a/b) (x^2 + a/b)</c>,
/// the square Rubi's 1.2.2 and 1.2.3 write. Every rule that reads a root of a quadratic
/// assumes it is not one:
/// <c>1/sqrt(1 + csch(x)^2)</c> under <c>u = tanh(x)</c> is <c>sqrt(u^2)/(u^2 - 1)</c>,
/// and the table rule for <c>sqrt(a w^2 + b w + c)</c> beside a linear, reached after the
/// partial fractions and a reciprocal, answered it with a logarithm of zero.
Expand All @@ -10467,7 +10470,8 @@ private static bool IsAPerfectSquareDiscriminant(Entity a, Entity b, Entity c)
internal static Entity? SolveByTakingARootOfAPerfectSquare(Entity expr, Entity.Variable x, bool integrateByParts)
{
// Asked of every integrand at every depth, so the radicand is read as a polynomial
// only where it is small enough to be a written square: x^2, or a x^2 + b x + c.
// only where it is small enough to be a written square: x^2, a x^2 + b x + c, or
// the same quadratic in a power of x, a x^(2k) + b x^k + c.
// At the top, any half-odd power of a perfect square is the power of the modulus;
// below it the square root only, since a sign written for a substitution's variable
// is a factor the rest of the search carries -- `(1 + 2u + u^2)^(5/2)` under
Expand All @@ -10493,12 +10497,20 @@ bool MayBeARootOfASquare(Entity node, Entity.Variable x)
var r = (Number.Rational)exponent;
if (!TreeAnalyzer.TryGetPolynomial(radicand, x, out var monomials) || monomials.Count == 0)
return node;
// A quadratic in w = x^k: a w^2 + b w + c, the square being of w + h. Beyond
// k = 1 only with the constant term written, since `a x^(2k)` alone is a power
// of a monomial, which another rule distributes.
var degree = monomials.Keys.Max()!;
if (!degree.Equals(EInteger.FromInt32(2)) || monomials.Keys.Any(k => k.Sign < 0))
if (degree.IsZero || !degree.IsEven || monomials.Keys.Any(k => k.Sign < 0))
return node;
var a = monomials.TryGetValue(EInteger.FromInt32(2), out var a2) ? a2 : Number.Integer.Zero;
var b = monomials.TryGetValue(EInteger.One, out var b1) ? b1 : Number.Integer.Zero;
var half = degree / EInteger.FromInt32(2);
if (monomials.Keys.Any(k => !k.IsZero && !k.Equals(half) && !k.Equals(degree))
|| !half.Equals(EInteger.One) && !monomials.ContainsKey(EInteger.Zero))
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;
Entity w = half.Equals(EInteger.One) ? x : MathS.Pow(x, Number.Integer.Create(half));
// A positive leading coefficient, since `sqrt(a (x + h)^2)` is `sqrt(a) |x + h|`:
// a positive number, or one positive for a real parameter -- `b^2` is, and
// `a^2 + 2 a b x + b^2 x^2` is the square Rubi writes -- whose condition travels
Expand All @@ -10517,17 +10529,22 @@ bool MayBeARootOfASquare(Entity node, Entity.Variable x)
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;
// a (x + h)^2 with h = b/(2a): (a (x + h)^2)^r is a^r |x + h|^(2r), and with 2r odd
// (a rational is held in lowest terms) that is a^r sgn(x + h) (x + h)^(2r).
// a (w + h)^2 with h = b/(2a): (a (w + h)^2)^r is a^r |w + h|^(2r), and with 2r odd
// (a rational is held in lowest terms) that is a^r sgn(w + h) (w + h)^(2r).
var h = (b / (Number.Integer.Create(2) * a)).InnerSimplified;
var linear = h == Number.Integer.Zero ? x : (x + h).InnerSimplified;
// `2 a b/(2 b^2)` is `a/b`, which the normalisation does not cancel.
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);
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.
signs = signs * MathS.Signum(linear);
// 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 }))
signs = signs * MathS.Signum(linear);
changed = true;
return leading == Number.Integer.One ? power : MathS.Pow(leading, r) * power;
});
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36 changes: 36 additions & 0 deletions Sources/Tests/UnitTests/Calculus/RootOfAPerfectSquareIntegralTest.cs
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Expand Up @@ -92,6 +92,42 @@ 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>
/// The same square in a power of the variable: <c>a^2 + 2 a b x^2 + b^2 x^4</c> is
/// <c>(a + b x^2)^2</c>, and its root is <c>sqrt(b^2) sgn(x^2 + a/b) (x^2 + a/b)</c>. Pinned
/// with <c>a</c> and <c>b</c> of opposite signs, so that the sign changes among the points.
/// Rubi's 1.2.2.7 and 1.2.3.2.
/// <a href="https://github.com/asc-community/AngouriMath/issues/718">#718</a>
/// </summary>
[Theory]
[InlineData("sqrt(c + pe*x + d*x^2)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)")]
[InlineData("x*sqrt(c + pe*x + d*x^2)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)")]
[InlineData("x/sqrt(a^2 + 2*a*b*x^3 + b^2*x^6)")]
public void ASquareInAPowerOfTheVariable(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).Substitute("c", 2.1)
.Substitute("pe", 0.3).Substitute("d", 1.3);
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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