\[ \int \frac {1-x^2}{a+b \left (-1+x^2\right )^4} \, dx \]
Optimal antiderivative \[ -\frac {\arctan \left (\frac {b^{\frac {1}{8}} x}{\sqrt {\left (-a \right )^{\frac {1}{4}}-b^{\frac {1}{4}}}}\right )}{4 b^{\frac {3}{8}} \sqrt {-a}\, \sqrt {\left (-a \right )^{\frac {1}{4}}-b^{\frac {1}{4}}}}+\frac {\arctanh \left (\frac {b^{\frac {1}{8}} x}{\sqrt {\left (-a \right )^{\frac {1}{4}}+b^{\frac {1}{4}}}}\right )}{4 b^{\frac {3}{8}} \sqrt {-a}\, \sqrt {\left (-a \right )^{\frac {1}{4}}+b^{\frac {1}{4}}}}-\frac {\arctan \left (\frac {-b^{\frac {1}{8}} x \sqrt {2}+\sqrt {b^{\frac {1}{4}}+\sqrt {\sqrt {-a}+\sqrt {b}}}}{\sqrt {-b^{\frac {1}{4}}+\sqrt {\sqrt {-a}+\sqrt {b}}}}\right ) \sqrt {-b^{\frac {1}{4}}+\sqrt {\sqrt {-a}+\sqrt {b}}}\, \sqrt {2}}{8 b^{\frac {3}{8}} \sqrt {-a}\, \sqrt {\sqrt {-a}+\sqrt {b}}}+\frac {\arctan \left (\frac {b^{\frac {1}{8}} x \sqrt {2}+\sqrt {b^{\frac {1}{4}}+\sqrt {\sqrt {-a}+\sqrt {b}}}}{\sqrt {-b^{\frac {1}{4}}+\sqrt {\sqrt {-a}+\sqrt {b}}}}\right ) \sqrt {-b^{\frac {1}{4}}+\sqrt {\sqrt {-a}+\sqrt {b}}}\, \sqrt {2}}{8 b^{\frac {3}{8}} \sqrt {-a}\, \sqrt {\sqrt {-a}+\sqrt {b}}}+\frac {\ln \left (b^{\frac {1}{4}} x^{2}+\sqrt {\sqrt {-a}+\sqrt {b}}-b^{\frac {1}{8}} x \sqrt {2}\, \sqrt {b^{\frac {1}{4}}+\sqrt {\sqrt {-a}+\sqrt {b}}}\right ) \sqrt {b^{\frac {1}{4}}+\sqrt {\sqrt {-a}+\sqrt {b}}}\, \sqrt {2}}{16 b^{\frac {3}{8}} \sqrt {-a}\, \sqrt {\sqrt {-a}+\sqrt {b}}}-\frac {\ln \left (b^{\frac {1}{4}} x^{2}+\sqrt {\sqrt {-a}+\sqrt {b}}+b^{\frac {1}{8}} x \sqrt {2}\, \sqrt {b^{\frac {1}{4}}+\sqrt {\sqrt {-a}+\sqrt {b}}}\right ) \sqrt {b^{\frac {1}{4}}+\sqrt {\sqrt {-a}+\sqrt {b}}}\, \sqrt {2}}{16 b^{\frac {3}{8}} \sqrt {-a}\, \sqrt {\sqrt {-a}+\sqrt {b}}} \]
command
integrate((-x^2+1)/(a+b*(x^2-1)^4),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \text {output too large to display} \]
Fricas 1.3.7 via sagemath 9.3 output \[ \text {Timed out} \]