Integrand size = 12, antiderivative size = 90 \[ \int \frac {1}{(b \text {csch}(c+d x))^{5/2}} \, dx=\frac {2 \cosh (c+d x)}{5 b d (b \text {csch}(c+d x))^{3/2}}+\frac {6 i E\left (\left .\frac {1}{2} \left (i c-\frac {\pi }{2}+i d x\right )\right |2\right )}{5 b^2 d \sqrt {b \text {csch}(c+d x)} \sqrt {i \sinh (c+d x)}} \]
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Time = 0.03 (sec) , antiderivative size = 90, 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} \[ \int \frac {1}{(b \text {csch}(c+d x))^{5/2}} \, dx=\frac {2 \cosh (c+d x)}{5 b d (b \text {csch}(c+d x))^{3/2}}+\frac {6 i E\left (\left .\frac {1}{2} \left (i c+i d x-\frac {\pi }{2}\right )\right |2\right )}{5 b^2 d \sqrt {i \sinh (c+d x)} \sqrt {b \text {csch}(c+d x)}} \]
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Rule 2719
Rule 3854
Rule 3856
Rubi steps \begin{align*} \text {integral}& = \frac {2 \cosh (c+d x)}{5 b d (b \text {csch}(c+d x))^{3/2}}-\frac {3 \int \frac {1}{\sqrt {b \text {csch}(c+d x)}} \, dx}{5 b^2} \\ & = \frac {2 \cosh (c+d x)}{5 b d (b \text {csch}(c+d x))^{3/2}}-\frac {3 \int \sqrt {i \sinh (c+d x)} \, dx}{5 b^2 \sqrt {b \text {csch}(c+d x)} \sqrt {i \sinh (c+d x)}} \\ & = \frac {2 \cosh (c+d x)}{5 b d (b \text {csch}(c+d x))^{3/2}}+\frac {6 i E\left (\left .\frac {1}{2} \left (i c-\frac {\pi }{2}+i d x\right )\right |2\right )}{5 b^2 d \sqrt {b \text {csch}(c+d x)} \sqrt {i \sinh (c+d x)}} \\ \end{align*}
Time = 0.11 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.76 \[ \int \frac {1}{(b \text {csch}(c+d x))^{5/2}} \, dx=\frac {-\frac {6 i E\left (\left .\frac {1}{4} (-2 i c+\pi -2 i d x)\right |2\right )}{\sqrt {i \sinh (c+d x)}}+\sinh (2 (c+d x))}{5 b^2 d \sqrt {b \text {csch}(c+d x)}} \]
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\[\int \frac {1}{\left (b \,\operatorname {csch}\left (d x +c \right )\right )^{\frac {5}{2}}}d x\]
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.09 (sec) , antiderivative size = 378, normalized size of antiderivative = 4.20 \[ \int \frac {1}{(b \text {csch}(c+d x))^{5/2}} \, dx=\frac {24 \, \sqrt {2} {\left (\cosh \left (d x + c\right )^{3} + 3 \, \cosh \left (d x + c\right )^{2} \sinh \left (d x + c\right ) + 3 \, \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{2} + \sinh \left (d x + c\right )^{3}\right )} \sqrt {b} {\rm weierstrassZeta}\left (4, 0, {\rm weierstrassPInverse}\left (4, 0, \cosh \left (d x + c\right ) + \sinh \left (d x + c\right )\right )\right ) + \sqrt {2} {\left (\cosh \left (d x + c\right )^{6} + 6 \, \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{5} + \sinh \left (d x + c\right )^{6} + {\left (15 \, \cosh \left (d x + c\right )^{2} + 11\right )} \sinh \left (d x + c\right )^{4} + 11 \, \cosh \left (d x + c\right )^{4} + 4 \, {\left (5 \, \cosh \left (d x + c\right )^{3} + 11 \, \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{3} + {\left (15 \, \cosh \left (d x + c\right )^{4} + 66 \, \cosh \left (d x + c\right )^{2} - 13\right )} \sinh \left (d x + c\right )^{2} - 13 \, \cosh \left (d x + c\right )^{2} + 2 \, {\left (3 \, \cosh \left (d x + c\right )^{5} + 22 \, \cosh \left (d x + c\right )^{3} - 13 \, \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right ) + 1\right )} \sqrt {\frac {b \cosh \left (d x + c\right ) + b \sinh \left (d x + c\right )}{\cosh \left (d x + c\right )^{2} + 2 \, \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + \sinh \left (d x + c\right )^{2} - 1}}}{20 \, {\left (b^{3} d \cosh \left (d x + c\right )^{3} + 3 \, b^{3} d \cosh \left (d x + c\right )^{2} \sinh \left (d x + c\right ) + 3 \, b^{3} d \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{2} + b^{3} d \sinh \left (d x + c\right )^{3}\right )}} \]
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\[ \int \frac {1}{(b \text {csch}(c+d x))^{5/2}} \, dx=\int \frac {1}{\left (b \operatorname {csch}{\left (c + d x \right )}\right )^{\frac {5}{2}}}\, dx \]
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\[ \int \frac {1}{(b \text {csch}(c+d x))^{5/2}} \, dx=\int { \frac {1}{\left (b \operatorname {csch}\left (d x + c\right )\right )^{\frac {5}{2}}} \,d x } \]
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\[ \int \frac {1}{(b \text {csch}(c+d x))^{5/2}} \, dx=\int { \frac {1}{\left (b \operatorname {csch}\left (d x + c\right )\right )^{\frac {5}{2}}} \,d x } \]
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Timed out. \[ \int \frac {1}{(b \text {csch}(c+d x))^{5/2}} \, dx=\int \frac {1}{{\left (\frac {b}{\mathrm {sinh}\left (c+d\,x\right )}\right )}^{5/2}} \,d x \]
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