\[ \int \frac {\text {sech}^5(c+d x)}{a+b \tanh ^2(c+d x)} \, dx \]
Optimal antiderivative \[ -\frac {\left (2 a +3 b \right ) \arctan \! \left (\sinh \! \left (d x +c \right )\right )}{2 b^{2} d}+\frac {\left (a +b \right )^{\frac {3}{2}} \arctan \! \left (\frac {\sinh \left (d x +c \right ) \sqrt {a +b}}{\sqrt {a}}\right )}{b^{2} d \sqrt {a}}-\frac {\mathrm {sech}\! \left (d x +c \right ) \tanh \! \left (d x +c \right )}{2 b d} \]
command
integrate(sech(d*x+c)^5/(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
\[ \frac {\frac {\sqrt {a^{2} - b^{2} + 2 \, \sqrt {-a b} {\left (a + b\right )}} {\left (a + \sqrt {-a b}\right )} {\left | a e^{\left (2 \, c\right )} + b e^{\left (2 \, c\right )} \right |} \arctan \left (\frac {e^{\left (d x\right )}}{\sqrt {\frac {a b^{2} e^{\left (2 \, c\right )} - b^{3} e^{\left (2 \, c\right )} + \sqrt {{\left (a b^{2} e^{\left (2 \, c\right )} - b^{3} e^{\left (2 \, c\right )}\right )}^{2} - {\left (a b^{2} e^{\left (4 \, c\right )} + b^{3} e^{\left (4 \, c\right )}\right )} {\left (a b^{2} + b^{3}\right )}}}{a b^{2} e^{\left (4 \, c\right )} + b^{3} e^{\left (4 \, c\right )}}}}\right ) e^{\left (-2 \, c\right )}}{a^{2} b^{2} - a b^{3} + 2 \, \sqrt {-a b} a b^{2}} + \frac {\sqrt {a^{2} - b^{2} - 2 \, \sqrt {-a b} {\left (a + b\right )}} {\left (a - \sqrt {-a b}\right )} {\left | a e^{\left (2 \, c\right )} + b e^{\left (2 \, c\right )} \right |} \arctan \left (\frac {e^{\left (d x\right )}}{\sqrt {\frac {a b^{2} e^{\left (2 \, c\right )} - b^{3} e^{\left (2 \, c\right )} - \sqrt {{\left (a b^{2} e^{\left (2 \, c\right )} - b^{3} e^{\left (2 \, c\right )}\right )}^{2} - {\left (a b^{2} e^{\left (4 \, c\right )} + b^{3} e^{\left (4 \, c\right )}\right )} {\left (a b^{2} + b^{3}\right )}}}{a b^{2} e^{\left (4 \, c\right )} + b^{3} e^{\left (4 \, c\right )}}}}\right ) e^{\left (-2 \, c\right )}}{a^{2} b^{2} - a b^{3} - 2 \, \sqrt {-a b} a b^{2}} - \frac {{\left (2 \, a e^{c} + 3 \, b e^{c}\right )} \arctan \left (e^{\left (d x + c\right )}\right ) e^{\left (-c\right )}}{b^{2}} - \frac {e^{\left (3 \, d x + 3 \, c\right )} - e^{\left (d x + c\right )}}{b {\left (e^{\left (2 \, d x + 2 \, c\right )} + 1\right )}^{2}}}{d} \]