35.5 Problem number 296

\[ \int \frac {e^{c (a+b x)}}{\cosh ^2(a c+b c x)^{3/2}} \, dx \]

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

\[ \int \frac {8 \,{\mathrm e}^{c \left (b x +a \right )}}{\left (2 \cosh \left (2 b c x +2 a c \right )+2\right )^{\frac {3}{2}}}\, dx \]________________________________________________________________________________________