\[ \int \frac {1}{(-1+x) \sqrt [4]{x+x^3}} \, dx \]
Optimal antiderivative \[ \frac {\arctan \! \left (\frac {2^{\frac {3}{4}} \left (x^{3}+x \right )^{\frac {1}{4}}}{1+x}\right ) 2^{\frac {3}{4}}}{4}-\frac {\operatorname {arctanh}\! \left (\frac {2^{\frac {3}{4}} \left (x^{3}+x \right )^{\frac {1}{4}}}{1+x}\right ) 2^{\frac {3}{4}}}{4} \]
command
Int[1/((-1 + x)*(x + x^3)^(1/4)),x]
Rubi 4.17.3 under Mathematica 13.3.1 output
\[ \int \frac {1}{(-1+x) \sqrt [4]{x+x^3}} \, dx \]
Rubi 4.16.1 under Mathematica 13.3.1 output
\[ -\frac {4 \sqrt [4]{x^2+1} x^2 \operatorname {AppellF1}\left (\frac {7}{8},1,\frac {1}{4},\frac {15}{8},x^2,-x^2\right )}{7 \sqrt [4]{x^3+x}}-\frac {4 \sqrt [4]{x^2+1} x \operatorname {AppellF1}\left (\frac {3}{8},1,\frac {1}{4},\frac {11}{8},x^2,-x^2\right )}{3 \sqrt [4]{x^3+x}} \]