\[ \int \frac {\coth ^3(c+d x)}{(a+b \text {sech}(c+d x))^{3/2}} \, dx \]
Optimal antiderivative \[ \frac {2 \arctanh \left (\frac {\sqrt {a +b \,\mathrm {sech}\left (d x +c \right )}}{\sqrt {a}}\right )}{a^{\frac {3}{2}} d}-\frac {\left (2 a -3 b \right ) \arctanh \left (\frac {\sqrt {a +b \,\mathrm {sech}\left (d x +c \right )}}{\sqrt {a -b}}\right )}{2 \left (a -b \right )^{\frac {5}{2}} d}+\frac {b \arctanh \left (\frac {\sqrt {a +b \,\mathrm {sech}\left (d x +c \right )}}{\sqrt {a -b}}\right )}{4 \left (a -b \right )^{\frac {5}{2}} d}-\frac {b \arctanh \left (\frac {\sqrt {a +b \,\mathrm {sech}\left (d x +c \right )}}{\sqrt {a +b}}\right )}{4 \left (a +b \right )^{\frac {5}{2}} d}-\frac {\left (2 a +3 b \right ) \arctanh \left (\frac {\sqrt {a +b \,\mathrm {sech}\left (d x +c \right )}}{\sqrt {a +b}}\right )}{2 \left (a +b \right )^{\frac {5}{2}} d}-\frac {2 b^{4}}{a \left (a^{2}-b^{2}\right )^{2} d \sqrt {a +b \,\mathrm {sech}\left (d x +c \right )}}-\frac {\sqrt {a +b \,\mathrm {sech}\left (d x +c \right )}}{4 \left (a +b \right )^{2} d \left (1-\mathrm {sech}\left (d x +c \right )\right )}-\frac {\sqrt {a +b \,\mathrm {sech}\left (d x +c \right )}}{4 \left (a -b \right )^{2} d \left (1+\mathrm {sech}\left (d x +c \right )\right )} \]
command
integrate(coth(d*x+c)^3/(a+b*sech(d*x+c))^(3/2),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 \[ \text {Timed out} \]