Another way to solve $\int \frac{\sin^4(x)}{1+\cos^2(x)}\ dx$ without the substitution $y=\tan\left(\frac{x}{2}\right)$? Is there another way to solve an integral $$\int \frac{\sin^4(x)}{1+\cos^2(x)}\ dx$$ without the substitution $y=\tan\left(\frac{x}{2}\right)$?
$\large \int \frac{\sin^3(x)}{1+\cos^2(x)}\ dx$ is easily solved using the substitution $y=\cos(x)$.
What if the power of sine is even?

If you enjoy special functions, using $t=\tan(x)$
$$I_n=\int \frac{\sin^n(x)}{1+\cos^2(x)}\ dx=\int \left(\frac{t}{\sqrt{t^2+1}}\right)^n\frac{dt}{t^2+2}$$
$$I_n=\frac {t^{n+1}}{2(n+1)}\,F_1\left(\frac{n+1}{2};\frac{n}{2},1;\frac{n+3}{2};-t^2,-\frac{t^2}{2}\right)$$ where appears the Appell hypergeometric function of two variables.
