Question

$\int {\tan }^{-1}\left(\frac{3x-x^3}{1-3x^2}\right)dx$.

Answer 1

$x=\tan m$

$dx={\sec }^{2}mdm$

$\int {\tan }^{-1}(\frac{3\tan m-{\tan }^{3}m}{1-3{\tan }^{2}m}){\sec }^{2}mdm$

$\int {\tan }^{-1}\left(tan3m\right){\sec }^{2}mdm$

$\int 3m{\sec }^{2}mdm$

$3m\tan m-3\int \tan mdm$

$3x{\tan }^{-1}x+3\ln |\cos \frac{1}{\sqrt{1+x^2}}|$