This question already has an answer here:
Because the class of function you consider easy to differentiate—namely, the elementary functions—is not mapped onto itself. You are accustomed to thinking of elementary functions as being exactly those on which you “do calculus” because they are easy to differentiate, but a randomly chosen one may not be in the image of the derivative function, and therefore, not easy to integrate. The standard example $f(x) = e^{x^2}$ applies here; no simple example of a hard-to-differentiate function exists, by definition.
Note that without the assurance that functions are given by explicit formulas of a convenient form, neither integration nor differentiation is particularly easy.
For special types of functions there are pretty clear cut methods for integrating (e.g. polynomials). Unfortunately, there is no general algorithm for creating integrals as there are functions that have no antiderivative. The classic example being $\ f(x)=e^{x^2}$.
There is an algorithm for computing indefinite integrals in terms of elementary functions, when such representations exist: Risch’s algorithm . It is a very complex algorithm which is now included in the major symbolic mathematics programs.
There are clear steps to follow when we want to compute the derivative of a function, only for a narrow class of functions. In general no algorithm is known for differentiation. The same holds for integrals.
EDIT: If in the search for further information, turn to the link, kindly suggested by Chris Janjigian.
In order to answer your question, one has to clarify it. You presumably mean
integrating explicitly elementary functions in terms of further elementary functions. Of course, a complete answer would required a precise definition of
this term but we can regard it informally as meaning one which is obtained from
the usual suspects (powers, trigonometric, exponential, logarithm, etc.) by simple operations (arithmetic, composition, etc.). The difference is then that while there are simple rules which describe what happens when differentiating,
there are no such ones for integration. For example, integration by parts merely
shifts the problem to a new integral—this might work in a particular case but not in general. You can see this effect when you use Mathematica which cannot integrate all elementary functions explicitly. By the way there is, rather surprisingly, a general theory which covers this topic and this is used in the algorithm employed by Mathematica.