\[ \int e^{c (a+b x)} \cosh ^2(a c+b c x)^{3/2} \, dx \]
Optimal antiderivative \[ -\frac {\mathrm {sech}\left (b c x +a c \right ) \sqrt {\frac {\cosh \left (2 b c x +2 a c \right )}{2}+\frac {1}{2}}\, {\mathrm e}^{-2 c \left (b x +a \right )}}{16 b c}+\frac {3 \,{\mathrm e}^{2 c \left (b x +a \right )} \mathrm {sech}\left (b c x +a c \right ) \sqrt {\frac {\cosh \left (2 b c x +2 a c \right )}{2}+\frac {1}{2}}}{16 b c}+\frac {{\mathrm e}^{4 c \left (b x +a \right )} \mathrm {sech}\left (b c x +a c \right ) \sqrt {\frac {\cosh \left (2 b c x +2 a c \right )}{2}+\frac {1}{2}}}{32 b c}+\frac {3 x \,\mathrm {sech}\left (b c x +a c \right ) \sqrt {\frac {\cosh \left (2 b c x +2 a c \right )}{2}+\frac {1}{2}}}{8} \]
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 | \(\frac {3 x \sqrt {\left (1+{\mathrm e}^{2 c \left (b x +a \right )}\right )^{2} {\mathrm e}^{-2 c \left (b x +a \right )}}\, {\mathrm e}^{c \left (b x +a \right )}}{8 \left (1+{\mathrm e}^{2 c \left (b x +a \right )}\right )}+\frac {\sqrt {\left (1+{\mathrm e}^{2 c \left (b x +a \right )}\right )^{2} {\mathrm e}^{-2 c \left (b x +a \right )}}\, {\mathrm e}^{5 c \left (b x +a \right )}}{32 c b \left (1+{\mathrm e}^{2 c \left (b x +a \right )}\right )}+\frac {3 \sqrt {\left (1+{\mathrm e}^{2 c \left (b x +a \right )}\right )^{2} {\mathrm e}^{-2 c \left (b x +a \right )}}\, {\mathrm e}^{3 c \left (b x +a \right )}}{16 c b \left (1+{\mathrm e}^{2 c \left (b x +a \right )}\right )}-\frac {\sqrt {\left (1+{\mathrm e}^{2 c \left (b x +a \right )}\right )^{2} {\mathrm e}^{-2 c \left (b x +a \right )}}\, {\mathrm e}^{-c \left (b x +a \right )}}{16 c b \left (1+{\mathrm e}^{2 c \left (b x +a \right )}\right )}\) | \(216\) |
Maple 2021.1 output
\[ \int {\mathrm e}^{c \left (b x +a \right )} \left (\frac {\cosh \left (2 b c x +2 a c \right )}{2}+\frac {1}{2}\right )^{\frac {3}{2}}\, dx \]________________________________________________________________________________________