\[ \int \frac {1+3 x^4+x^8}{x^2 \left (1+x^4\right )^{3/4} \left (1+3 x^4+3 x^8\right )} \, dx \]
Optimal antiderivative \[ -\frac {\left (x^{4}+1\right )^{\frac {1}{4}}}{x}-\frac {\arctan \left (\frac {\sqrt {3}\, x \left (x^{4}+1\right )^{\frac {1}{4}}}{-x^{2}+\sqrt {x^{4}+1}}\right ) \sqrt {3}}{3}+\frac {\arctanh \left (\frac {\sqrt {3}\, x \left (x^{4}+1\right )^{\frac {1}{4}}}{x^{2}+\sqrt {x^{4}+1}}\right ) \sqrt {3}}{3} \]
command
Integrate[(1 + 3*x^4 + x^8)/(x^2*(1 + x^4)^(3/4)*(1 + 3*x^4 + 3*x^8)),x]
Mathematica 13.1 output
\[ -\frac {\sqrt [4]{1+x^4}}{x}+\frac {\text {ArcTan}\left (\frac {\sqrt {3} x}{x-2 \sqrt [4]{1+x^4}}\right )}{\sqrt {3}}-\frac {\text {ArcTan}\left (\frac {\sqrt {3} x}{x+2 \sqrt [4]{1+x^4}}\right )}{\sqrt {3}}+\frac {\tanh ^{-1}\left (\frac {\sqrt {3} x \sqrt [4]{1+x^4}}{x^2+\sqrt {1+x^4}}\right )}{\sqrt {3}} \]
Mathematica 12.3 output
\[ \int \frac {1+3 x^4+x^8}{x^2 \left (1+x^4\right )^{3/4} \left (1+3 x^4+3 x^8\right )} \, dx \]________________________________________________________________________________________