Function derivativeDivides

src/compute-engine/rubi/rubi-utils.ts:3084–3110

DerivativeDivides[y, u, x] — see block comment. Returns the x-free quotient * or the `False` symbol.

(y: Expression, u: Expression, ctx: Ctx)

    if (!c.ops || polyDegreeX(c.ops[0], x) !== 1) return null;
    const coef =
      rest.length === 0
        ? t.engine.One
        : rest.length === 1
          ? rest[0]
          : t.engine.function('Multiply', rest);
    return { coef, arg: c.ops[0] };
  }
  return null;
}

/** `a + b·cos[arg]` → `a + b·sin[arg + π/2]` when the base's only x-dependence
 * is a single linear-argument cosine (the standalone-cosine clause); else null. */
function cosBaseToSin(
  ce: ComputeEngine,
  base: Expression,
  x: string
): Expression | null {
  const terms = base.operator === 'Add' && base.ops ? base.ops : [base];
  let a = ce.Zero;
  let cosCoef: Expression | null = null;
  let cosArg: Expression | null = null;
  for (const t of terms) {
    if (!t.has(x)) {
      a = a.add(t);
      continue;
    }
    const parts = cosTermParts(t, x);
    if (parts === null) return null; // an x-dependent non-cosine term ⇒ not standalone
    if (cosArg === null) {
      cosCoef = parts.coef;