/**
 * Symbolic indefinite integral of `expr` with respect to `variable` (default `'x'`),
 * returned as an expression string (without the constant of integration). If the
 * integrand is outside the supported subset, returns `integral(expr, variable)`.
 *
 * The direct recursion is tried first; on failure, partial-fraction integration
 * (for distinct-rational-root integrands), tabular integration by parts (for
 * polynomial·{exp,sin,cos}), and full rational-function integration (Risch
 * Layer 1) are attempted before giving up with the marker. Rational functions
 * with irreducible-quadratic, repeated, and degree-≥3 irreducible factors
 * are integrated (Layers 1–3: rational part + `log` + `arctan` / residues).
 * Nested transcendental towers that are not elementary still return the marker.
 *
 * @example symbolicIntegral('x^3')          // 'x^4 / 4'
 * @example symbolicIntegral('cos(3*x + 1)') // 'sin(3 * x + 1) / 3'
 * @example symbolicIntegral('1/(x^2 - 1)')  // partial fractions → sum of logs
 * @example symbolicIntegral('1/(x^2 + 1)')  // Risch Layer 1 → 'atan(...)'
 * @example symbolicIntegral('x * sin(x)')   // by parts → 'sin(x) - x*cos(x)'
 */
export declare function symbolicIntegral(expr: string, variable?: string): string;
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