Consider the rational function

$f(x)=\frac{x^{10}}{(x^2-1{)}^2(x^2+3)}$

$($a$)$ Use Sagemath to decompose $f(x)$ into partial fractions.

$($b$)$ Based on the decomposition in $($a$),$ compute the indefinite integral ${\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}f(x)dx$ by hand. Clearly show the steps for integrating each term.

$($c$)$ Verify your result from $($b$)$ using Sagemath by performing the integration numerically and symbolically.