\[ \int \coth ^3(c+d x) \sqrt {a+b \text {sech}(c+d x)} \, dx \]
Optimal antiderivative \[ \frac {2 \arctanh \left (\frac {\sqrt {a +b \,\mathrm {sech}\left (d x +c \right )}}{\sqrt {a}}\right ) \sqrt {a}}{d}-\frac {a \arctanh \left (\frac {\sqrt {a +b \,\mathrm {sech}\left (d x +c \right )}}{\sqrt {a -b}}\right )}{d \sqrt {a -b}}+\frac {3 b \arctanh \left (\frac {\sqrt {a +b \,\mathrm {sech}\left (d x +c \right )}}{\sqrt {a -b}}\right )}{4 d \sqrt {a -b}}-\frac {a \arctanh \left (\frac {\sqrt {a +b \,\mathrm {sech}\left (d x +c \right )}}{\sqrt {a +b}}\right )}{d \sqrt {a +b}}-\frac {3 b \arctanh \left (\frac {\sqrt {a +b \,\mathrm {sech}\left (d x +c \right )}}{\sqrt {a +b}}\right )}{4 d \sqrt {a +b}}-\frac {\left (\coth ^{2}\left (d x +c \right )\right ) \sqrt {a +b \,\mathrm {sech}\left (d x +c \right )}}{2 d} \]
command
integrate(coth(d*x+c)^3*(a+b*sech(d*x+c))^(1/2),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 \[ \text {Timed out} \]