<s>[INST] Integration questions on $\int \frac{x^4\left ( 1-x \right )^4}{1+x^2} \, dx$ and $\int \frac{x^4}{x^4+5x^2+4} \, dx$ I would appreciate any hints on how to solve the following integration problems, they are my homework questions btw:
$$\int \frac{x^4\left ( 1-x \right )^4}{1+x^2} \, dx$$
$$\int \frac{x^4}{x^4+5x^2+4} \, dx$$
Thank you very much in advance!
 [/INST] Here are a couple of off-hand suggestions. There are undoubtedly more efficient ways.
For the first integral, you could multiply out the top, and use polynomial long division. You will get something that has the shape $P(x)+\frac{Ax}{1+x^2}+\frac{B}{1+x^2}$.
Integrating the polynomial will be easy. For $\frac{Ax}{1+x^2}$, use $u=1+x^2$. 
For the second problem, divide. We get $1-\frac{5x^2+4}{x^4+5x^2+4}$. Then use the fact that $x^4+5x^2+4=(x^2+1)(x^2+4)$, and use partial fractions. We end up needing to integrate $\frac{A}{x^2+1}$ and $\frac{B}{x^2+4}$. The first is immediate. For the second, use $x=2u$. 
</s>