An expression is expressible in a closed form, if it can be converted (simplified) into an expression containing only elementary functions, combined by a finite amount of rational operations and compositions. Thus, such a closed form must not contain e.g. limit signs, integral signs, sum signs and ``...''.

For example, $$\int\!\!\frac{dx}{x^4\!+\!1},$$ may be expressed in the closed form $$\frac{1}{4\sqrt{2}}\ln\frac{x^2\!+\!x\sqrt{2}\!+\!1}{x^2\!-\!x\sqrt{2}\!+\!1}+ \frac{1}{2\sqrt{2}}\arctan\frac{x\sqrt{2}}{1\!-\!x^2}+C$$ but for $$\int\!\!\frac{dx}{\sqrt{x^4\!+\!1}}\,dx,$$ there exists no closed form.

In certain contexts, the scope of the ``elementary functions'' may be enlarged by allowing in it some other functions, e.g. the error function.