\[ \int \frac {\coth (c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^3} \, dx \]
Optimal antiderivative \[ -\frac {b^{3}}{4 a^{3} \left (a +b \right ) d \left (b +a \left (\cosh ^{2}\left (d x +c \right )\right )\right )^{2}}+\frac {b^{2} \left (3 a +2 b \right )}{2 a^{3} \left (a +b \right )^{2} d \left (b +a \left (\cosh ^{2}\left (d x +c \right )\right )\right )}+\frac {b \left (3 a^{2}+3 a b +b^{2}\right ) \ln \! \left (b +a \left (\cosh ^{2}\left (d x +c \right )\right )\right )}{2 a^{3} \left (a +b \right )^{3} d}+\frac {\ln \! \left (\sinh \! \left (d x +c \right )\right )}{\left (a +b \right )^{3} d} \]
command
integrate(coth(d*x+c)/(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
\[ \frac {\frac {2 \, {\left (3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} \log \left (a e^{\left (4 \, d x + 4 \, c\right )} + 2 \, a e^{\left (2 \, d x + 2 \, c\right )} + 4 \, b e^{\left (2 \, d x + 2 \, c\right )} + a\right )}{a^{6} + 3 \, a^{5} b + 3 \, a^{4} b^{2} + a^{3} b^{3}} + \frac {4 \, e^{\left (2 \, c\right )} \log \left ({\left | -e^{\left (2 \, d x + 2 \, c\right )} + 1 \right |}\right )}{a^{3} e^{\left (2 \, c\right )} + 3 \, a^{2} b e^{\left (2 \, c\right )} + 3 \, a b^{2} e^{\left (2 \, c\right )} + b^{3} e^{\left (2 \, c\right )}} - \frac {4 \, d x}{a^{3}} - \frac {9 \, a^{3} b e^{\left (8 \, d x + 8 \, c\right )} + 9 \, a^{2} b^{2} e^{\left (8 \, d x + 8 \, c\right )} + 3 \, a b^{3} e^{\left (8 \, d x + 8 \, c\right )} + 36 \, a^{3} b e^{\left (6 \, d x + 6 \, c\right )} + 84 \, a^{2} b^{2} e^{\left (6 \, d x + 6 \, c\right )} + 44 \, a b^{3} e^{\left (6 \, d x + 6 \, c\right )} + 8 \, b^{4} e^{\left (6 \, d x + 6 \, c\right )} + 54 \, a^{3} b e^{\left (4 \, d x + 4 \, c\right )} + 150 \, a^{2} b^{2} e^{\left (4 \, d x + 4 \, c\right )} + 146 \, a b^{3} e^{\left (4 \, d x + 4 \, c\right )} + 32 \, b^{4} e^{\left (4 \, d x + 4 \, c\right )} + 36 \, a^{3} b e^{\left (2 \, d x + 2 \, c\right )} + 84 \, a^{2} b^{2} e^{\left (2 \, d x + 2 \, c\right )} + 44 \, a b^{3} e^{\left (2 \, d x + 2 \, c\right )} + 8 \, b^{4} e^{\left (2 \, d x + 2 \, c\right )} + 9 \, a^{3} b + 9 \, a^{2} b^{2} + 3 \, a b^{3}}{{\left (a^{5} + 3 \, a^{4} b + 3 \, a^{3} b^{2} + a^{2} b^{3}\right )} {\left (a e^{\left (4 \, d x + 4 \, c\right )} + 2 \, a e^{\left (2 \, d x + 2 \, c\right )} + 4 \, b e^{\left (2 \, d x + 2 \, c\right )} + a\right )}^{2}}}{4 \, d} \]