\[ \int \frac {-3+x^4}{\left (1+x^4\right ) \sqrt [4]{-3 x+4 x^4-3 x^5}} \, dx \]
Optimal antiderivative \[ \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \left (-3 x^{5}+4 x^{4}-3 x \right )^{\frac {3}{4}}}{3 x^{4}-4 x^{3}+3}\right )+\sqrt {2}\, \arctanh \left (\frac {\sqrt {2}\, \left (-3 x^{5}+4 x^{4}-3 x \right )^{\frac {3}{4}}}{3 x^{4}-4 x^{3}+3}\right ) \]
command
Integrate[(-3 + x^4)/((1 + x^4)*(-3*x + 4*x^4 - 3*x^5)^(1/4)),x]
Mathematica 13.1 output
\[ -\frac {\sqrt [4]{x} \sqrt [4]{3-4 x^3+3 x^4} \left (\text {ArcTan}\left (\frac {2 x^{3/4} \sqrt [4]{3-4 x^3+3 x^4}}{-2 x^{3/2}+\sqrt {3-4 x^3+3 x^4}}\right )+\tanh ^{-1}\left (\frac {2 x^{3/4} \sqrt [4]{3-4 x^3+3 x^4}}{2 x^{3/2}+\sqrt {3-4 x^3+3 x^4}}\right )\right )}{\sqrt [4]{-3 x+4 x^4-3 x^5}} \]
Mathematica 12.3 output
\[ \int \frac {-3+x^4}{\left (1+x^4\right ) \sqrt [4]{-3 x+4 x^4-3 x^5}} \, dx \]________________________________________________________________________________________