Function xPowerExponent

src/compute-engine/rubi/rubi-utils.ts:3776–3791

The exponent k of a single `d·xᵏ` term (d x-free): bare `x`→1, `x^k`→k, * `d·x`/`d·x^k`→1/k. Null if `t` is not one x-free-scaled power of x.

(t: Expression, x: string)

  u: Expression,
  x: string
): Expression {
  const st: FoeState = { base: null, expon: null, flag: false };
  foeTest(u, x, st);
  if (st.base === null) return u;
  return foeFunctionAux(u, x, st);
}

/** If `u` is PURELY a function of a single exponential F^v with v linear in x
 *  (Rubi's `FunctionOfExponentialTest` returning true), return `{ v: F^v,
 *  g: u with F^v → x }` for the rule-2.3#97 substitution; otherwise null.
 *
 *  Unlike `functionOfExponential`, this REQUIRES the test to pass — so it
 *  rejects integrands with a bare-x factor (e.g. `Tanh[x]/x²`) or a non-linear
 *  hyperbolic argument (e.g. `Sech[c+d·x²]`), where the substitution would be
 *  invalid. It also drops Rubi's `$exponFlag$` gate (which demands an explicit
 *  exponential): the Chapter-6 fallback applies this to pure hyperbolics, whose
 *  bare-power reductions are not standalone corpus rules. Both v and g come from
 *  the SAME registered base/exponent, so they are consistent. */
export function functionOfExponentialSubstitution(