$$\int \frac{1}{1+3 \sin ^2 x+8 \cos ^2 x} d x$$ is equals to

$$\int \frac{\cos 2 x-\cos 2 \alpha}{\cos x-\cos \alpha} d x$$ is equal to

If $$\int \frac{d x}{(x+2)\left(x^2+1\right)}=a \log \left|1+x^2\right|+b \tan ^{-1} x +\frac{1}{5} \log |x+2|+c,$$ then

$$\int \frac{x^3 \sin \left(\tan ^{-1}\left(x^4\right)\right)}{1+x^8} d x$$ is equal to