Write a program to solve the following integral: $(5\mathrm{.}0$ Points$)$

I $=13$ sin$($x$2)$ cos$(2$x$)$dx

$1.$ Using the $3/8$ Simpson rule. The program should:

$($a$)$ Start with number of intervals $($N$)$ equal to $1.$

$($b$)$ Automatically triple the number of subintervals.

$($c$)$ At each iteration step the program must show the iteration step, subinterval number, integral value and the difference between a result and the preceding one.

$($d$)$ Stop when convergence is reached $($when the difference between a result and the$$preceding one is smaller than a threshold given by the user$).$ Choose a threshold$$value of $108.$

$2.$ Using the Gauss$-$Legendre method

$3.$ Compare the performance in terms of timing and precision of both methods