\[ \int \text {ArcCos}\left (a x^2\right ) \, dx \]
Optimal antiderivative \[ x \arccos \left (a \,x^{2}\right )+\frac {2 \EllipticE \left (x \sqrt {a}, i\right )}{\sqrt {a}}-\frac {2 \EllipticF \left (x \sqrt {a}, i\right )}{\sqrt {a}} \]
command
integrate(arccos(a*x^2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {a x^{2} \arccos \left (a x^{2}\right ) - 2 \, \sqrt {-a^{2} x^{4} + 1}}{a x} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\arccos \left (a x^{2}\right ), x\right ) \]