\[ \int \frac {\sqrt [4]{x^3+x^5} \left (1+x^4+x^8\right )}{x^4 \left (-1+x^4\right )} \, dx \]
Optimal antiderivative \[ \frac {4 \left (x^{4}+2 x^{2}+1\right ) \left (x^{5}+x^{3}\right )^{\frac {1}{4}}}{9 x^{3}}+\frac {3 \,2^{\frac {1}{4}} \arctan \left (\frac {2^{\frac {1}{4}} x}{\left (x^{5}+x^{3}\right )^{\frac {1}{4}}}\right )}{4}-\frac {3 \arctan \left (\frac {2^{\frac {3}{4}} x \left (x^{5}+x^{3}\right )^{\frac {1}{4}}}{x^{2} \sqrt {2}-\sqrt {x^{5}+x^{3}}}\right ) 2^{\frac {3}{4}}}{8}-\frac {3 \,2^{\frac {1}{4}} \arctanh \left (\frac {2^{\frac {1}{4}} x}{\left (x^{5}+x^{3}\right )^{\frac {1}{4}}}\right )}{4}-\frac {3 \arctanh \left (\frac {\frac {x^{2} 2^{\frac {3}{4}}}{2}+\frac {\sqrt {x^{5}+x^{3}}\, 2^{\frac {1}{4}}}{2}}{x \left (x^{5}+x^{3}\right )^{\frac {1}{4}}}\right ) 2^{\frac {3}{4}}}{8} \]
command
Integrate[((x^3 + x^5)^(1/4)*(1 + x^4 + x^8))/(x^4*(-1 + x^4)),x]
Mathematica 13.1 output
\[ \frac {\sqrt [4]{x^3+x^5} \left (32 \sqrt [4]{1+x^2}+64 x^2 \sqrt [4]{1+x^2}+32 x^4 \sqrt [4]{1+x^2}+54 \sqrt [4]{2} x^{9/4} \text {ArcTan}\left (\frac {\sqrt [4]{2} \sqrt [4]{x}}{\sqrt [4]{1+x^2}}\right )-27\ 2^{3/4} x^{9/4} \text {ArcTan}\left (\frac {2^{3/4} \sqrt [4]{x} \sqrt [4]{1+x^2}}{\sqrt {2} \sqrt {x}-\sqrt {1+x^2}}\right )-54 \sqrt [4]{2} x^{9/4} \tanh ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{x}}{\sqrt [4]{1+x^2}}\right )-27\ 2^{3/4} x^{9/4} \tanh ^{-1}\left (\frac {2 \sqrt [4]{2} \sqrt [4]{x} \sqrt [4]{1+x^2}}{2 \sqrt {x}+\sqrt {2} \sqrt {1+x^2}}\right )\right )}{72 x^3 \sqrt [4]{1+x^2}} \]
Mathematica 12.3 output
\[ \int \frac {\sqrt [4]{x^3+x^5} \left (1+x^4+x^8\right )}{x^4 \left (-1+x^4\right )} \, dx \]________________________________________________________________________________________