Integrand size = 10, antiderivative size = 89 \[ \int \frac {1}{\left (a \text {csch}^3(x)\right )^{3/2}} \, dx=-\frac {14 \cosh (x)}{45 a \sqrt {a \text {csch}^3(x)}}+\frac {14 i \text {csch}(x) E\left (\left .\frac {\pi }{4}-\frac {i x}{2}\right |2\right )}{15 a \sqrt {a \text {csch}^3(x)} \sqrt {i \sinh (x)}}+\frac {2 \cosh (x) \sinh ^2(x)}{9 a \sqrt {a \text {csch}^3(x)}} \]
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Time = 0.03 (sec) , antiderivative size = 89, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {4208, 3854, 3856, 2719} \[ \int \frac {1}{\left (a \text {csch}^3(x)\right )^{3/2}} \, dx=-\frac {14 \cosh (x)}{45 a \sqrt {a \text {csch}^3(x)}}+\frac {2 \sinh ^2(x) \cosh (x)}{9 a \sqrt {a \text {csch}^3(x)}}+\frac {14 i \text {csch}(x) E\left (\left .\frac {\pi }{4}-\frac {i x}{2}\right |2\right )}{15 a \sqrt {i \sinh (x)} \sqrt {a \text {csch}^3(x)}} \]
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Rule 2719
Rule 3854
Rule 3856
Rule 4208
Rubi steps \begin{align*} \text {integral}& = -\frac {\left (i (i \text {csch}(x))^{3/2}\right ) \int \frac {1}{(i \text {csch}(x))^{9/2}} \, dx}{a \sqrt {a \text {csch}^3(x)}} \\ & = \frac {2 \cosh (x) \sinh ^2(x)}{9 a \sqrt {a \text {csch}^3(x)}}-\frac {\left (7 i (i \text {csch}(x))^{3/2}\right ) \int \frac {1}{(i \text {csch}(x))^{5/2}} \, dx}{9 a \sqrt {a \text {csch}^3(x)}} \\ & = -\frac {14 \cosh (x)}{45 a \sqrt {a \text {csch}^3(x)}}+\frac {2 \cosh (x) \sinh ^2(x)}{9 a \sqrt {a \text {csch}^3(x)}}-\frac {\left (7 i (i \text {csch}(x))^{3/2}\right ) \int \frac {1}{\sqrt {i \text {csch}(x)}} \, dx}{15 a \sqrt {a \text {csch}^3(x)}} \\ & = -\frac {14 \cosh (x)}{45 a \sqrt {a \text {csch}^3(x)}}+\frac {2 \cosh (x) \sinh ^2(x)}{9 a \sqrt {a \text {csch}^3(x)}}+\frac {(7 \text {csch}(x)) \int \sqrt {i \sinh (x)} \, dx}{15 a \sqrt {a \text {csch}^3(x)} \sqrt {i \sinh (x)}} \\ & = -\frac {14 \cosh (x)}{45 a \sqrt {a \text {csch}^3(x)}}+\frac {14 i \text {csch}(x) E\left (\left .\frac {\pi }{4}-\frac {i x}{2}\right |2\right )}{15 a \sqrt {a \text {csch}^3(x)} \sqrt {i \sinh (x)}}+\frac {2 \cosh (x) \sinh ^2(x)}{9 a \sqrt {a \text {csch}^3(x)}} \\ \end{align*}
Time = 0.07 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.64 \[ \int \frac {1}{\left (a \text {csch}^3(x)\right )^{3/2}} \, dx=\frac {-33 \cosh (x)+5 \cosh (3 x)+84 \text {csch}^2(x) E\left (\left .\frac {1}{4} (\pi -2 i x)\right |2\right ) \sqrt {i \sinh (x)}}{90 a \sqrt {a \text {csch}^3(x)}} \]
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\[\int \frac {1}{\left (a \operatorname {csch}\left (x \right )^{3}\right )^{\frac {3}{2}}}d x\]
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.09 (sec) , antiderivative size = 407, normalized size of antiderivative = 4.57 \[ \int \frac {1}{\left (a \text {csch}^3(x)\right )^{3/2}} \, dx=-\frac {672 \, \sqrt {2} {\left (\cosh \left (x\right )^{5} + 5 \, \cosh \left (x\right )^{4} \sinh \left (x\right ) + 10 \, \cosh \left (x\right )^{3} \sinh \left (x\right )^{2} + 10 \, \cosh \left (x\right )^{2} \sinh \left (x\right )^{3} + 5 \, \cosh \left (x\right ) \sinh \left (x\right )^{4} + \sinh \left (x\right )^{5}\right )} \sqrt {a} {\rm weierstrassZeta}\left (4, 0, {\rm weierstrassPInverse}\left (4, 0, \cosh \left (x\right ) + \sinh \left (x\right )\right )\right ) - \sqrt {2} {\left (5 \, \cosh \left (x\right )^{10} + 50 \, \cosh \left (x\right ) \sinh \left (x\right )^{9} + 5 \, \sinh \left (x\right )^{10} + {\left (225 \, \cosh \left (x\right )^{2} - 43\right )} \sinh \left (x\right )^{8} - 43 \, \cosh \left (x\right )^{8} + 8 \, {\left (75 \, \cosh \left (x\right )^{3} - 43 \, \cosh \left (x\right )\right )} \sinh \left (x\right )^{7} + 2 \, {\left (525 \, \cosh \left (x\right )^{4} - 602 \, \cosh \left (x\right )^{2} - 149\right )} \sinh \left (x\right )^{6} - 298 \, \cosh \left (x\right )^{6} + 4 \, {\left (315 \, \cosh \left (x\right )^{5} - 602 \, \cosh \left (x\right )^{3} - 447 \, \cosh \left (x\right )\right )} \sinh \left (x\right )^{5} + 2 \, {\left (525 \, \cosh \left (x\right )^{6} - 1505 \, \cosh \left (x\right )^{4} - 2235 \, \cosh \left (x\right )^{2} + 187\right )} \sinh \left (x\right )^{4} + 374 \, \cosh \left (x\right )^{4} + 8 \, {\left (75 \, \cosh \left (x\right )^{7} - 301 \, \cosh \left (x\right )^{5} - 745 \, \cosh \left (x\right )^{3} + 187 \, \cosh \left (x\right )\right )} \sinh \left (x\right )^{3} + {\left (225 \, \cosh \left (x\right )^{8} - 1204 \, \cosh \left (x\right )^{6} - 4470 \, \cosh \left (x\right )^{4} + 2244 \, \cosh \left (x\right )^{2} - 43\right )} \sinh \left (x\right )^{2} - 43 \, \cosh \left (x\right )^{2} + 2 \, {\left (25 \, \cosh \left (x\right )^{9} - 172 \, \cosh \left (x\right )^{7} - 894 \, \cosh \left (x\right )^{5} + 748 \, \cosh \left (x\right )^{3} - 43 \, \cosh \left (x\right )\right )} \sinh \left (x\right ) + 5\right )} \sqrt {\frac {a \cosh \left (x\right ) + a \sinh \left (x\right )}{\cosh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2} - 1}}}{720 \, {\left (a^{2} \cosh \left (x\right )^{5} + 5 \, a^{2} \cosh \left (x\right )^{4} \sinh \left (x\right ) + 10 \, a^{2} \cosh \left (x\right )^{3} \sinh \left (x\right )^{2} + 10 \, a^{2} \cosh \left (x\right )^{2} \sinh \left (x\right )^{3} + 5 \, a^{2} \cosh \left (x\right ) \sinh \left (x\right )^{4} + a^{2} \sinh \left (x\right )^{5}\right )}} \]
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\[ \int \frac {1}{\left (a \text {csch}^3(x)\right )^{3/2}} \, dx=\int \frac {1}{\left (a \operatorname {csch}^{3}{\left (x \right )}\right )^{\frac {3}{2}}}\, dx \]
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\[ \int \frac {1}{\left (a \text {csch}^3(x)\right )^{3/2}} \, dx=\int { \frac {1}{\left (a \operatorname {csch}\left (x\right )^{3}\right )^{\frac {3}{2}}} \,d x } \]
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\[ \int \frac {1}{\left (a \text {csch}^3(x)\right )^{3/2}} \, dx=\int { \frac {1}{\left (a \operatorname {csch}\left (x\right )^{3}\right )^{\frac {3}{2}}} \,d x } \]
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Timed out. \[ \int \frac {1}{\left (a \text {csch}^3(x)\right )^{3/2}} \, dx=\int \frac {1}{{\left (\frac {a}{{\mathrm {sinh}\left (x\right )}^3}\right )}^{3/2}} \,d x \]
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