Question Number 223034 by MrGaster last updated on 13/Jul/25
![∫((x arctanxdx)/( (√((1+x^2 )(1+k′^2 x^2 )))))=(1/k^2 )[F((π/4),k)−(π/(2(√(2(1+k′2)))))]](https://www.tinkutara.com/question/Q223034.png)
$$ \\ $$$$\int\frac{{x}\:\mathrm{arctan}{xdx}}{\:\sqrt{\left(\mathrm{1}+{x}^{\mathrm{2}} \right)\left(\mathrm{1}+{k}'^{\mathrm{2}} {x}^{\mathrm{2}} \right)}}=\frac{\mathrm{1}}{{k}^{\mathrm{2}} }\left[{F}\left(\frac{\pi}{\mathrm{4}},{k}\right)−\frac{\pi}{\mathrm{2}\sqrt{\mathrm{2}\left(\mathrm{1}+{k}'\mathrm{2}\right)}}\right] \\ $$