سلسلة التكامل

$\large {\color{Red} \int (\frac{\tan2x-tanx}{\tan2x+tanx})dx=\int (\frac{\frac{2tanx}{1-tan^{2}x}-tanx}{\frac{2tanx}{1-tan^{2}x}+tanx})dx=\int (\frac{2tanx+tan^{3}x-tanx}{2tanx-tan^{3}x+tan})dx}$
${\color{Red} =\int (\frac{sec^{2}x}{3-tan^{2}x})dx=\int (\frac{dy}{3-y^{2}})=\frac{1}{\sqrt{3}}\tanh^{-1}(1/\sqrt{3})y+c}$