\[ \int \frac {\tanh ^5(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^2} \, dx \]
Optimal antiderivative \[ \frac {\left (a +b \right )^{2}}{2 a^{2} b d \left (b +a \left (\cosh ^{2}\left (d x +c \right )\right )\right )}+\frac {\ln \! \left (\cosh \! \left (d x +c \right )\right )}{b^{2} d}+\frac {\left (\frac {1}{a^{2}}-\frac {1}{b^{2}}\right ) \ln \! \left (b +a \left (\cosh ^{2}\left (d x +c \right )\right )\right )}{2 d} \]
command
integrate(tanh(d*x+c)^5/(a+b*sech(d*x+c)^2)^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
\[ -\frac {\frac {2 \, d x}{a^{2}} - \frac {2 \, \log \left (e^{\left (2 \, d x + 2 \, c\right )} + 1\right )}{b^{2}} + \frac {{\left (a^{2} - b^{2}\right )} \log \left (a e^{\left (4 \, d x + 4 \, c\right )} + 2 \, a e^{\left (2 \, d x + 2 \, c\right )} + 4 \, b e^{\left (2 \, d x + 2 \, c\right )} + a\right )}{a^{2} b^{2}} - \frac {a^{2} e^{\left (4 \, d x + 4 \, c\right )} - b^{2} e^{\left (4 \, d x + 4 \, c\right )} + 2 \, a^{2} e^{\left (2 \, d x + 2 \, c\right )} + 8 \, a b e^{\left (2 \, d x + 2 \, c\right )} + 6 \, b^{2} e^{\left (2 \, d x + 2 \, c\right )} + a^{2} - b^{2}}{{\left (a e^{\left (4 \, d x + 4 \, c\right )} + 2 \, a e^{\left (2 \, d x + 2 \, c\right )} + 4 \, b e^{\left (2 \, d x + 2 \, c\right )} + a\right )} a b^{2}}}{2 \, d} \]