Is there a rule of integration that corresponds to the quotient rule? When teaching the integration method of u-substitution, I like to emphasize its connection with the chain rule of integration. Likewise, the intimate connection between the product rule of derivatives and the method of integration by parts comes up in discussion. 

Is there an analogous rule of integration for the quotient rule?  

Of course, if you spot an integral of the form $\int \left (\frac{f(x)}{g(x)} \right )' = \int \frac{g(x) \cdot f(x)' - f(x) \cdot g(x)'}{\left [ g(x)\right ]^2 }$,
then the antiderivative is obvious. But is there another form/manipulation/"trick"?

As for me, I cannot see an advantage in introduction of such a rule since for any two functions $f,g$ it clearly holds that
$$
\frac fg = f\cdot\frac1g
$$
so the 'quotient rule' for derivatives is a product rule in disguise, and the same will also hold for the integration by parts. Indeed, when you are looking for the proper function to put under the differential sign integrating by parts, in case you have a bit of experince with such a procedure, you also will think about the 'quotients'. 
As an example:
$$
\int\frac{\sin\frac1x}{x^2}\,dx
$$
Of course you can present it as $\frac{f(x)}{x^2}$ and apply the new integration by parts based on the quotient rule, but I almost sure that a lot of the readers will rather think of the fact that $\frac1{x^2}\,dx = -d\frac1x$, by this seeing a product in the integrand rather than a quotient.
