\[ \int \frac {-2 b c+a c x^6}{\sqrt [4]{b+a x^6} \left (b-c^4 x^4+a x^6\right )} \, dx \]
Optimal antiderivative \[ -\arctan \left (\frac {c x}{\left (a \,x^{6}+b \right )^{\frac {1}{4}}}\right )-\arctanh \left (\frac {c x}{\left (a \,x^{6}+b \right )^{\frac {1}{4}}}\right ) \]
command
Integrate[(-2*b*c + a*c*x^6)/((b + a*x^6)^(1/4)*(b - c^4*x^4 + a*x^6)),x]
Mathematica 13.1 output
\[ -\text {ArcTan}\left (\frac {c x}{\sqrt [4]{b+a x^6}}\right )-\tanh ^{-1}\left (\frac {c x}{\sqrt [4]{b+a x^6}}\right ) \]
Mathematica 12.3 output
\[ \int \frac {-2 b c+a c x^6}{\sqrt [4]{b+a x^6} \left (b-c^4 x^4+a x^6\right )} \, dx \]________________________________________________________________________________________