[Mastermander]

$$\int \arcsin(\frac{2x}{1+x^2}) \, dx$$
$x=\tan u\quad ,dx=\sec^2u\ du$
$$\int \arcsin(\frac{2\tan u}{1+\tan^2u})\sec^2u\, du$$
$\sin 2u=\frac{2\tan u}{1+\tan^2u}$
$$\int 2u\, d(\tan u)\quad :by\; parts$$
$$2u\tan u-2\int\tan u\ du$$
$$2u\tan u -2\ln|\sec u|+c$$
$$2x\arctan x-\ln(x^2+1)+c$$