\[ \int \frac {-b+a x^8}{\left (b+a x^8\right ) \sqrt [4]{b-c x^4+a x^8}} \, dx \]
Optimal antiderivative \[ -\frac {\arctan \left (\frac {\sqrt {2}\, c^{\frac {1}{4}} x \left (a \,x^{8}-c \,x^{4}+b \right )^{\frac {1}{4}}}{-x^{2} \sqrt {c}+\sqrt {a \,x^{8}-c \,x^{4}+b}}\right ) \sqrt {2}}{4 c^{\frac {1}{4}}}-\frac {\arctanh \left (\frac {\frac {c^{\frac {1}{4}} x^{2} \sqrt {2}}{2}+\frac {\sqrt {a \,x^{8}-c \,x^{4}+b}\, \sqrt {2}}{2 c^{\frac {1}{4}}}}{x \left (a \,x^{8}-c \,x^{4}+b \right )^{\frac {1}{4}}}\right ) \sqrt {2}}{4 c^{\frac {1}{4}}} \]
command
Integrate[(-b + a*x^8)/((b + a*x^8)*(b - c*x^4 + a*x^8)^(1/4)),x]
Mathematica 13.1 output
\[ -\frac {\text {ArcTan}\left (\frac {\sqrt {2} \sqrt [4]{c} x \sqrt [4]{b-c x^4+a x^8}}{-\sqrt {c} x^2+\sqrt {b-c x^4+a x^8}}\right )+\tanh ^{-1}\left (\frac {\sqrt {c} x^2+\sqrt {b-c x^4+a x^8}}{\sqrt {2} \sqrt [4]{c} x \sqrt [4]{b-c x^4+a x^8}}\right )}{2 \sqrt {2} \sqrt [4]{c}} \]
Mathematica 12.3 output
\[ \int \frac {-b+a x^8}{\left (b+a x^8\right ) \sqrt [4]{b-c x^4+a x^8}} \, dx \]________________________________________________________________________________________