What did I do wrong?

$\displaystyle \int{\frac{dx}{x\sqrt{x^2+3}}}$

$\displaystyle \sin{\Theta}=\frac{\sqrt{3}}{\sqrt{x^2+3}}$

$\displaystyle \cos{\Theta}=\frac{x}{\sqrt{x^2+3}}$

$\displaystyle \tan{\Theta}=\frac{\sqrt{3}}{x}$

$\displaystyle =\int{\frac{1}{\frac{\sqrt{3}}{\tan{\Theta}}*\frac {x}{\cos{\Theta}}}*\frac{x^2*\sec^2{\Theta}}{-\sqrt{3}}d\Theta}$

$\displaystyle =\int{\frac{\tan{\Theta}}{\sqrt{3}}*\frac{\cos{\Th eta}}{x}*\frac{x^2*\sec^2{\Theta}}{-\sqrt{3}}d\Theta}$

$\displaystyle =\int{\frac{\tan{\Theta}*\cos{\Theta}*x*\sec^2{\Th eta}}{-3}d\Theta}$

$\displaystyle =\frac{-1}{3}\int{\tan{\Theta}*{\sec{\Theta}{d\Theta}}}$

The correct answer is $\displaystyle \frac{1}{\sqrt{3}}\int{\csc{\Theta}d\Theta}$