\[ \int e^{\text {sech}^{-1}\left (a x^2\right )} x^2 \, dx \]
Optimal antiderivative \[ \frac {2 x}{3 a}+\frac {\left (\frac {1}{a \,x^{2}}+\sqrt {\frac {1}{a \,x^{2}}-1}\, \sqrt {\frac {1}{a \,x^{2}}+1}\right ) x^{3}}{3}+\frac {2 \EllipticF \left (x \sqrt {a}, i\right ) \sqrt {\frac {1}{a \,x^{2}+1}}\, \sqrt {a \,x^{2}+1}}{3 a^{\frac {3}{2}}} \]
command
integrate((1/a/x^2+(1/a/x^2-1)^(1/2)*(1/a/x^2+1)^(1/2))*x^2,x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {a x^{3} \sqrt {\frac {a x^{2} + 1}{a x^{2}}} \sqrt {-\frac {a x^{2} - 1}{a x^{2}}} + 3 \, x}{3 \, a} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {a x^{2} \sqrt {\frac {a x^{2} + 1}{a x^{2}}} \sqrt {-\frac {a x^{2} - 1}{a x^{2}}} + 1}{a}, x\right ) \]