16.133 Problem number 950

\[ \int \frac {1}{\left (1+x^4\right )^{3/2}} \, dx \]

Optimal antiderivative \[ \frac {x}{2 \sqrt {x^{4}+1}}+\frac {\left (x^{2}+1\right ) \sqrt {\frac {\cos \left (4 \arctan \left (x \right )\right )}{2}+\frac {1}{2}}\, \EllipticF \left (\sin \left (2 \arctan \left (x \right )\right ), \frac {\sqrt {2}}{2}\right ) \sqrt {\frac {x^{4}+1}{\left (x^{2}+1\right )^{2}}}}{4 \cos \left (2 \arctan \left (x \right )\right ) \sqrt {x^{4}+1}} \]

command

integrate(1/(x^4+1)^(3/2),x, algorithm="fricas")

Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output

\[ \frac {\sqrt {i} {\left (-i \, x^{4} - i\right )} {\rm ellipticF}\left (\sqrt {i} x, -1\right ) + \sqrt {x^{4} + 1} x}{2 \, {\left (x^{4} + 1\right )}} \]

Fricas 1.3.7 via sagemath 9.3 output

\[ {\rm integral}\left (\frac {\sqrt {x^{4} + 1}}{x^{8} + 2 \, x^{4} + 1}, x\right ) \]