\[ \int \frac {1+x^{10}}{\sqrt {1+x^4} \left (-1+x^{10}\right )} \, dx \]
Optimal antiderivative \[ -\frac {\sqrt {-2+2 \sqrt {5}}\, \arctan \left (\frac {\sqrt {2+2 \sqrt {5}}\, x}{2 \sqrt {x^{4}+1}}\right )}{5}-\frac {\operatorname {arctanh}\left (\frac {\sqrt {2}\, x}{\sqrt {x^{4}+1}}\right ) \sqrt {2}}{10}-\frac {\sqrt {2+2 \sqrt {5}}\, \operatorname {arctanh}\left (\frac {\sqrt {-2+2 \sqrt {5}}\, x}{2 \sqrt {x^{4}+1}}\right )}{5} \]
command
Int[(1 + x^10)/(Sqrt[1 + x^4]*(-1 + x^10)),x]
Rubi 4.17.3 under Mathematica 13.3.1 output
\[ \frac {(-1)^{4/5} \left (1+\sqrt [5]{-1}\right ) \left (1+(-1)^{4/5}\right ) \arctan \left (\frac {\sqrt {\sqrt [5]{-1}-(-1)^{4/5}} x}{\sqrt {x^4+1}}\right )}{5 \left (1+(-1)^{3/5}\right ) \sqrt {\sqrt [5]{-1}-(-1)^{4/5}}}-\frac {\arctan \left (\frac {\sqrt {\sqrt [5]{-1}-(-1)^{4/5}} x}{\sqrt {x^4+1}}\right )}{5 \sqrt {\sqrt [5]{-1}-(-1)^{4/5}}}-\frac {\text {arctanh}\left (\frac {\sqrt {2} x}{\sqrt {x^4+1}}\right )}{5 \sqrt {2}}-\frac {\left (1-\sqrt [5]{-1}+(-1)^{2/5}\right ) \left (1-(-1)^{3/5}\right ) \text {arctanh}\left (\frac {\sqrt {(-1)^{2/5}-(-1)^{3/5}} x}{\sqrt {x^4+1}}\right )}{5 \sqrt {(-1)^{2/5}-(-1)^{3/5}}}-\frac {\text {arctanh}\left (\frac {\sqrt {(-1)^{2/5}-(-1)^{3/5}} x}{\sqrt {x^4+1}}\right )}{5 \sqrt {(-1)^{2/5}-(-1)^{3/5}}}-\frac {\left (1-(-1)^{4/5}\right ) \left (x^2+1\right ) \sqrt {\frac {x^4+1}{\left (x^2+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan (x),\frac {1}{2}\right )}{5 \left (1+(-1)^{3/5}\right ) \sqrt {x^4+1}}-\frac {\left (1-\sqrt [5]{-1}+(-1)^{2/5}\right ) \left (x^2+1\right ) \sqrt {\frac {x^4+1}{\left (x^2+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan (x),\frac {1}{2}\right )}{5 \sqrt {x^4+1}}-\frac {\left (x^2+1\right ) \sqrt {\frac {x^4+1}{\left (x^2+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan (x),\frac {1}{2}\right )}{5 \left (1+(-1)^{2/5}\right ) \sqrt {x^4+1}}-\frac {\left (x^2+1\right ) \sqrt {\frac {x^4+1}{\left (x^2+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan (x),\frac {1}{2}\right )}{5 \left (1-\sqrt [5]{-1}\right ) \sqrt {x^4+1}}+\frac {2 \left (x^2+1\right ) \sqrt {\frac {x^4+1}{\left (x^2+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan (x),\frac {1}{2}\right )}{5 \sqrt {x^4+1}}+\frac {\sqrt [5]{-1} \left (1+\sqrt [5]{-1}\right ) \left (1+(-1)^{2/5}\right ) \left (x^2+1\right ) \sqrt {\frac {x^4+1}{\left (x^2+1\right )^2}} \operatorname {EllipticPi}\left (\frac {1}{4} (-1)^{4/5} \left (1-\sqrt [5]{-1}\right )^2,2 \arctan (x),\frac {1}{2}\right )}{10 \sqrt {x^4+1}}+\frac {\left (1-(-1)^{2/5}\right ) \left (x^2+1\right ) \sqrt {\frac {x^4+1}{\left (x^2+1\right )^2}} \operatorname {EllipticPi}\left (-\frac {1}{4} (-1)^{3/5} \left (1+(-1)^{2/5}\right )^2,2 \arctan (x),\frac {1}{2}\right )}{10 \left (1+(-1)^{2/5}\right ) \sqrt {x^4+1}}+\frac {(-1)^{3/5} \left (1-\sqrt [5]{-1}\right ) \left (1+(-1)^{3/5}\right ) \left (x^2+1\right ) \sqrt {\frac {x^4+1}{\left (x^2+1\right )^2}} \operatorname {EllipticPi}\left (\frac {1}{4} (-1)^{2/5} \left (1-(-1)^{3/5}\right )^2,2 \arctan (x),\frac {1}{2}\right )}{10 \sqrt {x^4+1}}+\frac {\left (1-(-1)^{4/5}\right )^2 \left (x^2+1\right ) \sqrt {\frac {x^4+1}{\left (x^2+1\right )^2}} \operatorname {EllipticPi}\left (-\frac {1}{4} \sqrt [5]{-1} \left (1+(-1)^{4/5}\right )^2,2 \arctan (x),\frac {1}{2}\right )}{10 \left (1+(-1)^{3/5}\right ) \sqrt {x^4+1}} \]
Rubi 4.16.1 under Mathematica 13.3.1 output
\[ \int \frac {1+x^{10}}{\sqrt {1+x^4} \left (-1+x^{10}\right )} \, dx \]________________________________________________________________________________________