\[ \int \frac {e^{c (a+b x)}}{\cosh ^2(a c+b c x)^{7/2}} \, dx \]
Optimal antiderivative \[ \frac {64 \cosh \left (b c x +a c \right )}{3 b c \left (1+{\mathrm e}^{2 c \left (b x +a \right )}\right )^{6} \sqrt {2 \cosh \left (2 b c x +2 a c \right )+2}}-\frac {384 \cosh \left (b c x +a c \right )}{5 b c \left (1+{\mathrm e}^{2 c \left (b x +a \right )}\right )^{5} \sqrt {2 \cosh \left (2 b c x +2 a c \right )+2}}+\frac {96 \cosh \left (b c x +a c \right )}{b c \left (1+{\mathrm e}^{2 c \left (b x +a \right )}\right )^{4} \sqrt {2 \cosh \left (2 b c x +2 a c \right )+2}}-\frac {128 \cosh \left (b c x +a c \right )}{3 b c \left (1+{\mathrm e}^{2 c \left (b x +a \right )}\right )^{3} \sqrt {2 \cosh \left (2 b c x +2 a c \right )+2}} \]
command
int(exp(c*(b*x+a))/(cosh(b*c*x+a*c)^2)^(7/2),x,method=_RETURNVERBOSE)
Maple 2022.1 output
| method | result | size |
| risch | \(-\frac {16 \left (20 \,{\mathrm e}^{6 c \left (b x +a \right )}+15 \,{\mathrm e}^{4 c \left (b x +a \right )}+6 \,{\mathrm e}^{2 c \left (b x +a \right )}+1\right ) {\mathrm e}^{-c \left (b x +a \right )}}{15 c b \sqrt {\left (1+{\mathrm e}^{2 c \left (b x +a \right )}\right )^{2} {\mathrm e}^{-2 c \left (b x +a \right )}}\, \left (1+{\mathrm e}^{2 c \left (b x +a \right )}\right )^{5}}\) | \(91\) |
Maple 2021.1 output
\[ \int \frac {128 \,{\mathrm e}^{c \left (b x +a \right )}}{\left (2 \cosh \left (2 b c x +2 a c \right )+2\right )^{\frac {7}{2}}}\, dx \]________________________________________________________________________________________