symbolic integration
There are two halves of symbolic calculus, and they are dramatically unequal. Symbolic differentiation — finding the exact derivative of a formula — is mechanical and easy: a short set of rules (product, quotient, chain) applied to the expression tree turns sin(x^2) into 2x cos(x^2) every time, and it is the seed from which automatic differentiation grew. Symbolic integration — finding an exact antiderivative, a formula whose derivative is the function you started with — is the hard, deep, and sometimes impossible sibling.
Differentiation is so reliable because every rule shrinks the problem toward known derivatives of elementary pieces; a computer-algebra system simply walks the tree and assembles the answer. Integration has no such guaranteed march. The classic approach mixes a table of known integrals, substitution, integration by parts, and partial-fraction decomposition (which is why the polynomial factorization of denominators matters); the modern, complete approach for elementary functions is the Risch algorithm, a decision procedure that either produces an elementary antiderivative or proves that none exists. The most jarring fact for a newcomer is that proof of non-existence: many perfectly innocent-looking functions simply have no antiderivative expressible in elementary terms.
Symbolic integration matters because a closed-form integral can be analyzed, differentiated back to check, evaluated to arbitrary precision, and reused — far more than a single number. But the honest reality is stark. Differentiation always succeeds; integration may legitimately answer 'no elementary antiderivative exists', as it does for exp(-x^2), the bell curve whose integral underlies all of statistics, or for sin(x)/x. When that happens you fall back to numerical quadrature for a number, or to special functions (the error function erf, the sine integral Si) that name the otherwise-unwritable answer. Symbolic and numerical integration are partners, not rivals.
The integral of 1/(1 + x^2) is arctan(x) — a clean elementary answer a CAS returns instantly. But the integral of exp(-x^2) has no elementary antiderivative at all; the best a CAS can do is name it, returning (sqrt(pi)/2) erf(x), where erf is the special error function defined precisely as that integral.
Some elementary functions have no elementary antiderivative — only a named special function.
Differentiation is always possible by a fixed set of rules; integration is not. The Risch algorithm can prove that an elementary antiderivative does not exist — so a 'no answer' from a CAS may be a theorem, not a failure. When that happens, switch to numerical quadrature or accept a special function.