\[ \int \frac {\text {sech}^7(c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^3} \, dx \]
Optimal antiderivative \[ -\frac {\arctan \! \left (\sinh \! \left (d x +c \right )\right )}{b^{3} d}+\frac {\left (a +b \right ) \sinh \! \left (d x +c \right )}{4 a b d \left (a +\left (a +b \right ) \left (\sinh ^{2}\left (d x +c \right )\right )\right )^{2}}-\frac {\left (4 a -3 b \right ) \left (a +b \right ) \sinh \! \left (d x +c \right )}{8 a^{2} b^{2} d \left (a +\left (a +b \right ) \left (\sinh ^{2}\left (d x +c \right )\right )\right )}+\frac {\left (8 a^{2}-4 a b +3 b^{2}\right ) \arctan \! \left (\frac {\sinh \left (d x +c \right ) \sqrt {a +b}}{\sqrt {a}}\right ) \sqrt {a +b}}{8 a^{\frac {5}{2}} b^{3} d} \]
command
integrate(sech(d*x+c)^7/(a+b*tanh(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
\[ \text {output too large to display} \]