Monte Carlo 4D integral with variable limit

So, i have this wonderfull function f(th,k,p,n,W) that i need to integrate over 4 variables, but one of them, the variable k, goes from 0 to p (then i integrate over p). I've tried doing this integration using nquad, but it is taking forever... so i was wondering if i can do this integral using monte carlo, but i don't know how to do this with a variable limit. Does any one know how can i do this ?

import numpy as npimport sympy as symfrom scipy.special import gamma, gammalnfrom scipy.integrate import nquadimport matplotlib.pyplot as pltimport pandas as pdfrom scipy.optimize import curve_fitimport globlanbda_bar = 0.205x0 = 0.01sigma0= 23/0.389lanbda_QCD = 0.226 #GeVQ0 = 0.6 #Gevrapidez= [0.5, 1.0, 1.5, 2.0, 2.4, 3.0, 3.4]W=5020 #GeVcsv_files = glob.glob('path/*.csv')list_n=[] list_Pb=[]for i in csv_files:    df = pd.read_csv(i)    n = df['N'].to_numpy()    PB = df['PN'].to_numpy()    list_n.append(n)     list_Pb.append(PB)def u(W): return 0.24/(0.13 + 0.32*((W/1000)**0.115))def y(n,W): return (1/2)*np.log((np.sqrt(((np.cosh(n))**2)+((u(W))**2)) + np.sinh(n))/(np.sqrt(((np.cosh(n))**2)+((u(W))**2)) - np.sinh(n)))def J(n,W): return np.cosh(n)/(np.sqrt((np.cosh(n))**2 + (u(W))**2))def mod_p_2(p,k,th): return 0.25*(p**2 + k**2 + 2*p*k*np.cos(th))def mod_m_2(p,k,th): return 0.25*(p**2 + k**2 - 2*p*k*np.cos(th))def Q2s_1(n,W): return (Q0**2)*(x0*(W/Q0)*np.exp(y(n,W)))**lanbda_bardef Q2s_2(n,W): return (208**(1/3))*(Q0**2)*(x0*(W/Q0)*np.exp(-y(n,W)))**lanbda_bardef phi1(p,k,th,n,W): return ((3*sigma0)/(4*(np.pi**2)))*(mod_p_2(p,k,th)/Q2s_1(n,W))*np.exp(-(mod_p_2(p,k,th)/Q2s_1(n,W)))*(1/0.52 if (12*np.pi)/(27*np.log(Q2s_1(n,W)/lanbda_QCD**2))>0.52 else (27*np.log(Q2s_1(n,W)/lanbda_QCD**2))/(12*np.pi))def phi2(p,k,th,n,W): return ((3*sigma0)/(4*(np.pi**2)))*(mod_m_2(p,k,th)/Q2s_2(n,W))*np.exp(-(mod_m_2(p,k,th)/Q2s_2(n,W)))*(1/0.52 if (12*np.pi)/(27*np.log(Q2s_2(n,W)/lanbda_QCD**2))>0.52 else (27*np.log(Q2s_2(n,W)/lanbda_QCD**2))/(12*np.pi))def f(th,k,p,n,W): return (3/8)*(2*np.pi)*J(n,W)*(k/p)*phi1(p,k,th,n,W)*phi2(p,k,th,n,W)*0.3range_p = [0, np.inf]range_th = [0, 2*np.pi]range_n=[[-1/2,1/2],[-1,1],[-1.5,1.5],[-2,2],[-2.4,2.4],[-3,3],[-3.4,3.4]]def range_k(p,n,W):    return [0, p]#----------Here's where i use nquad, i want to change this to monte carlo-----------list_n_medio=[]for range_eta,rap in zip(range_n,rapidez):    result, erro = nquad(f, [range_th,range_k,range_p,range_eta],args=(W,))    print('|η|<%.1f: OK!'%rap)    list_n_medio.append(result)#-----------------------------------------------------------------------------------