To prove $\int_0^1dx\int_x^{1/x}\frac{ydy}{(1+xy)^2(1+y^2)}=\frac{\pi-1}{4}$

Question

Prove by changing order of integration $$\int_0^1dx\int_x^{1/x}\frac{ydy}{(1+xy)^2(1+y^2)}=\frac{\pi-1}{4}$$

When I tried to draw the region of integration for this,I'm not getting a closed boundary.I double checked and I'm not getting where I went wrong

Answer 1

$\int_0^1dx\int_x^{1/x}\cfrac{ydy}{(1+xy)^2(1+y^2)}$

$\implies \int_0^1\int_o^y \cfrac{y}{(1+xy)^2(1+y^2)}dxdy+\int_0^{\infty}\int_0^{1/y}\cfrac{y}{(1+xy)^2(1+y^2)}dxdy$

$\implies \cfrac{\pi-1}{4}$