Integrand size = 10, antiderivative size = 71 \[ \int \left (a \cosh ^3(x)\right )^{3/2} \, dx=-\frac {14 i a \sqrt {a \cosh ^3(x)} E\left (\left .\frac {i x}{2}\right |2\right )}{15 \cosh ^{\frac {3}{2}}(x)}+\frac {14}{45} a \sqrt {a \cosh ^3(x)} \sinh (x)+\frac {2}{9} a \cosh ^2(x) \sqrt {a \cosh ^3(x)} \sinh (x) \]
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Time = 0.03 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {3286, 2715, 2719} \[ \int \left (a \cosh ^3(x)\right )^{3/2} \, dx=\frac {14}{45} a \sinh (x) \sqrt {a \cosh ^3(x)}-\frac {14 i a E\left (\left .\frac {i x}{2}\right |2\right ) \sqrt {a \cosh ^3(x)}}{15 \cosh ^{\frac {3}{2}}(x)}+\frac {2}{9} a \sinh (x) \cosh ^2(x) \sqrt {a \cosh ^3(x)} \]
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Rule 2715
Rule 2719
Rule 3286
Rubi steps \begin{align*} \text {integral}& = \frac {\left (a \sqrt {a \cosh ^3(x)}\right ) \int \cosh ^{\frac {9}{2}}(x) \, dx}{\cosh ^{\frac {3}{2}}(x)} \\ & = \frac {2}{9} a \cosh ^2(x) \sqrt {a \cosh ^3(x)} \sinh (x)+\frac {\left (7 a \sqrt {a \cosh ^3(x)}\right ) \int \cosh ^{\frac {5}{2}}(x) \, dx}{9 \cosh ^{\frac {3}{2}}(x)} \\ & = \frac {14}{45} a \sqrt {a \cosh ^3(x)} \sinh (x)+\frac {2}{9} a \cosh ^2(x) \sqrt {a \cosh ^3(x)} \sinh (x)+\frac {\left (7 a \sqrt {a \cosh ^3(x)}\right ) \int \sqrt {\cosh (x)} \, dx}{15 \cosh ^{\frac {3}{2}}(x)} \\ & = -\frac {14 i a \sqrt {a \cosh ^3(x)} E\left (\left .\frac {i x}{2}\right |2\right )}{15 \cosh ^{\frac {3}{2}}(x)}+\frac {14}{45} a \sqrt {a \cosh ^3(x)} \sinh (x)+\frac {2}{9} a \cosh ^2(x) \sqrt {a \cosh ^3(x)} \sinh (x) \\ \end{align*}
Time = 0.09 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.76 \[ \int \left (a \cosh ^3(x)\right )^{3/2} \, dx=\frac {\left (a \cosh ^3(x)\right )^{3/2} \left (-168 i E\left (\left .\frac {i x}{2}\right |2\right )+\sqrt {\cosh (x)} (38 \sinh (2 x)+5 \sinh (4 x))\right )}{180 \cosh ^{\frac {9}{2}}(x)} \]
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\[\int \left (a \cosh \left (x \right )^{3}\right )^{\frac {3}{2}}d x\]
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.09 (sec) , antiderivative size = 317, normalized size of antiderivative = 4.46 \[ \int \left (a \cosh ^3(x)\right )^{3/2} \, dx=-\frac {336 \, {\left (\sqrt {2} a \cosh \left (x\right )^{4} + 4 \, \sqrt {2} a \cosh \left (x\right )^{3} \sinh \left (x\right ) + 6 \, \sqrt {2} a \cosh \left (x\right )^{2} \sinh \left (x\right )^{2} + 4 \, \sqrt {2} a \cosh \left (x\right ) \sinh \left (x\right )^{3} + \sqrt {2} a \sinh \left (x\right )^{4}\right )} \sqrt {a} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cosh \left (x\right ) + \sinh \left (x\right )\right )\right ) - {\left (5 \, a \cosh \left (x\right )^{8} + 40 \, a \cosh \left (x\right ) \sinh \left (x\right )^{7} + 5 \, a \sinh \left (x\right )^{8} + 38 \, a \cosh \left (x\right )^{6} + 2 \, {\left (70 \, a \cosh \left (x\right )^{2} + 19 \, a\right )} \sinh \left (x\right )^{6} + 4 \, {\left (70 \, a \cosh \left (x\right )^{3} + 57 \, a \cosh \left (x\right )\right )} \sinh \left (x\right )^{5} - 336 \, a \cosh \left (x\right )^{4} + 2 \, {\left (175 \, a \cosh \left (x\right )^{4} + 285 \, a \cosh \left (x\right )^{2} - 168 \, a\right )} \sinh \left (x\right )^{4} + 8 \, {\left (35 \, a \cosh \left (x\right )^{5} + 95 \, a \cosh \left (x\right )^{3} - 168 \, a \cosh \left (x\right )\right )} \sinh \left (x\right )^{3} - 38 \, a \cosh \left (x\right )^{2} + 2 \, {\left (70 \, a \cosh \left (x\right )^{6} + 285 \, a \cosh \left (x\right )^{4} - 1008 \, a \cosh \left (x\right )^{2} - 19 \, a\right )} \sinh \left (x\right )^{2} + 4 \, {\left (10 \, a \cosh \left (x\right )^{7} + 57 \, a \cosh \left (x\right )^{5} - 336 \, a \cosh \left (x\right )^{3} - 19 \, a \cosh \left (x\right )\right )} \sinh \left (x\right ) - 5 \, a\right )} \sqrt {a \cosh \left (x\right )}}{360 \, {\left (\cosh \left (x\right )^{4} + 4 \, \cosh \left (x\right )^{3} \sinh \left (x\right ) + 6 \, \cosh \left (x\right )^{2} \sinh \left (x\right )^{2} + 4 \, \cosh \left (x\right ) \sinh \left (x\right )^{3} + \sinh \left (x\right )^{4}\right )}} \]
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Timed out. \[ \int \left (a \cosh ^3(x)\right )^{3/2} \, dx=\text {Timed out} \]
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\[ \int \left (a \cosh ^3(x)\right )^{3/2} \, dx=\int { \left (a \cosh \left (x\right )^{3}\right )^{\frac {3}{2}} \,d x } \]
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\[ \int \left (a \cosh ^3(x)\right )^{3/2} \, dx=\int { \left (a \cosh \left (x\right )^{3}\right )^{\frac {3}{2}} \,d x } \]
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Timed out. \[ \int \left (a \cosh ^3(x)\right )^{3/2} \, dx=\int {\left (a\,{\mathrm {cosh}\left (x\right )}^3\right )}^{3/2} \,d x \]
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