\[ \int e^{\text {sech}^{-1}\left (a x^2\right )} x^6 \, dx \]
Optimal antiderivative \[ \frac {2 x^{5}}{35 a}+\frac {\left (\frac {1}{a \,x^{2}}+\sqrt {\frac {1}{a \,x^{2}}-1}\, \sqrt {\frac {1}{a \,x^{2}}+1}\right ) x^{7}}{7}+\frac {2 \EllipticF \left (x \sqrt {a}, i\right ) \sqrt {\frac {1}{a \,x^{2}+1}}\, \sqrt {a \,x^{2}+1}}{21 a^{\frac {7}{2}}}-\frac {2 x \sqrt {\frac {1}{a \,x^{2}+1}}\, \sqrt {a \,x^{2}+1}\, \sqrt {-a^{2} x^{4}+1}}{21 a^{3}} \]
command
integrate((1/a/x^2+(1/a/x^2-1)^(1/2)*(1/a/x^2+1)^(1/2))*x^6,x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {21 \, a x^{5} + 5 \, {\left (3 \, a^{2} x^{7} - 2 \, x^{3}\right )} \sqrt {\frac {a x^{2} + 1}{a x^{2}}} \sqrt {-\frac {a x^{2} - 1}{a x^{2}}}}{105 \, a^{2}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {a x^{6} \sqrt {\frac {a x^{2} + 1}{a x^{2}}} \sqrt {-\frac {a x^{2} - 1}{a x^{2}}} + x^{4}}{a}, x\right ) \]