I came across this integral when I was studying indefinite integrals.

So far I've had many unsuccessful attempts, including partial attempts, but I couldn't find a way to continue.

I even tried to divide the numerator and denominator by $ x ^ 4 $ give in $$ int frac {(1 + x ^ {- 2}) (1 + 2x ^ {- 2})} {( cos x + { sin x via x}) ^ 4} dx $$ still unable to move forward.

I even tried cheating a bit by deriving options, still nothing.

And many more..

So the problem remains. Can someone tell me how to do it?

(Note: This is a problem from a very elementary calculus course, i.e. no contour integrals, no multivariables and the like. However, I think that a differentiation under integration would be fine. It would also be helpful if answers are one of the answers available in options ( see image). .)