Is there a rule of integration that corresponds to the quotient rule?

I guess you could arrange an analog to integration by parts, but making students learn it would be superfluous.

$$ \int \frac{du}{v} = \frac{u}{v} + \int \frac{u}{v^2} dv.$$

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.

It's worth emphasizing that a "quotient rule" does play a role in Hermite's algorithm for integrating rational functions. It works as follows. By squarefree decomposing the denominator and partial fraction expanding, we reduce to integrating $\rm\:A/D^k\in \mathbb Q(x)\:,\:$ where $\rm\:\deg\:A < \deg\:D^k,\:$ and where $\rm\:D\:$ is squarefree, so $\rm\:\gcd(D,D') = 1\:.\:$ Thus by Bezout (extended Euclidean algorithm) there are $\rm\:B,C\in \mathbb Q[x]\:$ such that $\rm\ B\ D' + C\ D\ =\ A/(1-k)\:.\:$ Then a little algebra shows that

$$\rm\int \frac{A}{D^k}\ =\ \frac{B}{D^{k-1}}\ +\ \int \frac{(1-k)\ C - B'}{D^{k-1}} $$

Iterating the above rule we eventually reduce to the case $\rm\:k=1\:,\:$ i.e. squarefree denominator $\rm\:D\:.\:$ Thus using the above "quotient rule" and nothing deeper than Euclid's algorithm for polynomials (without requiring any factorization) one can mechanically compute the "rational part" of the integral of a rational function, i.e. the part of the integral not involving logarithms. This Hermite reduction rule is the basis of an algorithm due to Hermite (1872). It plays a fundamental role in the transcendental case of some integration algorithms.