Integrand size = 10, antiderivative size = 83 \[ \int \left (a \sinh ^3(x)\right )^{3/2} \, dx=-\frac {14}{45} a \cosh (x) \sqrt {a \sinh ^3(x)}+\frac {14 i a \text {csch}(x) E\left (\left .\frac {\pi }{4}-\frac {i x}{2}\right |2\right ) \sqrt {a \sinh ^3(x)}}{15 \sqrt {i \sinh (x)}}+\frac {2}{9} a \cosh (x) \sinh ^2(x) \sqrt {a \sinh ^3(x)} \]
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Time = 0.03 (sec) , antiderivative size = 83, 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 = {3286, 2715, 2721, 2719} \[ \int \left (a \sinh ^3(x)\right )^{3/2} \, dx=-\frac {14}{45} a \cosh (x) \sqrt {a \sinh ^3(x)}+\frac {2}{9} a \sinh ^2(x) \cosh (x) \sqrt {a \sinh ^3(x)}+\frac {14 i a \text {csch}(x) E\left (\left .\frac {\pi }{4}-\frac {i x}{2}\right |2\right ) \sqrt {a \sinh ^3(x)}}{15 \sqrt {i \sinh (x)}} \]
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Rule 2715
Rule 2719
Rule 2721
Rule 3286
Rubi steps \begin{align*} \text {integral}& = \frac {\left (a \sqrt {a \sinh ^3(x)}\right ) \int \sinh ^{\frac {9}{2}}(x) \, dx}{\sinh ^{\frac {3}{2}}(x)} \\ & = \frac {2}{9} a \cosh (x) \sinh ^2(x) \sqrt {a \sinh ^3(x)}-\frac {\left (7 a \sqrt {a \sinh ^3(x)}\right ) \int \sinh ^{\frac {5}{2}}(x) \, dx}{9 \sinh ^{\frac {3}{2}}(x)} \\ & = -\frac {14}{45} a \cosh (x) \sqrt {a \sinh ^3(x)}+\frac {2}{9} a \cosh (x) \sinh ^2(x) \sqrt {a \sinh ^3(x)}+\frac {\left (7 a \sqrt {a \sinh ^3(x)}\right ) \int \sqrt {\sinh (x)} \, dx}{15 \sinh ^{\frac {3}{2}}(x)} \\ & = -\frac {14}{45} a \cosh (x) \sqrt {a \sinh ^3(x)}+\frac {2}{9} a \cosh (x) \sinh ^2(x) \sqrt {a \sinh ^3(x)}+\frac {\left (7 a \text {csch}(x) \sqrt {a \sinh ^3(x)}\right ) \int \sqrt {i \sinh (x)} \, dx}{15 \sqrt {i \sinh (x)}} \\ & = -\frac {14}{45} a \cosh (x) \sqrt {a \sinh ^3(x)}+\frac {14 i a \text {csch}(x) E\left (\left .\frac {\pi }{4}-\frac {i x}{2}\right |2\right ) \sqrt {a \sinh ^3(x)}}{15 \sqrt {i \sinh (x)}}+\frac {2}{9} a \cosh (x) \sinh ^2(x) \sqrt {a \sinh ^3(x)} \\ \end{align*}
Time = 0.06 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.69 \[ \int \left (a \sinh ^3(x)\right )^{3/2} \, dx=\frac {1}{180} a \text {csch}(x) \sqrt {a \sinh ^3(x)} \left (168 \text {csch}(x) E\left (\left .\frac {1}{4} (\pi -2 i x)\right |2\right ) \sqrt {i \sinh (x)}-38 \sinh (2 x)+5 \sinh (4 x)\right ) \]
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\[\int \left (a \sinh \left (x \right )^{3}\right )^{\frac {3}{2}}d x\]
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.10 (sec) , antiderivative size = 317, normalized size of antiderivative = 3.82 \[ \int \left (a \sinh ^3(x)\right )^{3/2} \, dx=-\frac {336 \, {\left (\sqrt {2} a \cosh \left (x\right )^{4} + 4 \, \sqrt {2} a \cosh \left (x\right )^{3} \sinh \left (x\right ) + 6 \, \sqrt {2} a \cosh \left (x\right )^{2} \sinh \left (x\right )^{2} + 4 \, \sqrt {2} a \cosh \left (x\right ) \sinh \left (x\right )^{3} + \sqrt {2} a \sinh \left (x\right )^{4}\right )} \sqrt {a} {\rm weierstrassZeta}\left (4, 0, {\rm weierstrassPInverse}\left (4, 0, \cosh \left (x\right ) + \sinh \left (x\right )\right )\right ) - {\left (5 \, a \cosh \left (x\right )^{8} + 40 \, a \cosh \left (x\right ) \sinh \left (x\right )^{7} + 5 \, a \sinh \left (x\right )^{8} - 38 \, a \cosh \left (x\right )^{6} + 2 \, {\left (70 \, a \cosh \left (x\right )^{2} - 19 \, a\right )} \sinh \left (x\right )^{6} + 4 \, {\left (70 \, a \cosh \left (x\right )^{3} - 57 \, a \cosh \left (x\right )\right )} \sinh \left (x\right )^{5} - 336 \, a \cosh \left (x\right )^{4} + 2 \, {\left (175 \, a \cosh \left (x\right )^{4} - 285 \, a \cosh \left (x\right )^{2} - 168 \, a\right )} \sinh \left (x\right )^{4} + 8 \, {\left (35 \, a \cosh \left (x\right )^{5} - 95 \, a \cosh \left (x\right )^{3} - 168 \, a \cosh \left (x\right )\right )} \sinh \left (x\right )^{3} + 38 \, a \cosh \left (x\right )^{2} + 2 \, {\left (70 \, a \cosh \left (x\right )^{6} - 285 \, a \cosh \left (x\right )^{4} - 1008 \, a \cosh \left (x\right )^{2} + 19 \, a\right )} \sinh \left (x\right )^{2} + 4 \, {\left (10 \, a \cosh \left (x\right )^{7} - 57 \, a \cosh \left (x\right )^{5} - 336 \, a \cosh \left (x\right )^{3} + 19 \, a \cosh \left (x\right )\right )} \sinh \left (x\right ) - 5 \, a\right )} \sqrt {a \sinh \left (x\right )}}{360 \, {\left (\cosh \left (x\right )^{4} + 4 \, \cosh \left (x\right )^{3} \sinh \left (x\right ) + 6 \, \cosh \left (x\right )^{2} \sinh \left (x\right )^{2} + 4 \, \cosh \left (x\right ) \sinh \left (x\right )^{3} + \sinh \left (x\right )^{4}\right )}} \]
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\[ \int \left (a \sinh ^3(x)\right )^{3/2} \, dx=\int \left (a \sinh ^{3}{\left (x \right )}\right )^{\frac {3}{2}}\, dx \]
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\[ \int \left (a \sinh ^3(x)\right )^{3/2} \, dx=\int { \left (a \sinh \left (x\right )^{3}\right )^{\frac {3}{2}} \,d x } \]
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\[ \int \left (a \sinh ^3(x)\right )^{3/2} \, dx=\int { \left (a \sinh \left (x\right )^{3}\right )^{\frac {3}{2}} \,d x } \]
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Timed out. \[ \int \left (a \sinh ^3(x)\right )^{3/2} \, dx=\int {\left (a\,{\mathrm {sinh}\left (x\right )}^3\right )}^{3/2} \,d x \]
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