\[ \int \frac {e^{c (a+b x)}}{\sinh ^2(a c+b c x)^{5/2}} \, dx \]
Optimal antiderivative \[ -\frac {8 \sinh \left (b c x +a c \right )}{b c \left (1-{\mathrm e}^{2 c \left (b x +a \right )}\right )^{4} \sqrt {-2+2 \cosh \left (2 b c x +2 a c \right )}}+\frac {64 \sinh \left (b c x +a c \right )}{3 b c \left (1-{\mathrm e}^{2 c \left (b x +a \right )}\right )^{3} \sqrt {-2+2 \cosh \left (2 b c x +2 a c \right )}}-\frac {16 \sinh \left (b c x +a c \right )}{b c \left (1-{\mathrm e}^{2 c \left (b x +a \right )}\right )^{2} \sqrt {-2+2 \cosh \left (2 b c x +2 a c \right )}} \]
command
int(exp(c*(b*x+a))/(sinh(b*c*x+a*c)^2)^(5/2),x,method=_RETURNVERBOSE)
Maple 2022.1 output
| method | result | size |
| risch | \(-\frac {4 \left (6 \,{\mathrm e}^{4 c \left (b x +a \right )}-4 \,{\mathrm e}^{2 c \left (b x +a \right )}+1\right ) {\mathrm e}^{-c \left (b x +a \right )}}{3 c b \sqrt {\left ({\mathrm e}^{2 c \left (b x +a \right )}-1\right )^{2} {\mathrm e}^{-2 c \left (b x +a \right )}}\, \left ({\mathrm e}^{2 c \left (b x +a \right )}-1\right )^{3}}\) | \(80\) |
Maple 2021.1 output
\[ \int \frac {32 \,{\mathrm e}^{c \left (b x +a \right )}}{\left (-2+2 \cosh \left (2 b c x +2 a c \right )\right )^{\frac {5}{2}}}\, dx \]________________________________________________________________________________________