\(\int \sin ^{-1}\left(\frac{2 x}{1+x^2}\right) \mathrm{d} x=\)

\(I=\int \sin ^{-1}\left(\frac{2 x}{1+x^2}\right) d x\)
\(\text {Let } x=\tan t\)
\(\therefore d x=\sec ^2 t d t\)
\(\therefore \mathrm{I} =\int \sin ^{-1}\left(\frac{2 \tan t}{1+\tan ^2 t}\right) \sec ^2 t \mathrm{dt} \)
\(=\int\left(\sin ^{-1}(\sin 2 \mathrm{t}) \sec ^2 \mathrm{t}\right) \mathrm{dt} \)
\( =\int 2 \mathrm{t} \sec ^2 \mathrm{t} d \mathrm{t} \)
\( =2\left[\mathrm{t} \int \sec ^2 \mathrm{t} d \mathrm{t}-\int \frac{\mathrm{dt}}{\mathrm{dt}}\left(\int \sec ^2 \mathrm{t} d \mathrm{dt}\right) \mathrm{dt}\right]+\mathrm{c} \)
\( =2\left[\mathrm{t} \tan \mathrm{t}-\int \tan \mathrm{t} \mathrm{dt}\right]+\mathrm{c} \)
\( =2 \mathrm{t} \tan \mathrm{t}+2 \log |\cos \mathrm{t}|+\mathrm{c} \)
\( =2 x \tan ^{-1} x+2 \log \left|\frac{1}{\sqrt{1+x^2}}\right|+\mathrm{c} \)
\( =2 x \tan ^{-1} x-\log \left(1+x^2\right)+\mathrm{c}\)