\[ \int \frac {1}{(a+a \cosh (c+d x))^{5/2}} \, dx \]
Optimal antiderivative \[ \frac {\sinh \! \left (d x +c \right )}{4 d \left (a +a \cosh \! \left (d x +c \right )\right )^{\frac {5}{2}}}+\frac {3 \sinh \! \left (d x +c \right )}{16 a d \left (a +a \cosh \! \left (d x +c \right )\right )^{\frac {3}{2}}}+\frac {3 \arctan \! \left (\frac {\sinh \left (d x +c \right ) \sqrt {a}\, \sqrt {2}}{2 \sqrt {a +a \cosh \left (d x +c \right )}}\right ) \sqrt {2}}{32 a^{\frac {5}{2}} d} \]
command
integrate(1/(a+a*cosh(d*x+c))^(5/2),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
\[ \frac {\sqrt {2} {\left (\frac {3 \, \arctan \left (e^{\left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}\right )}{a^{\frac {5}{2}}} + \frac {3 \, a^{\frac {7}{2}} e^{\left (\frac {7}{2} \, d x + \frac {7}{2} \, c\right )} + 11 \, a^{\frac {7}{2}} e^{\left (\frac {5}{2} \, d x + \frac {5}{2} \, c\right )} - 11 \, a^{\frac {7}{2}} e^{\left (\frac {3}{2} \, d x + \frac {3}{2} \, c\right )} - 3 \, a^{\frac {7}{2}} e^{\left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}}{{\left (a e^{\left (d x + c\right )} + a\right )}^{4} a^{2}}\right )}}{16 \, d} \]