<s>[INST] Integral of $\int \frac{\sin(x)}{3\cos^3(x)+\sin^2(x)\cdot\cos(x)}\,dx$ So, from here $$\int \frac{\sin(x)}{3\cos^3(x)+\sin^2(x)\cdot\cos(x)} dx$$
I divided by cos(x) and I got
$$\int \frac{\tan(x)}{2\cos^2(x)+1} dx$$
But I'm stuck here.
I tried to substitute $t=\cos(x)$
$$\int \frac{-1}{t\cdot(2t^2+1)} dt$$
Any help would be greatly appreciated.
 [/INST] $$
\begin{aligned}
& \int \frac{\sin x}{3 \cos ^{3} x+\sin ^{2} x \cos x} d x \\
=& \int \frac{\sin x d x}{\cos x\left(3 \cos ^{2} x+\sin ^{2} x\right)} \\
=& \int \frac{d(\cos x)}{\cos x\left(2 \cos ^{2} x+1\right)} \\
=& \int\left(\frac{1}{\cos x}-\frac{2 \cos x}{2 \cos ^{2} x+1}\right) d(\cos x) \\
=& \ln (\cos x)-\frac{1}{2} \int \frac{d\left(2\cos ^{2} x+1\right)}{2 \cos ^{2} x+1} \\
=& \ln |\cos x|-\frac{1}{2} \ln \left(2 \cos ^{2} x+1\right)+C
\end{aligned}
$$
</s>