Soft question:
I recently discovered Weierstrass substitutions (tangent half-angle, $t = \tan(\frac{x}{2}$)) and became intrigued with substitutions arising from unit circle re-parametrization. So I did some more research and found some weird substitutions for indefinite integrals like $$\int \frac{\cos x}{1+\csc^2x}\,dx$$ I'm wondering if anyone wants to share any amusing/weird/unconventional substitutions for relatively simple integrals?
Recently I found an interesting indefinite integral: $$\int{\frac{x^2+2\cos x-1}{(1+x^2)\sin x-2x}dx}$$For there are both $x$,$\sin x and \cos x$ in the integral,I think your Weierstrass substitutions may be invalid.
Here is the answer： Let $p=x\cos \frac{x}{2}-\sin \frac{x}{2}$ and $q=x\sin \frac{x}{2}-\cos \frac{x}{2}$
Why do we have so strange p and q？In fact we have the factorization that $2pq=(1+x^2)\sin x-2x$ and $2(p\ dq-q\ dp)=(x^2+2\cos x-1)\ dx$,and now we have the result:$$\int{\frac{x^2+2\cos x-1}{(1+x^2)\sin x-2x}dx}=\int\frac{p\ dq-q\ dp}{pq}=log\frac{q}{p}+C=log(x\tan (\frac{x}{2})-1)-log(x-\tan \frac{x}{2})+C$$