\[ \int \frac {\coth ^3(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^3} \, dx \]
Optimal antiderivative \[ -\frac {b^{4}}{4 a^{3} \left (a +b \right )^{2} d \left (b +a \left (\cosh ^{2}\left (d x +c \right )\right )\right )^{2}}+\frac {b^{3} \left (2 a +b \right )}{a^{3} \left (a +b \right )^{3} d \left (b +a \left (\cosh ^{2}\left (d x +c \right )\right )\right )}-\frac {\mathrm {csch}\! \left (d x +c \right )^{2}}{2 \left (a +b \right )^{3} d}+\frac {b^{2} \left (6 a^{2}+4 a b +b^{2}\right ) \ln \! \left (b +a \left (\cosh ^{2}\left (d x +c \right )\right )\right )}{2 a^{3} \left (a +b \right )^{4} d}+\frac {\left (a +4 b \right ) \ln \! \left (\sinh \! \left (d x +c \right )\right )}{\left (a +b \right )^{4} d} \]
command
integrate(coth(d*x+c)^3/(a+b*sech(d*x+c)^2)^3,x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \text {Exception raised: TypeError} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \frac {\frac {{\left (6 \, a^{2} b^{2} + 4 \, a b^{3} + b^{4}\right )} \log \left (a e^{\left (4 \, d x + 4 \, c\right )} + 2 \, a e^{\left (2 \, d x + 2 \, c\right )} + 4 \, b e^{\left (2 \, d x + 2 \, c\right )} + a\right )}{a^{7} + 4 \, a^{6} b + 6 \, a^{5} b^{2} + 4 \, a^{4} b^{3} + a^{3} b^{4}} + \frac {2 \, {\left (a e^{\left (2 \, c\right )} + 4 \, b e^{\left (2 \, c\right )}\right )} \log \left ({\left | e^{\left (2 \, d x + 2 \, c\right )} - 1 \right |}\right )}{a^{4} e^{\left (2 \, c\right )} + 4 \, a^{3} b e^{\left (2 \, c\right )} + 6 \, a^{2} b^{2} e^{\left (2 \, c\right )} + 4 \, a b^{3} e^{\left (2 \, c\right )} + b^{4} e^{\left (2 \, c\right )}} - \frac {2 \, d x}{a^{3}} - \frac {a^{5} e^{\left (12 \, d x + 12 \, c\right )} + 3 \, a^{4} b e^{\left (12 \, d x + 12 \, c\right )} + 3 \, a^{3} b^{2} e^{\left (12 \, d x + 12 \, c\right )} + a^{2} b^{3} e^{\left (12 \, d x + 12 \, c\right )} + 6 \, a^{5} e^{\left (10 \, d x + 10 \, c\right )} + 14 \, a^{4} b e^{\left (10 \, d x + 10 \, c\right )} + 30 \, a^{3} b^{2} e^{\left (10 \, d x + 10 \, c\right )} + 10 \, a^{2} b^{3} e^{\left (10 \, d x + 10 \, c\right )} + 15 \, a^{5} e^{\left (8 \, d x + 8 \, c\right )} + 29 \, a^{4} b e^{\left (8 \, d x + 8 \, c\right )} + 13 \, a^{3} b^{2} e^{\left (8 \, d x + 8 \, c\right )} + 47 \, a^{2} b^{3} e^{\left (8 \, d x + 8 \, c\right )} - 8 \, a b^{4} e^{\left (8 \, d x + 8 \, c\right )} - 8 \, b^{5} e^{\left (8 \, d x + 8 \, c\right )} + 20 \, a^{5} e^{\left (6 \, d x + 6 \, c\right )} + 36 \, a^{4} b e^{\left (6 \, d x + 6 \, c\right )} - 28 \, a^{3} b^{2} e^{\left (6 \, d x + 6 \, c\right )} - 116 \, a^{2} b^{3} e^{\left (6 \, d x + 6 \, c\right )} + 16 \, a b^{4} e^{\left (6 \, d x + 6 \, c\right )} + 16 \, b^{5} e^{\left (6 \, d x + 6 \, c\right )} + 15 \, a^{5} e^{\left (4 \, d x + 4 \, c\right )} + 29 \, a^{4} b e^{\left (4 \, d x + 4 \, c\right )} + 13 \, a^{3} b^{2} e^{\left (4 \, d x + 4 \, c\right )} + 47 \, a^{2} b^{3} e^{\left (4 \, d x + 4 \, c\right )} - 8 \, a b^{4} e^{\left (4 \, d x + 4 \, c\right )} - 8 \, b^{5} e^{\left (4 \, d x + 4 \, c\right )} + 6 \, a^{5} e^{\left (2 \, d x + 2 \, c\right )} + 14 \, a^{4} b e^{\left (2 \, d x + 2 \, c\right )} + 30 \, a^{3} b^{2} e^{\left (2 \, d x + 2 \, c\right )} + 10 \, a^{2} b^{3} e^{\left (2 \, d x + 2 \, c\right )} + a^{5} + 3 \, a^{4} b + 3 \, a^{3} b^{2} + a^{2} b^{3}}{{\left (a^{6} + 3 \, a^{5} b + 3 \, a^{4} b^{2} + a^{3} b^{3}\right )} {\left (a e^{\left (6 \, d x + 6 \, c\right )} + a e^{\left (4 \, d x + 4 \, c\right )} + 4 \, b e^{\left (4 \, d x + 4 \, c\right )} - a e^{\left (2 \, d x + 2 \, c\right )} - 4 \, b e^{\left (2 \, d x + 2 \, c\right )} - a\right )}^{2}}}{2 \, d} \]