\[ \int \frac {\text {sech}^3(c+d x)}{a+b \tanh ^2(c+d x)} \, dx \]
Optimal antiderivative \[ -\frac {\arctan \! \left (\sinh \! \left (d x +c \right )\right )}{b d}+\frac {\arctan \! \left (\frac {\sinh \left (d x +c \right ) \sqrt {a +b}}{\sqrt {a}}\right ) \sqrt {a +b}}{b d \sqrt {a}} \]
command
integrate(sech(d*x+c)^3/(a+b*tanh(d*x+c)^2),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 (2 \, \sqrt {a^{2} - b^{2} + 2 \, \sqrt {-a b} {\left (a + b\right )}} a b^{2} {\left | a e^{\left (2 \, c\right )} + b e^{\left (2 \, c\right )} \right |} + \sqrt {a^{2} - b^{2} + 2 \, \sqrt {-a b} {\left (a + b\right )}} \sqrt {-a b} {\left (a + b\right )} {\left | a e^{\left (2 \, c\right )} + b e^{\left (2 \, c\right )} \right |} {\left | b \right |} - {\left (a b^{2} - b^{3}\right )} \sqrt {a^{2} - b^{2} + 2 \, \sqrt {-a b} {\left (a + b\right )}} {\left | a e^{\left (2 \, c\right )} + b e^{\left (2 \, c\right )} \right |}\right )} \arctan \left (\frac {e^{\left (d x\right )}}{\sqrt {\frac {a b e^{\left (2 \, c\right )} - b^{2} e^{\left (2 \, c\right )} + \sqrt {{\left (a b e^{\left (2 \, c\right )} - b^{2} e^{\left (2 \, c\right )}\right )}^{2} - {\left (a b e^{\left (4 \, c\right )} + b^{2} e^{\left (4 \, c\right )}\right )} {\left (a b + b^{2}\right )}}}{a b e^{\left (4 \, c\right )} + b^{2} e^{\left (4 \, c\right )}}}}\right ) e^{\left (-2 \, c\right )}}{{\left (a^{3} b + 3 \, a^{2} b^{2} + 3 \, a b^{3} + b^{4}\right )} \sqrt {-a b} {\left | b \right |}} - \frac {{\left (2 \, \sqrt {a^{2} - b^{2} - 2 \, \sqrt {-a b} {\left (a + b\right )}} a b^{2} {\left | a e^{\left (2 \, c\right )} + b e^{\left (2 \, c\right )} \right |} - \sqrt {a^{2} - b^{2} - 2 \, \sqrt {-a b} {\left (a + b\right )}} \sqrt {-a b} {\left (a + b\right )} {\left | a e^{\left (2 \, c\right )} + b e^{\left (2 \, c\right )} \right |} {\left | b \right |} - {\left (a b^{2} - b^{3}\right )} \sqrt {a^{2} - b^{2} - 2 \, \sqrt {-a b} {\left (a + b\right )}} {\left | a e^{\left (2 \, c\right )} + b e^{\left (2 \, c\right )} \right |}\right )} \arctan \left (\frac {e^{\left (d x\right )}}{\sqrt {\frac {a b e^{\left (2 \, c\right )} - b^{2} e^{\left (2 \, c\right )} - \sqrt {{\left (a b e^{\left (2 \, c\right )} - b^{2} e^{\left (2 \, c\right )}\right )}^{2} - {\left (a b e^{\left (4 \, c\right )} + b^{2} e^{\left (4 \, c\right )}\right )} {\left (a b + b^{2}\right )}}}{a b e^{\left (4 \, c\right )} + b^{2} e^{\left (4 \, c\right )}}}}\right ) e^{\left (-2 \, c\right )}}{{\left (a^{3} b + 3 \, a^{2} b^{2} + 3 \, a b^{3} + b^{4}\right )} \sqrt {-a b} {\left | b \right |}} - \frac {2 \, \arctan \left (e^{\left (d x + c\right )}\right )}{b}}{d} \]