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"?

I guess you could arrange an analog to integration by parts, but making students learn it would be superfluous.
$$ \int \frac{du}{v} = \frac{u}{v} + \int \frac{u}{v^2} dv.$$
