What is indefinite $ \int\frac{\sin\left(x\right)\cos\left(x\right)}{\sqrt{3 - x^{4}}}{\rm d}x . $ $$
\int\frac{\sin\left(x\right)\cos\left(x\right)}{\sqrt{3 - x^{4}}}{\rm d}x .
$$
Hello.
Today, in exam, we had to evaluate this integral. Noone was able to do it. Appreciate any help.

As said in comments, the integral you posted does not seem to be doable.
Let us admit a typo in the problem and let us consider $$I=\int\frac{\sin\left(x\right)\cos\left(x\right)}{\sqrt{3 - \sin^4(x)}}\,{\rm d}x $$ The denominator looking more or less as $\sqrt{1-y^2}$, I thought that $\sin^{-1}(...)$. So, let us try a change of variable such that $$\frac{\sin ^2(x)}{\sqrt{3}}=y$$ that is to say $$x=\sin ^{-1}\left(\sqrt[4]{3} \sqrt{y}\right)$$ $$dx=\frac{\sqrt[4]{3}}{2 \sqrt{y} \sqrt{1-\sqrt{3} y}}$$ $$\sin(x)=\sqrt[4]{3} \sqrt{y}$$ $$\cos(x)=\sqrt{1-\sqrt{3} y}$$ Now, replacing $$I=\frac{1}{2}\int\frac{1}{ \sqrt{1-y^2}}dy=\frac{1}{2} \sin^{-1}(y)=\frac{1}{2} \sin ^{-1}\left(\frac{\sin ^2(x)}{\sqrt{3}}\right)$$
