Evaluation of definite integrals via the method of differentiation under the integral sign has this character. This probably needs no introduction, but the idea is to view the integrand $f(x)$ of $\int_a^b f(x)\; dx$ as belonging to a (judiciously chosen) parametrized family of functions $F(x, t)$, say at $t = t_0$. (So the parameter $t$ is the 18th camel.) Under mild hypotheses the function

$$G(t) = \int_a^b F(x, t)\; dx$$

has derivative

$$G'(t) = \int_a^b \frac{\partial}{\partial t} F(x, t)\; dx$$

and in many cases this will be simple enough to allow one to solve for $G(t)$. Then the original integral is $G(t_0)$. The Wikipedia article gives some examples of this technique (made popular by Feynman through his book "Surely You're Joking, Mr. Feynman!").

My impression is that the method of creative telescoping is similar in nature; perhaps others more knowledgeable about this can weigh in.