\[ \int \frac {\left (1+x^4\right ) \sqrt [4]{-x^3+x^4}}{x^4 \left (-1+x^4\right )} \, dx \]
Optimal antiderivative \[ \mathit {Unintegrable} \]
command
Integrate[((1 + x^4)*(-x^3 + x^4)^(1/4))/(x^4*(-1 + x^4)),x]
Mathematica 13.1 output
\[ \frac {(-1+x)^{3/4} \left (-8 \left (4 \sqrt [4]{-1+x} \left (-5+x+4 x^2\right )-45 \sqrt [4]{2} x^{9/4} \text {ArcTan}\left (\frac {\sqrt [4]{2}}{\sqrt [4]{\frac {-1+x}{x}}}\right )+45 \sqrt [4]{2} x^{9/4} \tanh ^{-1}\left (\frac {\sqrt [4]{2}}{\sqrt [4]{\frac {-1+x}{x}}}\right )\right )+45 x^{9/4} \text {RootSum}\left [2-2 \text {$\#$1}^4+\text {$\#$1}^8\&,\frac {-2 \log (x)+8 \log \left (\sqrt [4]{-1+x}-\sqrt [4]{x} \text {$\#$1}\right )+\log (x) \text {$\#$1}^4-4 \log \left (\sqrt [4]{-1+x}-\sqrt [4]{x} \text {$\#$1}\right ) \text {$\#$1}^4}{-\text {$\#$1}^3+\text {$\#$1}^7}\&\right ]\right )}{360 \left ((-1+x) x^3\right )^{3/4}} \]
Mathematica 12.3 output
\[ \int \frac {\left (1+x^4\right ) \sqrt [4]{-x^3+x^4}}{x^4 \left (-1+x^4\right )} \, dx \]________________________________________________________________________________________