Integration of rational functions of the form $\frac{p(x)}{\left[q(x)\right]^\alpha}$ when $\deg p < \deg q$ and $q(x)$ cannot be factored

Question

What is the general method to evaluate integrals of the form $$\int\frac{p(x)}{\left[q(x)\right]^\alpha}\mathrm dx$$ when $\deg p < \deg q$ and $q(x)$ is irreducible? I don't see how partial fraction decomposition can help here, since there is nothing to decompose.

For example, I'm having a hard time with $$\int\frac1{\left(1+t^2\right)^2}\mathrm dt$$

For this one I also tried some substitutions, from which I was not able to conclude anything.

Answer 1

Hint:

For solving the integral $$ \int \frac{1}{(1+t^2)^2}dt,$$ do the substitution

$ t =$ tan$(\theta) \Rightarrow dt = $sec$^2 (\theta) d\theta.$ Notice that cos$^2(\theta) = \frac{1}{1 + \text{tan}^2(\theta) }.$ Hence,

$$ \int \frac{1}{(1+t^2)^2}dt = \int \text{cos}^2(\theta)d\theta.$$