85.2 Problem number 157

\[ \int \frac {e^{c (a+b x)}}{\text {sech}^2(a c+b c x)^{5/2}} \, dx \]

Optimal antiderivative \[ -\frac {\mathrm {sech}\left (b c x +a c \right ) {\mathrm e}^{-4 c \left (b x +a \right )}}{128 b c \sqrt {\mathrm {sech}\left (b c x +a c \right )^{2}}}-\frac {5 \,\mathrm {sech}\left (b c x +a c \right ) {\mathrm e}^{-2 c \left (b x +a \right )}}{64 b c \sqrt {\mathrm {sech}\left (b c x +a c \right )^{2}}}+\frac {5 \,{\mathrm e}^{2 c \left (b x +a \right )} \mathrm {sech}\left (b c x +a c \right )}{32 b c \sqrt {\mathrm {sech}\left (b c x +a c \right )^{2}}}+\frac {5 \,{\mathrm e}^{4 c \left (b x +a \right )} \mathrm {sech}\left (b c x +a c \right )}{128 b c \sqrt {\mathrm {sech}\left (b c x +a c \right )^{2}}}+\frac {{\mathrm e}^{6 c \left (b x +a \right )} \mathrm {sech}\left (b c x +a c \right )}{192 b c \sqrt {\mathrm {sech}\left (b c x +a c \right )^{2}}}+\frac {5 x \,\mathrm {sech}\left (b c x +a c \right )}{16 \sqrt {\mathrm {sech}\left (b c x +a c \right )^{2}}} \]

command

integrate(exp(c*(b*x+a))/(sech(b*c*x+a*c)**2)**(5/2),x)

Sympy 1.10.1 under Python 3.10.4 output

\[ \begin {cases} x & \text {for}\: c = 0 \wedge \left (b = 0 \vee c = 0\right ) \\\frac {x e^{a c}}{\left (\operatorname {sech}^{2}{\left (a c \right )}\right )^{\frac {5}{2}}} & \text {for}\: b = 0 \\- \frac {5 x e^{a c} e^{b c x} \tanh ^{5}{\left (a c + b c x \right )}}{16 \left (\operatorname {sech}^{2}{\left (a c + b c x \right )}\right )^{\frac {5}{2}}} + \frac {5 x e^{a c} e^{b c x} \tanh ^{4}{\left (a c + b c x \right )}}{16 \left (\operatorname {sech}^{2}{\left (a c + b c x \right )}\right )^{\frac {5}{2}}} + \frac {5 x e^{a c} e^{b c x} \tanh ^{3}{\left (a c + b c x \right )}}{8 \left (\operatorname {sech}^{2}{\left (a c + b c x \right )}\right )^{\frac {5}{2}}} - \frac {5 x e^{a c} e^{b c x} \tanh ^{2}{\left (a c + b c x \right )}}{8 \left (\operatorname {sech}^{2}{\left (a c + b c x \right )}\right )^{\frac {5}{2}}} - \frac {5 x e^{a c} e^{b c x} \tanh {\left (a c + b c x \right )}}{16 \left (\operatorname {sech}^{2}{\left (a c + b c x \right )}\right )^{\frac {5}{2}}} + \frac {5 x e^{a c} e^{b c x}}{16 \left (\operatorname {sech}^{2}{\left (a c + b c x \right )}\right )^{\frac {5}{2}}} + \frac {8 e^{a c} e^{b c x} \tanh ^{5}{\left (a c + b c x \right )}}{15 b c \left (\operatorname {sech}^{2}{\left (a c + b c x \right )}\right )^{\frac {5}{2}}} - \frac {53 e^{a c} e^{b c x} \tanh ^{4}{\left (a c + b c x \right )}}{240 b c \left (\operatorname {sech}^{2}{\left (a c + b c x \right )}\right )^{\frac {5}{2}}} - \frac {331 e^{a c} e^{b c x} \tanh ^{3}{\left (a c + b c x \right )}}{240 b c \left (\operatorname {sech}^{2}{\left (a c + b c x \right )}\right )^{\frac {5}{2}}} + \frac {131 e^{a c} e^{b c x} \tanh ^{2}{\left (a c + b c x \right )}}{240 b c \left (\operatorname {sech}^{2}{\left (a c + b c x \right )}\right )^{\frac {5}{2}}} + \frac {253 e^{a c} e^{b c x} \tanh {\left (a c + b c x \right )}}{240 b c \left (\operatorname {sech}^{2}{\left (a c + b c x \right )}\right )^{\frac {5}{2}}} - \frac {11 e^{a c} e^{b c x}}{30 b c \left (\operatorname {sech}^{2}{\left (a c + b c x \right )}\right )^{\frac {5}{2}}} & \text {otherwise} \end {cases} \]

Sympy 1.8 under Python 3.8.8 output

\[ \text {Timed out} \]________________________________________________________________________________________