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