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