\[ \int \frac {1}{(a+a \text {sech}(c+d x))^{3/2}} \, dx \]
Optimal antiderivative \[ \frac {2 \arctanh \left (\frac {\sqrt {a}\, \tanh \left (d x +c \right )}{\sqrt {a +a \,\mathrm {sech}\left (d x +c \right )}}\right )}{a^{\frac {3}{2}} d}-\frac {5 \arctanh \left (\frac {\sqrt {a}\, \tanh \left (d x +c \right ) \sqrt {2}}{2 \sqrt {a +a \,\mathrm {sech}\left (d x +c \right )}}\right ) \sqrt {2}}{4 a^{\frac {3}{2}} d}-\frac {\tanh \left (d x +c \right )}{2 d \left (a +a \,\mathrm {sech}\left (d x +c \right )\right )^{\frac {3}{2}}} \]
command
integrate(1/(a+a*sech(d*x+c))^(3/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ -\frac {\frac {5 \, \sqrt {2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {a} e^{\left (d x + c\right )} - \sqrt {a e^{\left (2 \, d x + 2 \, c\right )} + a} + \sqrt {a}\right )}}{2 \, \sqrt {-a}}\right )}{\sqrt {-a} a} - \frac {4 \, \arctan \left (-\frac {\sqrt {a} e^{\left (d x + c\right )} - \sqrt {a e^{\left (2 \, d x + 2 \, c\right )} + a}}{\sqrt {-a}}\right )}{\sqrt {-a} a} + \frac {2 \, \log \left ({\left | -\sqrt {a} e^{\left (d x + c\right )} + \sqrt {a e^{\left (2 \, d x + 2 \, c\right )} + a} \right |}\right )}{a^{\frac {3}{2}}} + \frac {2 \, {\left (3 \, {\left (\sqrt {a} e^{\left (d x + c\right )} - \sqrt {a e^{\left (2 \, d x + 2 \, c\right )} + a}\right )}^{3} + {\left (\sqrt {a} e^{\left (d x + c\right )} - \sqrt {a e^{\left (2 \, d x + 2 \, c\right )} + a}\right )}^{2} \sqrt {a} - {\left (\sqrt {a} e^{\left (d x + c\right )} - \sqrt {a e^{\left (2 \, d x + 2 \, c\right )} + a}\right )} a + a^{\frac {3}{2}}\right )}}{{\left ({\left (\sqrt {a} e^{\left (d x + c\right )} - \sqrt {a e^{\left (2 \, d x + 2 \, c\right )} + a}\right )}^{2} + 2 \, {\left (\sqrt {a} e^{\left (d x + c\right )} - \sqrt {a e^{\left (2 \, d x + 2 \, c\right )} + a}\right )} \sqrt {a} - a\right )}^{2} a}}{2 \, d} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: TypeError} \]________________________________________________________________________________________