The basis of factors that I got for the given Maple example.

basis.txt

[x*cos(y)**2, cos(y), 1/(x*log(x)*sin(y)), sin(y), x*sin(y)*cos(y), -1/(x**2*log(x)*sin(y)) - 1/(x**2*log(x)**2*sin(y)), cos(y)/(x*log(x)*sin(y)**2), cos(y)**2, x**2*cos(y)**4, cos(y)**2/(x**2*log(x)**2*sin(y)**4), 1/(x**4*log(x)**2*sin(y)**2) + 2/(x**4*log(x)**3*sin(y)**2) + 1/(x**4*log(x)**4*sin(y)**2), 1/(x**2*log(x)**2*sin(y)**2), cos(y)**4, x**2*sin(y)**2*cos(y)**2, sin(y)**2, x**2*sin(y)*cos(y)**3, cos(y)/(x*log(x)*sin(y)), -1/(x**2*log(x)) - 1/(x**2*log(x)**2), x*sin(y)**2*cos(y), cos(y)/log(x), cos(y)**3/(x*log(x)*sin(y)**2), cos(y)**3, cos(y)/(x**2*log(x)**2*sin(y)**3), x*sin(y)*cos(y)**3, -cos(y)**2/(x**2*log(x)*sin(y)) - cos(y)**2/(x**2*log(x)**2*sin(y)), -cos(y)**2/(x*log(x)*sin(y)) - cos(y)**2/(x*log(x)**2*sin(y)), x*cos(y)**4, cos(y)**2/(x*log(x)*sin(y)**2), cos(y)**2/(log(x)*sin(y)), x*cos(y)**3, -cos(y)/(x**2*log(x)*sin(y)) - cos(y)/(x**2*log(x)**2*sin(y)), -1/(x**3*log(x)**2*sin(y)**2) - 1/(x**3*log(x)**3*sin(y)**2), cos(y)**3/(log(x)*sin(y)**2), -cos(y)/(x*log(x)) - cos(y)/(x*log(x)**2), sin(y)*cos(y)**2, cos(y)**2/(x*log(x)*sin(y)), sin(y)*cos(y), x*sin(y)*cos(y)**2, -cos(y)/(x**3*log(x)**2*sin(y)**3) - cos(y)/(x**3*log(x)**3*sin(y)**3), 1/(x*log(x))]