Integrand size = 12, antiderivative size = 76 \[ \int \frac {1}{(b \text {sech}(c+d x))^{5/2}} \, dx=-\frac {6 i E\left (\left .\frac {1}{2} i (c+d x)\right |2\right )}{5 b^2 d \sqrt {\cosh (c+d x)} \sqrt {b \text {sech}(c+d x)}}+\frac {2 \sinh (c+d x)}{5 b d (b \text {sech}(c+d x))^{3/2}} \]
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Time = 0.03 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3854, 3856, 2719} \[ \int \frac {1}{(b \text {sech}(c+d x))^{5/2}} \, dx=\frac {2 \sinh (c+d x)}{5 b d (b \text {sech}(c+d x))^{3/2}}-\frac {6 i E\left (\left .\frac {1}{2} i (c+d x)\right |2\right )}{5 b^2 d \sqrt {\cosh (c+d x)} \sqrt {b \text {sech}(c+d x)}} \]
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Rule 2719
Rule 3854
Rule 3856
Rubi steps \begin{align*} \text {integral}& = \frac {2 \sinh (c+d x)}{5 b d (b \text {sech}(c+d x))^{3/2}}+\frac {3 \int \frac {1}{\sqrt {b \text {sech}(c+d x)}} \, dx}{5 b^2} \\ & = \frac {2 \sinh (c+d x)}{5 b d (b \text {sech}(c+d x))^{3/2}}+\frac {3 \int \sqrt {\cosh (c+d x)} \, dx}{5 b^2 \sqrt {\cosh (c+d x)} \sqrt {b \text {sech}(c+d x)}} \\ & = -\frac {6 i E\left (\left .\frac {1}{2} i (c+d x)\right |2\right )}{5 b^2 d \sqrt {\cosh (c+d x)} \sqrt {b \text {sech}(c+d x)}}+\frac {2 \sinh (c+d x)}{5 b d (b \text {sech}(c+d x))^{3/2}} \\ \end{align*}
Time = 0.11 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.84 \[ \int \frac {1}{(b \text {sech}(c+d x))^{5/2}} \, dx=\frac {\sqrt {b \text {sech}(c+d x)} \left (-12 i \sqrt {\cosh (c+d x)} E\left (\left .\frac {1}{2} i (c+d x)\right |2\right )+\sinh (c+d x)+\sinh (3 (c+d x))\right )}{10 b^3 d} \]
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\[\int \frac {1}{\left (b \,\operatorname {sech}\left (d x +c \right )\right )^{\frac {5}{2}}}d x\]
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.09 (sec) , antiderivative size = 379, normalized size of antiderivative = 4.99 \[ \int \frac {1}{(b \text {sech}(c+d x))^{5/2}} \, dx=-\frac {24 \, \sqrt {2} {\left (\cosh \left (d x + c\right )^{3} + 3 \, \cosh \left (d x + c\right )^{2} \sinh \left (d x + c\right ) + 3 \, \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{2} + \sinh \left (d x + c\right )^{3}\right )} \sqrt {b} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cosh \left (d x + c\right ) + \sinh \left (d x + c\right )\right )\right ) - \sqrt {2} {\left (\cosh \left (d x + c\right )^{6} + 6 \, \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{5} + \sinh \left (d x + c\right )^{6} + {\left (15 \, \cosh \left (d x + c\right )^{2} - 11\right )} \sinh \left (d x + c\right )^{4} - 11 \, \cosh \left (d x + c\right )^{4} + 4 \, {\left (5 \, \cosh \left (d x + c\right )^{3} - 11 \, \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{3} + {\left (15 \, \cosh \left (d x + c\right )^{4} - 66 \, \cosh \left (d x + c\right )^{2} - 13\right )} \sinh \left (d x + c\right )^{2} - 13 \, \cosh \left (d x + c\right )^{2} + 2 \, {\left (3 \, \cosh \left (d x + c\right )^{5} - 22 \, \cosh \left (d x + c\right )^{3} - 13 \, \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right ) - 1\right )} \sqrt {\frac {b \cosh \left (d x + c\right ) + b \sinh \left (d x + c\right )}{\cosh \left (d x + c\right )^{2} + 2 \, \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + \sinh \left (d x + c\right )^{2} + 1}}}{20 \, {\left (b^{3} d \cosh \left (d x + c\right )^{3} + 3 \, b^{3} d \cosh \left (d x + c\right )^{2} \sinh \left (d x + c\right ) + 3 \, b^{3} d \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{2} + b^{3} d \sinh \left (d x + c\right )^{3}\right )}} \]
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\[ \int \frac {1}{(b \text {sech}(c+d x))^{5/2}} \, dx=\int \frac {1}{\left (b \operatorname {sech}{\left (c + d x \right )}\right )^{\frac {5}{2}}}\, dx \]
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\[ \int \frac {1}{(b \text {sech}(c+d x))^{5/2}} \, dx=\int { \frac {1}{\left (b \operatorname {sech}\left (d x + c\right )\right )^{\frac {5}{2}}} \,d x } \]
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\[ \int \frac {1}{(b \text {sech}(c+d x))^{5/2}} \, dx=\int { \frac {1}{\left (b \operatorname {sech}\left (d x + c\right )\right )^{\frac {5}{2}}} \,d x } \]
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Timed out. \[ \int \frac {1}{(b \text {sech}(c+d x))^{5/2}} \, dx=\int \frac {1}{{\left (\frac {b}{\mathrm {cosh}\left (c+d\,x\right )}\right )}^{5/2}} \,d x \]
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