<s>[INST] How do I calculate: $\int \frac{dx}{3\sin^2 x+5\cos^2x}?$ How can I calculate this integral ?
$$\int \frac{dx}{3\sin^2 x+5\cos^2x}=\text{?}$$
Thank you! I've tried using universal substitution but the result was too complicated to be somehow integrated. Can you please give me a useful hint ?
 [/INST] $$\begin{gathered}
  \int {\frac{1}
{{3{{\sin }^2}x + 5{{\cos }^2}x}}dx}  = \int {\frac{1}
{{\left( {3\frac{{{{\sin }^2}x}}
{{{{\cos }^2}x}} + 5} \right){{\cos }^2}x}}dx}  = \int {\frac{1}
{{3{{\tan }^2}x + 5}}d\left( {\tan x} \right)}  \hfill \\
   = \frac{1}
{3}\int {\frac{1}
{{{{\tan }^2}x + \frac{5}
{3}}}d\left( {\tan x} \right)}  = \frac{1}
{3}.\frac{1}
{{\sqrt {5/3} }}\arctan \left( {\frac{{\tan x}}
{{\sqrt {5/3} }}} \right) + C = \frac{1}
{{\sqrt {15} }}\arctan \left( {\sqrt {\frac{3}
{5}} \tan x} \right) + C \hfill \\ 
\end{gathered} $$
</s>