Looking this question Indefinite integral $\int \frac{1}{1+\sin^4\left(x\right)}dx$ and the comment of the nice user @Bernard for the use of the substitution
$$t=\tan x$$
On Wikipedia there are some examples when it is necessary to use the
If $$\omega (-t)=\omega (t)$$ a good change of variables is $u=\cos t$ (even function).
If $$\omega (\pi -t)=\omega (t)$$ a good change of variables is $u=\sin t$ (odd function). If $$\omega (\pi +t)=\omega (t)$$ a good change of variables is $u=\tan t$.
If two of the preceding relations both hold, a good change of variables is $$u=\cos 2t$$
In all other cases, use $$u=\tan(t/2)$$
Why @Bernard have used $t=\tan x$ if have an even integrand function? Shouldn't you use cosine $u=\cos t$?
PS: Never I have studied the Bioche's rules in the integration during the period at the university.
As Wikipedia notes, $\omega$ includes the $dt$ factor, say $\omega=fdt$, so $\omega(c-t)=\omega(t)$ if $f(c-t)=-f(t)$.