$\arctan(y/x) = \ln(\sqrt{x^2+y^2}) + C$

$\frac{1}{2} \ln(x^2+y^2) = \arctan(y/x) + C$

$\ln(x^2+y^2) = 2\arctan(y/x) + C$

$\arctan(y/x) = \ln(x^2+y^2) + C$