calculus - How to quickly solve partial fractions equation?

Often I am dealing with an integral of let's say:

$$\int\frac{dt}{(t-2)(t+3)}$$

or

$$\int \frac{dt}{t(t-4)}$$

or to make this a more general case in which I am interested the most:

$$\int \frac{dt}{(t+\alpha)(t+\beta)} \quad \quad \alpha, \beta \in \mathbb{R}$$

Basically any integral with decomposed polynomial of second degree in the denominator. Obviously this leads to sum of two natural logarithms.

What I do is write down partial fractions equation and then solve system of linear equations (2 variables here):

$$\frac{1}{(t+\alpha)(t+\beta)} = \frac{A}{t+\alpha} + \frac{B}{t+\beta}$$

After solving this I end up with some $A, B$ coefficients and I can solve the integral.

Is there faster way to find $A, B$ ? Some algorithm or anything that I can follow and would always work for such case? Surely, solving that does not take much time but I just wonder if it could be done even faster.

(bonus question: what if there were more variables, like 3 variables?)

I would greatly appreciate all feedback as it could help me save countless number of minutes in the future.

Answer

Here's your answer
for general $n$.

$\dfrac1{\prod_{k=1}^n (x-a_k)} =\sum_{k=1}^n \dfrac{b_k}{x-a_k}$.

Therefore
$1 =\sum_{k=1}^n \dfrac{b_k\prod_{j=1}^n (x-a_j)}{x-a_k} =\sum_{k=1}^n b_k\prod_{j=1, j\ne k}^n (x-a_j)$.

Setting
$x = a_i$
for each $i$,
all the terms
except the one with $k=i$
have the factor $a_i-a_i$,
so
$1 = b_i\prod_{j=1, j\ne i}^n (a_i-a_j)$

so that
$b_i =\dfrac1{\prod_{j=1, j\ne i}^n (a_i-a_j)}$.

For $n=2$,
$b_1 =\dfrac1{a_1-a_2}$,
$b_2 =\dfrac1{a_2-a_1}$.

For $n=3$,
$b_1 =\dfrac1{(a_1-a_2)(a_1-a_3)}$,
$b_2 =\dfrac1{(a_2-a_1)(a_2-a_3)}$,

$b_3 =\dfrac1{(a_3-a_1)(a_3-a_2)}$.