A triple integral is only a triple integral and nothing more difficult
Yep, a tautology. What reason do you have to think this is an easy integral to approximate?
my formula is the easiest one to calculate Pi(x) because the number of step to calculate it does not depend on the value of x
Not even close to true. Even without analyzing the complexity of approximating the integral, notice that your equation looks like:
Pi(t) = t^4 * triple integral
Which means that every time you double
t, the error of approximation gets magnified 16 times. This means that you need to do extra work approximating the integral to decrease the error by a factor of 16.
In the case of a single integral and we were using Simpson's rule, this would mean that you need to double the number of subdivisions every time you double
t, so that your algorithm is Ω(x), already worse than existing methods.
However, you're not using a single integral, you're using a triple integral, which takes much more effort to approximate well.
This is also for proper integrals: yours are improper, which are much harder to approximate well than ordinary integrals. One would probably truncate the domain of integration, which generates error. To get the greater accuracy needed with higher
t, you will have to consider ever larger portions of the domain of integration.
Furthermore, the bigger
t is, the bigger the integrand. In particular, the integrand grows much faster than
t. The bigger the integrand, the more work you need to do to get a good approximation.
But wait, there's more. As the numbers get larger, the number of operations increases, and the greater precision required, you need to store more and more significant figures, which means more time per basic operation.