\[ \int \frac {1}{\left (a \text {sech}^3(x)\right )^{5/2}} \, dx \]
Optimal antiderivative \[ -\frac {26 i \sqrt {\frac {\cosh \left (x \right )}{2}+\frac {1}{2}}\, \EllipticF \left (i \sinh \left (\frac {x}{2}\right ), \sqrt {2}\right )}{77 \cosh \left (\frac {x}{2}\right ) a^{2} \cosh \left (x \right )^{\frac {3}{2}} \sqrt {a \mathrm {sech}\left (x \right )^{3}}}+\frac {78 \cosh \left (x \right ) \sinh \left (x \right )}{385 a^{2} \sqrt {a \mathrm {sech}\left (x \right )^{3}}}+\frac {26 \left (\cosh ^{3}\left (x \right )\right ) \sinh \left (x \right )}{165 a^{2} \sqrt {a \mathrm {sech}\left (x \right )^{3}}}+\frac {2 \left (\cosh ^{5}\left (x \right )\right ) \sinh \left (x \right )}{15 a^{2} \sqrt {a \mathrm {sech}\left (x \right )^{3}}}+\frac {26 \tanh \left (x \right )}{77 a^{2} \sqrt {a \mathrm {sech}\left (x \right )^{3}}} \]
command
integrate(1/(a*sech(x)^3)^(5/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {49920 \, \sqrt {2} {\left (\cosh \left (x\right )^{8} + 8 \, \cosh \left (x\right )^{7} \sinh \left (x\right ) + 28 \, \cosh \left (x\right )^{6} \sinh \left (x\right )^{2} + 56 \, \cosh \left (x\right )^{5} \sinh \left (x\right )^{3} + 70 \, \cosh \left (x\right )^{4} \sinh \left (x\right )^{4} + 56 \, \cosh \left (x\right )^{3} \sinh \left (x\right )^{5} + 28 \, \cosh \left (x\right )^{2} \sinh \left (x\right )^{6} + 8 \, \cosh \left (x\right ) \sinh \left (x\right )^{7} + \sinh \left (x\right )^{8}\right )} \sqrt {a} {\rm weierstrassPInverse}\left (-4, 0, \cosh \left (x\right ) + \sinh \left (x\right )\right ) + \sqrt {2} {\left (77 \, \cosh \left (x\right )^{16} + 1232 \, \cosh \left (x\right ) \sinh \left (x\right )^{15} + 77 \, \sinh \left (x\right )^{16} + 14 \, {\left (660 \, \cosh \left (x\right )^{2} + 59\right )} \sinh \left (x\right )^{14} + 826 \, \cosh \left (x\right )^{14} + 196 \, {\left (220 \, \cosh \left (x\right )^{3} + 59 \, \cosh \left (x\right )\right )} \sinh \left (x\right )^{13} + 2 \, {\left (70070 \, \cosh \left (x\right )^{4} + 37583 \, \cosh \left (x\right )^{2} + 2203\right )} \sinh \left (x\right )^{12} + 4406 \, \cosh \left (x\right )^{12} + 8 \, {\left (42042 \, \cosh \left (x\right )^{5} + 37583 \, \cosh \left (x\right )^{3} + 6609 \, \cosh \left (x\right )\right )} \sinh \left (x\right )^{11} + 2 \, {\left (308308 \, \cosh \left (x\right )^{6} + 413413 \, \cosh \left (x\right )^{4} + 145398 \, \cosh \left (x\right )^{2} + 9561\right )} \sinh \left (x\right )^{10} + 19122 \, \cosh \left (x\right )^{10} + 4 \, {\left (220220 \, \cosh \left (x\right )^{7} + 413413 \, \cosh \left (x\right )^{5} + 242330 \, \cosh \left (x\right )^{3} + 47805 \, \cosh \left (x\right )\right )} \sinh \left (x\right )^{9} + 6 \, {\left (165165 \, \cosh \left (x\right )^{8} + 413413 \, \cosh \left (x\right )^{6} + 363495 \, \cosh \left (x\right )^{4} + 143415 \, \cosh \left (x\right )^{2}\right )} \sinh \left (x\right )^{8} + 16 \, {\left (55055 \, \cosh \left (x\right )^{9} + 177177 \, \cosh \left (x\right )^{7} + 218097 \, \cosh \left (x\right )^{5} + 143415 \, \cosh \left (x\right )^{3}\right )} \sinh \left (x\right )^{7} + 2 \, {\left (308308 \, \cosh \left (x\right )^{10} + 1240239 \, \cosh \left (x\right )^{8} + 2035572 \, \cosh \left (x\right )^{6} + 2007810 \, \cosh \left (x\right )^{4} - 9561\right )} \sinh \left (x\right )^{6} - 19122 \, \cosh \left (x\right )^{6} + 4 \, {\left (84084 \, \cosh \left (x\right )^{11} + 413413 \, \cosh \left (x\right )^{9} + 872388 \, \cosh \left (x\right )^{7} + 1204686 \, \cosh \left (x\right )^{5} - 28683 \, \cosh \left (x\right )\right )} \sinh \left (x\right )^{5} + 2 \, {\left (70070 \, \cosh \left (x\right )^{12} + 413413 \, \cosh \left (x\right )^{10} + 1090485 \, \cosh \left (x\right )^{8} + 2007810 \, \cosh \left (x\right )^{6} - 143415 \, \cosh \left (x\right )^{2} - 2203\right )} \sinh \left (x\right )^{4} - 4406 \, \cosh \left (x\right )^{4} + 8 \, {\left (5390 \, \cosh \left (x\right )^{13} + 37583 \, \cosh \left (x\right )^{11} + 121165 \, \cosh \left (x\right )^{9} + 286830 \, \cosh \left (x\right )^{7} - 47805 \, \cosh \left (x\right )^{3} - 2203 \, \cosh \left (x\right )\right )} \sinh \left (x\right )^{3} + 2 \, {\left (4620 \, \cosh \left (x\right )^{14} + 37583 \, \cosh \left (x\right )^{12} + 145398 \, \cosh \left (x\right )^{10} + 430245 \, \cosh \left (x\right )^{8} - 143415 \, \cosh \left (x\right )^{4} - 13218 \, \cosh \left (x\right )^{2} - 413\right )} \sinh \left (x\right )^{2} - 826 \, \cosh \left (x\right )^{2} + 4 \, {\left (308 \, \cosh \left (x\right )^{15} + 2891 \, \cosh \left (x\right )^{13} + 13218 \, \cosh \left (x\right )^{11} + 47805 \, \cosh \left (x\right )^{9} - 28683 \, \cosh \left (x\right )^{5} - 4406 \, \cosh \left (x\right )^{3} - 413 \, \cosh \left (x\right )\right )} \sinh \left (x\right ) - 77\right )} \sqrt {\frac {a \cosh \left (x\right ) + a \sinh \left (x\right )}{\cosh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2} + 1}}}{147840 \, {\left (a^{3} \cosh \left (x\right )^{8} + 8 \, a^{3} \cosh \left (x\right )^{7} \sinh \left (x\right ) + 28 \, a^{3} \cosh \left (x\right )^{6} \sinh \left (x\right )^{2} + 56 \, a^{3} \cosh \left (x\right )^{5} \sinh \left (x\right )^{3} + 70 \, a^{3} \cosh \left (x\right )^{4} \sinh \left (x\right )^{4} + 56 \, a^{3} \cosh \left (x\right )^{3} \sinh \left (x\right )^{5} + 28 \, a^{3} \cosh \left (x\right )^{2} \sinh \left (x\right )^{6} + 8 \, a^{3} \cosh \left (x\right ) \sinh \left (x\right )^{7} + a^{3} \sinh \left (x\right )^{8}\right )}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {\sqrt {a \operatorname {sech}\left (x\right )^{3}}}{a^{3} \operatorname {sech}\left (x\right )^{9}}, x\right ) \]