\[ \int \frac {1}{\cosh ^{\frac {5}{2}}(a+b x)} \, dx \]
Optimal antiderivative \[ -\frac {2 i \sqrt {\frac {\cosh \left (b x +a \right )}{2}+\frac {1}{2}}\, \EllipticF \left (i \sinh \left (\frac {a}{2}+\frac {b x}{2}\right ), \sqrt {2}\right )}{3 \cosh \left (\frac {a}{2}+\frac {b x}{2}\right ) b}+\frac {2 \sinh \left (b x +a \right )}{3 b \cosh \left (b x +a \right )^{\frac {3}{2}}} \]
command
integrate(1/cosh(b*x+a)^(5/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {2 \, {\left ({\left (\sqrt {2} \cosh \left (b x + a\right )^{4} + 4 \, \sqrt {2} \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{3} + \sqrt {2} \sinh \left (b x + a\right )^{4} + 2 \, {\left (3 \, \sqrt {2} \cosh \left (b x + a\right )^{2} + \sqrt {2}\right )} \sinh \left (b x + a\right )^{2} + 2 \, \sqrt {2} \cosh \left (b x + a\right )^{2} + 4 \, {\left (\sqrt {2} \cosh \left (b x + a\right )^{3} + \sqrt {2} \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right ) + \sqrt {2}\right )} {\rm weierstrassPInverse}\left (-4, 0, \cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right ) + 2 \, {\left (\cosh \left (b x + a\right )^{3} + 3 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{2} + \sinh \left (b x + a\right )^{3} + {\left (3 \, \cosh \left (b x + a\right )^{2} - 1\right )} \sinh \left (b x + a\right ) - \cosh \left (b x + a\right )\right )} \sqrt {\cosh \left (b x + a\right )}\right )}}{3 \, {\left (b \cosh \left (b x + a\right )^{4} + 4 \, b \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{3} + b \sinh \left (b x + a\right )^{4} + 2 \, b \cosh \left (b x + a\right )^{2} + 2 \, {\left (3 \, b \cosh \left (b x + a\right )^{2} + b\right )} \sinh \left (b x + a\right )^{2} + 4 \, {\left (b \cosh \left (b x + a\right )^{3} + b \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right ) + b\right )}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {1}{\cosh \left (b x + a\right )^{\frac {5}{2}}}, x\right ) \]