\[ \int \frac {e^{c (a+b x)}}{\sqrt {\cosh ^2(a c+b c x)}} \, dx \]
Optimal antiderivative \[ \frac {2 \cosh \left (b c x +a c \right ) \ln \left (1+{\mathrm e}^{2 c \left (b x +a \right )}\right )}{b c \sqrt {2 \cosh \left (2 b c x +2 a c \right )+2}} \]
command
int(exp(c*(b*x+a))/(cosh(b*c*x+a*c)^2)^(1/2),x,method=_RETURNVERBOSE)
Maple 2022.1 output
| method | result | size |
| risch | \(\frac {\ln \left ({\mathrm e}^{2 b c x}+{\mathrm e}^{-2 a c}\right ) \left (1+{\mathrm e}^{2 c \left (b x +a \right )}\right ) {\mathrm e}^{-c \left (b x +a \right )}}{c b \sqrt {\left (1+{\mathrm e}^{2 c \left (b x +a \right )}\right )^{2} {\mathrm e}^{-2 c \left (b x +a \right )}}}\) | \(66\) |
Maple 2021.1 output
\[ \int \frac {2 \,{\mathrm e}^{c \left (b x +a \right )}}{\sqrt {2 \cosh \left (2 b c x +2 a c \right )+2}}\, dx \]________________________________________________________________________________________