\[ \int \frac {e^{\text {sech}^{-1}\left (a x^2\right )}}{x^2} \, dx \]
Optimal antiderivative \[ \frac {2}{3 a \,x^{3}}-\frac {\frac {1}{a \,x^{2}}+\sqrt {\frac {1}{a \,x^{2}}-1}\, \sqrt {\frac {1}{a \,x^{2}}+1}}{x}-\frac {2 \EllipticF \left (x \sqrt {a}, i\right ) \sqrt {a}\, \sqrt {\frac {1}{a \,x^{2}+1}}\, \sqrt {a \,x^{2}+1}}{3}+\frac {2 \sqrt {\frac {1}{a \,x^{2}+1}}\, \sqrt {a \,x^{2}+1}\, \sqrt {-a^{2} x^{4}+1}}{3 a \,x^{3}} \]
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
\[ -\frac {2 \, a^{\frac {3}{2}} x^{3} {\rm ellipticF}\left (\sqrt {a} x, -1\right ) + a x^{2} \sqrt {\frac {a x^{2} + 1}{a x^{2}}} \sqrt {-\frac {a x^{2} - 1}{a x^{2}}} + 1}{3 \, a x^{3}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {a x^{2} \sqrt {\frac {a x^{2} + 1}{a x^{2}}} \sqrt {-\frac {a x^{2} - 1}{a x^{2}}} + 1}{a x^{4}}, x\right ) \]