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