$\int \bigg(x^3-x+\large\frac{x+1}{x^2+1}\bigg)$$dx$
$\int (x^3-x) dx +\int \large\frac{x+1}{x^3+1}$$dx$
$x^2 +1 =t$
$2xdx =dt$
$\int (x^3-x) dx +\int \large\frac{dt}{2(t)}+\int \large\frac{1}{x^2+1}$$dx$
=>$ \large\frac{x^4}{4}-\frac{x^2}{2}+\frac{1}{2}$$ \log (t)+\tan ^{-1}(x)+c$
=>$ \large\frac{x^4}{4}-\frac{x^2}{2}+\frac{1}{2}$$ \log (x^2+1)+\tan ^{-1} (x^2+1) +c$
Hence a is the correct answer.