\[ \int \text {sech}^2(a+b x) \tanh ^n(a+b x) \, dx \]
Optimal antiderivative \[ \frac {\tanh ^{1+n}\left (b x +a \right )}{b \left (1+n \right )} \]
command
integrate(sech(b*x+a)^2*tanh(b*x+a)^n,x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {\left (\frac {e^{\left (2 \, b x + 2 \, a\right )} - 1}{e^{\left (2 \, b x + 2 \, a\right )} + 1}\right )^{n + 1}}{b {\left (n + 1\right )}} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \int \tanh \left (b x + a\right )^{n} \operatorname {sech}\left (b x + a\right )^{2}\,{d x} \]________________________________________________________________________________________