Maxima provides a compact set of commands for rewriting algebraic expressions into cleaner equivalent forms, including rational simplification, polynomial factorization/expansion, fraction manipulation, and common trigonometric or exponential rewrites.

Rational and Radical Simplification

For rational expressions, ratsimp(expr) performs standard rational simplification, while fullratsimp(expr) applies a more aggressive form of rational simplification when intermediate cancellations are nontrivial. For radicals, radcan(expr) attempts canonical simplification by reducing and combining radical components.

  • ratsimp(expr); simplifies rational expressions.
  • fullratsimp(expr); performs deeper rational simplification.
  • radcan(expr); simplifies expressions involving radicals.
ratsimp((x^2 - 1)/(x - 1));
fullratsimp((x^2 + 2*x + 1)/(x + 1));
radcan(sqrt(8) + sqrt(18));

Polynomial Factorization, Expansion, and Term Collection

Polynomial structure is often made explicit by either factoring or expanding. When expressions involve multiple parameters and powers of a variable, collectterms(expr, x) organizes the result as a polynomial in $x$, which is useful for comparison, coefficient extraction, and symbolic matching.

  • factor(expr); factors an algebraic expression.
  • expand(expr); expands products and powers.
  • collectterms(expr, x); collects terms in powers of $x$.
factor(x^2 - 5*x + 6);
expand((x + 1)^3);
collectterms(a*x^2 + b*x + c + x^2, x);
factor(x^3 - 3*x^2 - 4*x + 12);
expand((x - 2)*(x + 2)*(x + 1));

Fraction Restructuring and Partial Fractions

When an expression is presented as a sum of rational terms, combine(expr) merges it into a single rational form. For integration and decomposition tasks, apart(expr, x) produces a partial fraction decomposition with respect to $x$. The functions num(expr) and denom(expr) extract the numerator and denominator of a rational expression, supporting algebraic inspection and further transformations.

  • combine(expr); combines terms into a single rational expression.
  • apart(expr, x); gives partial fraction decomposition with respect to $x$.
  • num(expr); extracts the numerator.
  • denom(expr); extracts the denominator.
combine(1/x + 1/(x+1));
apart((2*x+3)/(x^2+x), x);
num((x^2 - 1)/(x + 1));
denom((x^2 - 1)/(x + 1));

Trigonometric and Exponential Rewrites

Maxima can simplify standard trigonometric identities and switch between trigonometric and exponential representations. trigsimp(expr) reduces trigonometric expressions using identities (e.g., $\sin^2 x + \cos^2 x = 1$). exponentialize(expr) rewrites trigonometric functions in exponential form, while demoivre(expr) applies De Moivre-style rewriting for complex exponentials.

  • trigsimp(expr); simplifies trigonometric expressions.
  • exponentialize(expr); converts trig functions to exponential form.
  • demoivre(expr); rewrites complex exponentials using trigonometric form.
trigsimp(sin(x)^2 + cos(x)^2);
exponentialize(sin(x));
demoivre(exp(%i*x));

kill(all)$
expr1 : (x^2 - 1)/(x - 1)$
expr2 : (x + 1)^3$
expr3 : x^2 - 5*x + 6$
expr4 : 1/x + 1/(x+1)$

ratsimp(expr1);
expand(expr2);
factor(expr3);
combine(expr4);