\[ \int \left (a+b \text {sech}^2(c+d x)\right )^{5/2} \, dx \]
Optimal antiderivative \[ \frac {a^{\frac {5}{2}} \arctanh \left (\frac {\sqrt {a}\, \tanh \left (d x +c \right )}{\sqrt {a +b -b \left (\tanh ^{2}\left (d x +c \right )\right )}}\right )}{d}+\frac {\left (15 a^{2}+10 a b +3 b^{2}\right ) \arctan \left (\frac {\sqrt {b}\, \tanh \left (d x +c \right )}{\sqrt {a +b -b \left (\tanh ^{2}\left (d x +c \right )\right )}}\right ) \sqrt {b}}{8 d}+\frac {b \left (7 a +3 b \right ) \sqrt {a +b -b \left (\tanh ^{2}\left (d x +c \right )\right )}\, \tanh \left (d x +c \right )}{8 d}+\frac {b \tanh \left (d x +c \right ) \left (a +b -b \left (\tanh ^{2}\left (d x +c \right )\right )\right )^{\frac {3}{2}}}{4 d} \]
command
integrate((a+b*sech(d*x+c)^2)^(5/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \text {output too large to display} \]
Fricas 1.3.7 via sagemath 9.3 output \[ \text {Timed out} \]