I got here throughout this integral whereas finding out indefinite integrals.

Thus far I’ve had many unsuccessful makes an attempt which embody – making an attempt by elements however I couldn’t discover a strategy to procced by it.

I even tried dividing each numerator and denominator by $x^4$ to yield $$int frac {(1+x^{-2})(1+2x^{-2})}{( cosx+{sinx over x})^4}dx$$ nonetheless not capable of transfer ahead.

I even tried to cheat just a little bit by taking spinoff of choices, nonetheless nothing.

And lots of extra..

So the issue nonetheless stands. Can somebody inform me tips on how to proceed?

(Be aware: it is a drawback from very elementary calculus course so no contour integrals, no multivariable and such stuff, nevertheless I feel differentiation underneath integration can be high quality. Additionally it will assist if solutions are a kind of current in choices (see picture).)