\[ \int \frac {e^{c (a+b x)}}{\cosh ^2(a c+b c x)^{3/2}} \, dx \]
Optimal antiderivative \[ \frac {4 \,{\mathrm e}^{4 c \left (b x +a \right )} \cosh \left (b c x +a c \right )}{b c \left (1+{\mathrm e}^{2 c \left (b x +a \right )}\right )^{2} \sqrt {2 \cosh \left (2 b c x +2 a c \right )+2}} \]
command
int(exp(c*(b*x+a))/(cosh(b*c*x+a*c)^2)^(3/2),x,method=_RETURNVERBOSE)
Maple 2022.1 output
method | result | size |
risch | \(-\frac {2 \left (2 \,{\mathrm e}^{2 c \left (b x +a \right )}+1\right ) {\mathrm e}^{-c \left (b x +a \right )}}{c b \sqrt {\left (1+{\mathrm e}^{2 c \left (b x +a \right )}\right )^{2} {\mathrm e}^{-2 c \left (b x +a \right )}}\, \left (1+{\mathrm e}^{2 c \left (b x +a \right )}\right )}\) | \(69\) |
Maple 2021.1 output
\[ \int \frac {8 \,{\mathrm e}^{c \left (b x +a \right )}}{\left (2 \cosh \left (2 b c x +2 a c \right )+2\right )^{\frac {3}{2}}}\, dx \]________________________________________________________________________________________