$$I=\int_0^\infty \frac{\tan^{-1} ax \tan^{-1}bx}{x^2}\mathrm dx$$
$$\frac{dI}{da}=\int_0^{\infty} \frac{\tan^{-1} bx}{x^2} \cdot \frac{x}{(ax)^2+1}\mathrm dx$$
$$\frac{d^2I}{\mathrm da \mathrm db} =\int_0^\infty \frac{x}{x((ax)^2+1)((bx)^2+1)}\mathrm dx$$
that's what I got. But, my book says there should be a $x^2$ in numerator and, no $x$ in denominator. But, I see that there's no $x$. Why? (ignoring $(ax)^2+1)((bx)^2+1)$ it in the context).
My book says
$$\frac{d^2I}{\mathrm da\mathrm db}=\int_0^\infty \frac{x^2}{((ax)^2+1)((bx)^2+1)}\mathrm dx$$
I can't find my mistake.
