\[ \int \frac {e^{\text {csch}^{-1}\left (a x^2\right )}}{x^2} \, dx \]
Optimal antiderivative \[ -\frac {1}{3 a \,x^{3}}-\frac {\sqrt {1+\frac {1}{a^{2} x^{4}}}}{3 x}-\frac {\left (a +\frac {1}{x^{2}}\right ) \sqrt {\frac {\cos \left (4 \,\mathrm {arccot}\left (x \sqrt {a}\right )\right )}{2}+\frac {1}{2}}\, \EllipticF \left (\sin \left (2 \,\mathrm {arccot}\left (x \sqrt {a}\right )\right ), \frac {\sqrt {2}}{2}\right ) \sqrt {\frac {a^{2}+\frac {1}{x^{4}}}{\left (a +\frac {1}{x^{2}}\right )^{2}}}}{3 \cos \left (2 \,\mathrm {arccot}\left (x \sqrt {a}\right )\right ) \sqrt {a}\, \sqrt {1+\frac {1}{a^{2} x^{4}}}} \]
command
integrate((1/a/x^2+(1+1/a^2/x^4)^(1/2))/x^2,x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ -\frac {2 \, \left (-a^{2}\right )^{\frac {3}{4}} x^{3} {\rm ellipticF}\left (\left (-a^{2}\right )^{\frac {1}{4}} x, -1\right ) + a x^{2} \sqrt {\frac {a^{2} x^{4} + 1}{a^{2} x^{4}}} + 1}{3 \, a x^{3}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {a x^{2} \sqrt {\frac {a^{2} x^{4} + 1}{a^{2} x^{4}}} + 1}{a x^{4}}, x\right ) \]