$\int \frac{1}{a^2{\sin }^{2}x+b^2{\cos }^{2}x}dx=$

$\int \frac{1}{a^2{\sin }^{2}x+b^2{\cos }^{2}x}dx$

$=\int \frac{{\sec }^{2}xdx}{a^2{\tan }^{2}x+b^2}$

Put tanx =t

$\Rightarrow {\sec }^{2}dx=dt$

$=\int \frac{dt}{a^2{t}^2+b^2}=\frac{1}{a^2}\int \frac{dt}{t^2+{\left(\frac{b}{a}\right)}^2}$

$=\frac{1}{a^2}\times \frac{1}{\frac{b}{a}}{\tan }^{-1}\left(\frac{{\displaystyle t}}{{\displaystyle b/a}}\right)+c$

$=\frac{1}{ab}{\tan }^{-1}\left(\frac{{\displaystyle a\tan x}}{{\displaystyle b}}\right)+c$