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Rafael-SOWNet merged 1 commit into from
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Integrate a symbolic secant power, a half-odd power of a ± a sin, and an even cosine power #1975

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24 changes: 24 additions & 0 deletions BREAKING-CHANGES.md
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Expand Up @@ -350,6 +350,30 @@ cache is now keyed on the downcasting setting too
|---|---|---|
| `MathS.FromString("1/3").Evaled` under downcasting off, after the same parse with it on | `1/3` | `0.3333…` (one hundred digits) |

### A symbolic power of the secant, a half-odd power of `a ± a sin` beside other linears, and an even power of the cosine over a sine's binomial are integrated

**Answers where there were none.** `(b sec(g + f x))^m tan(g + f x)^3` and three more of Rubi's 4.3.0 were declined only
because `m` was not a number: in `u = sec(y)` it is `(b u)^m (u^2 - 1)/u`, and the rule for an odd power of the tangent
beside a function of the secant now reads a power free of x whatever its exponent.
`1/(sqrt(a + a sin(y)) (c + d sin(y))^2)`, `1/((a + a sin(y))^2 sqrt(c - c sin(y)))` and fourteen more of 4.1.2.1 and
4.1.3.1 are answered by the half angle at which `a + a sin(y)` is a square, which they never reached: the polynomial
term wrote the reciprocal as a power and asked the substitution search, which spent the budget. A half-odd power of
`a ± a sin(y)` beside at least one other linear in the same sine or cosine is asked it first now. And
`cos(y)^4 sin(y)^3/(a + b sin(y))^2` and six more of 4.1.2.2 are read by the rule below, an even whole power of the
cosine being a power of `1 - sin(y)^2` exactly, where a power of the sine stands above the bar beside it. That file's
neighbours the unreleased master answered are respelled, some shorter and some longer ([#233](https://github.com/asc-community/AngouriMath/issues/233)).

| Input | Was (2.5.0) | Now |
|---|---|---|
| `"(b*sec(g+f*x))^m*tan(g+f*x)^3".ToEntity().Integrate("x")` | `integral(...)` | 144 characters |
| `"cot(g+f*x)^5*(b*csc(g+f*x))^m".ToEntity().Integrate("x")` | `integral(...)` | 221 characters |
| `"1/(sqrt(a+a*sin(g+f*x))*(c+d*sin(g+f*x))^2)".ToEntity().Integrate("x")` | `integral(...)` | 2,679 characters |
| `"1/((a+a*sin(g+f*x))^2*sqrt(c-c*sin(g+f*x)))".ToEntity().Integrate("x")` | `integral(...)` | 329 characters |
| `"cos(c+d*x)^4*sin(c+d*x)^3/(a+b*sin(c+d*x))^2".ToEntity().Integrate("x")` | `integral(...)` | 4,174 characters |
| `"cos(c+d*x)^6*sin(c+d*x)^3/(a+b*sin(c+d*x))^3".ToEntity().Integrate("x")` | `integral(...)` | 9,029 characters |
| `"cos(c+d*x)^4*sin(c+d*x)^2/(a+b*sin(c+d*x))^2".ToEntity().Integrate("x")` | `integral(...)`; 7,068 characters, in 1.3 s, on the unreleased master | 3,677 characters, in 154 ms |
| `"cos(c+d*x)^2*sin(c+d*x)^2/(a+b*sin(c+d*x))^3".ToEntity().Integrate("x")` | `integral(...)`; 4,621 characters, in 431 ms, on the unreleased master | 6,584 characters, in 570 ms |

### A polynomial in a sine over a power of a binomial in it is written in powers of the binomial

**Answers where there were none.** `(A + B cos(c + d x) + C cos(c + d x)^2)/(a + b cos(c + d x))^4` ran out of time, and
Expand Down
190 changes: 179 additions & 11 deletions Sources/AngouriMath/Functions/Continuous/Integration/IndefiniteIntegralSolver.cs
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Expand Up @@ -5364,7 +5364,8 @@ Entity Integral(ERational power)
/// </para>
/// https://github.com/asc-community/AngouriMath/issues/718
/// </remarks>
internal static Entity? SolveByTheHalfAngleWhereOnePlusASineIsASquare(Entity expr, Entity.Variable x, bool integrateByParts)
internal static Entity? SolveByTheHalfAngleWhereOnePlusASineIsASquare(Entity expr, Entity.Variable x, bool integrateByParts,
bool collectTheLinears = false)
{
if (!Integration.AnsweringTheQuestionAsked)
return null;
Expand Down Expand Up @@ -5445,6 +5446,41 @@ Entity Integral(ERational power)
constant = MathS.Pow(a, exponent) * constant;
return constant * MathS.Pow(square, Number.Integer.Create(twoP));
});
// `c + d sin(y)` as `(c - d) + 2d sin(u)^2`, collected, where asked: written
// `c + d (2 sin(u)^2 - 1)`, `sin(u)^5/(c + d (2 sin(u)^2 - 1))^2` ran past the budget.
if (collectTheLinears)
{
var (function, square) = sineKind ? (sine, MathS.Sqr(sinU)) : (cosine, MathS.Sqr(cosU));
rewritten = rewritten.Replace(node =>
{
if (node is not Sumf || !node.ContainsNode(function))
return node;
Entity constant = Number.Integer.Zero, coefficient = Number.Integer.Zero;
foreach (var term in Sumf.LinearChildren(node))
{
if (!term.ContainsNode(x))
{
constant += term;
continue;
}
Entity factor = Number.Integer.One;
var seen = false;
foreach (var part in Mulf.LinearChildren(term))
if (part == function && !seen)
seen = true;
else if (part.ContainsNode(x))
return node;
else
factor *= part;
if (!seen)
return node;
coefficient += factor;
}
var shifted = (constant - coefficient).InnerSimplified;
var collected = (2 * coefficient).InnerSimplified * square;
return VanishesIdentically(shifted) ? collected : shifted + collected;
});
}
var inU = rewritten.Substitute(sine, sineInU).Substitute(cosine, cosineInU);
// What is left of the variable is the variable itself: the angle is `rate x + offset`
// and `u` is half of it (a quarter turn on for the sine), so `x` is
Expand All @@ -5468,6 +5504,82 @@ Entity Integral(ERational power)
return answer.Nodes.Any(node => node == MathS.NaN) ? null : answer;
}

/// <summary>
/// A half-odd power of <c>a ± a sin(y)</c> times whole or half-odd powers of other linears
/// in the same sine, <c>c + d sin(y)</c>, and nothing else of <c>x</c>: by
/// <see cref="SolveByTheHalfAngleWhereOnePlusASineIsASquare"/>, asked first. The cosine's
/// the same.
/// </summary>
/// <remarks>
/// <para>
/// Rubi's <c>1/(sqrt(a + a sin(e + f x)) (c + d sin(e + f x))^2)</c>,
/// <c>1/((a + a sin(e + f x))^2 sqrt(c - c sin(e + f x)))</c> and their kin in 4.1.2.1
/// ran past the budget. The half angle answers each, its question in <c>u</c> taking
/// tens of milliseconds, but it was not reached: the polynomial term above it wrote the
/// reciprocal as a power, took the constant out of the root, and asked the substitution
/// search below that, whose simplification of what it found took the whole budget.
/// </para>
/// <para>
/// The answer is that rule's, so exact on the same terms. Read narrowly: a product of
/// powers of linears in one sine or one cosine, one of them half-odd and of
/// <c>a ± a sin(y)</c>, and at least two of them, since one alone is
/// <see cref="SolveAHalfPowerOfOnePlusASine"/>'s.
/// https://github.com/asc-community/AngouriMath/issues/233
/// </para>
/// </remarks>
internal static Entity? SolveAHalfOddPowerOfOnePlusMinusASineBesidePowersOfLinearsInIt(Entity expr, Entity.Variable x, bool integrateByParts)
{
if (!Integration.AnsweringTheQuestionAsked)
return null;
if (!TryReadAsQuotient(expr, out var above, out var below))
(above, below) = (expr, Number.Integer.One);
Entity? function = null;
var (factors, halfOdd) = (0, false);
foreach (var side in new[] { above, below })
foreach (var factor in Mulf.LinearChildren(side))
{
if (!factor.ContainsNode(x))
continue;
var (@base, exponent) = factor is Powf(var inner, Number.Rational power) ? (inner, power.ERational) : (factor, ERational.One);
var isHalfOdd = exponent.Denominator.Equals(EInteger.FromInt32(2));
if (!isHalfOdd && !exponent.Denominator.Equals(EInteger.One))
return null;
// c + d f(y), the one f throughout, and c there: a power of f itself is not this shape.
Entity? found = null;
var constant = false;
foreach (var term in Sumf.LinearChildren(@base))
{
if (!term.ContainsNode(x))
{
constant = true;
continue;
}
foreach (var part in Mulf.LinearChildren(term))
if (part.ContainsNode(x))
{
if (found is not null || part is not (Sinf or Cosf))
return null;
found = part;
}
}
if (found is null || !constant || function is not null && found != function)
return null;
function = found;
if (isHalfOdd)
{
if (ReadAsOnePlusMinusAFunction(@base, function is Sinf ? function : MathS.Sin(function.DirectChildren.First()),
function is Cosf ? function : MathS.Cos(function.DirectChildren.First())) is null)
return null;
halfOdd = true;
}
factors++;
}
if (!halfOdd || factors < 2 || function is null
|| !TreeAnalyzer.TryGetPolyLinear(function.DirectChildren.First(), x, out var rate, out _) || rate.ContainsNode(x))
return null;
return SolveByTheHalfAngleWhereOnePlusASineIsASquare(expr, x, integrateByParts, collectTheLinears: true);
}

/// <summary>
/// Powers of <c>a ± a sin(y)</c> with a symbolic exponent, or beside a fractional power of
/// <c>g cos(y)</c>, and anything else in the sine alone, under <c>u = sin(y)</c>:
Expand Down Expand Up @@ -6154,8 +6266,12 @@ static bool IsHalfOdd(Entity exponent) => exponent is Number.Rational rational a
/// <remarks>
/// Rubi's <c>cot(c + d x)/(a + b sec(c + d x))^(3/2)</c> was declined and
/// <c>cot(c + d x)^3 sqrt(a + b sec(c + d x))</c> ran past thirty seconds, where in the
/// secant each is a rational function beside the root of a linear.
/// secant each is a rational function beside the root of a linear. A power free of
/// <c>x</c> is read whatever its exponent: Rubi's <c>(b sec(e + f x))^m tan(e + f x)^3</c>,
/// 4.3.0, was declined only because <c>m</c> was not a number, where in <c>u</c> it is
/// <c>(b u)^m (u^2 - 1)/u</c>.
/// https://github.com/asc-community/AngouriMath/issues/718
/// https://github.com/asc-community/AngouriMath/issues/233
/// </remarks>
internal static Entity? SolveAnOddPowerOfTheTangentBesideAFunctionOfTheSecant(Entity expr, Entity.Variable x, bool integrateByParts)
{
Expand All @@ -6174,10 +6290,24 @@ static bool IsHalfOdd(Entity exponent) => exponent is Number.Rational rational a
if (argument is null || !TreeAnalyzer.TryGetPolyLinear(argument, x, out var rate, out _)
|| rate.ContainsNode(x) || TreeAnalyzer.IsZero(rate))
return null;
if (!expr.Nodes.All(node => !node.ContainsNode(x)
|| node is Variable or Sumf or Minusf or Mulf or Divf or Sinf or Cosf or Secantf or Cosecantf or Tanf or Cotanf
|| node is Powf(_, Number.Rational)))
return null;
// A symbolic power too, of a product of constants and secants or cosecants:
// `(b sec(y))^m` is `(b u)^m`. Of a sum it brings `(a + b u)^m/u`, not elementary.
var z = Variable.CreateUnique(expr, "z_secant");
bool IsAProductOfSecants(Entity @base)
{
var inZ = @base.Replace(node => node is Secantf(var a) && a == argument || node is Cosecantf(var b) && b == argument ? z : node);
return !inZ.ContainsNode(x) && inZ.Nodes.All(node => !node.ContainsNode(z) || node == z || node is Mulf or Divf or Powf(_, Number.Rational));
}
var symbolic = false;
foreach (var node in expr.Nodes)
{
if (!node.ContainsNode(x) || node is Variable or Sumf or Minusf or Mulf or Divf or Sinf or Cosf or Secantf or Cosecantf or Tanf or Cotanf
|| node is Powf(_, Number.Rational))
continue;
if (node is not Powf(var @base, var exponent) || exponent.ContainsNode(x) || !IsAProductOfSecants(@base))
return null;
symbolic = true;
}
var t = Variable.CreateUnique(expr, "t_odd_tangent");
var u = Variable.CreateUnique(expr, "u_secant");
foreach (var ofTheSecant in new[] { true, false })
Expand All @@ -6193,6 +6323,9 @@ static bool IsHalfOdd(Entity exponent) => exponent is Number.Rational rational a
var n = exponents[0];
if (rest.ContainsNode(t) || !n.IsInteger() || n.ToEInteger().IsEven || !n.ToEInteger().CanFitInInt32())
continue;
// Beside a symbolic power a negative one leaves `u^m/(u (u^2 - 1)^k)`, hypergeometric.
if (symbolic && n.Sign < 0)
continue;
var inU = rest.Replace(node => node switch
{
Secantf(var a) when ofTheSecant && a == argument => u,
Expand Down Expand Up @@ -27949,22 +28082,57 @@ when two.Equals(EInteger.FromInt32(2)) && e1 == MathS.e && e2 == MathS.e
/// The rules after it took the whole quotient through the half-angle and wrote 30 million
/// characters for <c>(A + B cos(x) + C cos(x)^2)/(a + b cos(x))^4</c> in eighteen seconds, where
/// <c>1/(a + b cos(x))^4</c> alone is 44 thousand. Rubi's 4.1.4 and 4.2.4
/// <c>(a + b cos)^m (A + B cos + C cos^2)</c> at <c>m = -4</c> timed out.
/// <c>(a + b cos)^m (A + B cos + C cos^2)</c> at <c>m = -4</c> timed out. An even whole power
/// of the other function above the bar is a power of <c>1 - f^2</c>, exactly, and read as
/// one: Rubi's 4.1.2.2 <c>cos(c + d x)^6 sin(c + d x)^3/(a + b sin(c + d x))^2</c> and its
/// kin timed out too, where in the sine each is the polynomial over the power.
/// https://github.com/asc-community/AngouriMath/issues/718
/// https://github.com/asc-community/AngouriMath/issues/233
/// </remarks>
internal static Entity? SolveAPolynomialInASineOverAPowerOfABinomialInItInPowersOfTheBinomial(Entity expr, Entity.Variable x)
{
if (!Integration.AnsweringTheQuestionAsked)
return null;
var functions = expr.Nodes.Where(node => node is Sinf or Cosf && node.ContainsNode(x)).Distinct().ToList();
if (functions.Count != 1)
if (functions.Count is not (1 or 2))
return null;
var s = Variable.CreateUnique(expr, "s_pow");
Entity? function = null, inS = null;
foreach (var candidate in functions)
{
// The other of the two only to even whole powers above the bar, each a power of
// 1 - f^2: `cos(y)^4 sin(y)^3/(a + b sin(y))^2` is a polynomial in the sine over the power.
var written = candidate;
if (functions.Count == 2)
{
var other = functions.Single(f => f != candidate);
if (candidate is Sinf && other is not Cosf || candidate is Cosf && other is not Sinf
|| other.DirectChildren.First() != candidate.DirectChildren.First())
return null;
// And only beside the function itself above the bar: `cos(y)^4/(a + b sin(y))^8`, with
// no sine above it, is answered by the rules after it in 40 thousand characters, and
// was 106 thousand here, in 60% more time.
var (top, _) = Functions.SingleQuotient.Of(expr);
if (!top.ContainsNode(candidate))
continue;
written = expr.Replace(node => node is Powf(var @base, Number.Integer { EInteger: var power }) && @base == other
&& power.Sign > 0 && power.IsEven && power.CompareTo(EInteger.FromInt32(12)) <= 0
? MathS.Pow(1 - MathS.Sqr(candidate), Number.Integer.Create(power / 2))
: node);
if (written.ContainsNode(other))
continue;
}
else
written = expr;
function = candidate;
inS = written.Substitute(candidate, s);
break;
}
if (function is null || inS is null)
return null;
var function = functions[0];
var argument = function switch { Sinf(var y) => y, Cosf(var y) => y, _ => null! };
if (!TreeAnalyzer.TryGetPolyLinear(argument, x, out var slope, out _) || Functions.PartialFractions.IsZeroAsAValue(slope))
return null;
var s = Variable.CreateUnique(expr, "s_pow");
var inS = expr.Substitute(function, s);
if (inS.ContainsNode(x))
return null;
var (above, below) = Functions.SingleQuotient.Of(inS);
Expand Down
3 changes: 3 additions & 0 deletions Sources/AngouriMath/Functions/Continuous/Integration/Integration.Definition.cs
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Expand Up @@ -772,6 +772,9 @@ private static Entity WithItsPiecewiseSumsCombined(Entity expr)
// A polynomial beside error functions of one shifted linear, written in it: by parts in
// x leaves a Gaussian of the shifted linear beside a polynomial, which nothing reads.
if ((answer = IndefiniteIntegralSolver.SolveAPolynomialBesideErrorFunctionsOfAShiftedLinear(expr, x, integrateByParts)) is { }) return answer;
// A half-odd power of a ± a sin(y) beside powers of other linears in the sine, by the half
// angle, before the polynomial term: beneath it the substitution search ran out of time.
if ((answer = IndefiniteIntegralSolver.SolveAHalfOddPowerOfOnePlusMinusASineBesidePowersOfLinearsInIt(expr, x, integrateByParts)) is { }) return answer;
if ((answer = IndefiniteIntegralSolver.SolveAsPolynomialTerm(expr, x, integrateByParts)) is { }) return answer;
// A product of powers of the variable with a power of a constant multiple of it
// among them, `(c x)^m x^n`, by the power rule with the written power kept as
Expand Down
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