Integrand size = 10, antiderivative size = 135 \[ \int \left (a \sinh ^3(x)\right )^{5/2} \, dx=-\frac {26}{77} a^2 \coth (x) \sqrt {a \sinh ^3(x)}+\frac {26}{77} i a^2 \text {csch}^2(x) \operatorname {EllipticF}\left (\frac {\pi }{4}-\frac {i x}{2},2\right ) \sqrt {i \sinh (x)} \sqrt {a \sinh ^3(x)}+\frac {78}{385} a^2 \cosh (x) \sinh (x) \sqrt {a \sinh ^3(x)}-\frac {26}{165} a^2 \cosh (x) \sinh ^3(x) \sqrt {a \sinh ^3(x)}+\frac {2}{15} a^2 \cosh (x) \sinh ^5(x) \sqrt {a \sinh ^3(x)} \]
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Time = 0.04 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {3286, 2715, 2721, 2720} \[ \int \left (a \sinh ^3(x)\right )^{5/2} \, dx=-\frac {26}{165} a^2 \sinh ^3(x) \cosh (x) \sqrt {a \sinh ^3(x)}+\frac {78}{385} a^2 \sinh (x) \cosh (x) \sqrt {a \sinh ^3(x)}+\frac {2}{15} a^2 \sinh ^5(x) \cosh (x) \sqrt {a \sinh ^3(x)}-\frac {26}{77} a^2 \coth (x) \sqrt {a \sinh ^3(x)}+\frac {26}{77} i a^2 \sqrt {i \sinh (x)} \text {csch}^2(x) \operatorname {EllipticF}\left (\frac {\pi }{4}-\frac {i x}{2},2\right ) \sqrt {a \sinh ^3(x)} \]
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Rule 2715
Rule 2720
Rule 2721
Rule 3286
Rubi steps \begin{align*} \text {integral}& = \frac {\left (a^2 \sqrt {a \sinh ^3(x)}\right ) \int \sinh ^{\frac {15}{2}}(x) \, dx}{\sinh ^{\frac {3}{2}}(x)} \\ & = \frac {2}{15} a^2 \cosh (x) \sinh ^5(x) \sqrt {a \sinh ^3(x)}-\frac {\left (13 a^2 \sqrt {a \sinh ^3(x)}\right ) \int \sinh ^{\frac {11}{2}}(x) \, dx}{15 \sinh ^{\frac {3}{2}}(x)} \\ & = -\frac {26}{165} a^2 \cosh (x) \sinh ^3(x) \sqrt {a \sinh ^3(x)}+\frac {2}{15} a^2 \cosh (x) \sinh ^5(x) \sqrt {a \sinh ^3(x)}+\frac {\left (39 a^2 \sqrt {a \sinh ^3(x)}\right ) \int \sinh ^{\frac {7}{2}}(x) \, dx}{55 \sinh ^{\frac {3}{2}}(x)} \\ & = \frac {78}{385} a^2 \cosh (x) \sinh (x) \sqrt {a \sinh ^3(x)}-\frac {26}{165} a^2 \cosh (x) \sinh ^3(x) \sqrt {a \sinh ^3(x)}+\frac {2}{15} a^2 \cosh (x) \sinh ^5(x) \sqrt {a \sinh ^3(x)}-\frac {\left (39 a^2 \sqrt {a \sinh ^3(x)}\right ) \int \sinh ^{\frac {3}{2}}(x) \, dx}{77 \sinh ^{\frac {3}{2}}(x)} \\ & = -\frac {26}{77} a^2 \coth (x) \sqrt {a \sinh ^3(x)}+\frac {78}{385} a^2 \cosh (x) \sinh (x) \sqrt {a \sinh ^3(x)}-\frac {26}{165} a^2 \cosh (x) \sinh ^3(x) \sqrt {a \sinh ^3(x)}+\frac {2}{15} a^2 \cosh (x) \sinh ^5(x) \sqrt {a \sinh ^3(x)}+\frac {\left (13 a^2 \sqrt {a \sinh ^3(x)}\right ) \int \frac {1}{\sqrt {\sinh (x)}} \, dx}{77 \sinh ^{\frac {3}{2}}(x)} \\ & = -\frac {26}{77} a^2 \coth (x) \sqrt {a \sinh ^3(x)}+\frac {78}{385} a^2 \cosh (x) \sinh (x) \sqrt {a \sinh ^3(x)}-\frac {26}{165} a^2 \cosh (x) \sinh ^3(x) \sqrt {a \sinh ^3(x)}+\frac {2}{15} a^2 \cosh (x) \sinh ^5(x) \sqrt {a \sinh ^3(x)}+\frac {1}{77} \left (13 a^2 \text {csch}^2(x) \sqrt {i \sinh (x)} \sqrt {a \sinh ^3(x)}\right ) \int \frac {1}{\sqrt {i \sinh (x)}} \, dx \\ & = -\frac {26}{77} a^2 \coth (x) \sqrt {a \sinh ^3(x)}+\frac {26}{77} i a^2 \text {csch}^2(x) \operatorname {EllipticF}\left (\frac {\pi }{4}-\frac {i x}{2},2\right ) \sqrt {i \sinh (x)} \sqrt {a \sinh ^3(x)}+\frac {78}{385} a^2 \cosh (x) \sinh (x) \sqrt {a \sinh ^3(x)}-\frac {26}{165} a^2 \cosh (x) \sinh ^3(x) \sqrt {a \sinh ^3(x)}+\frac {2}{15} a^2 \cosh (x) \sinh ^5(x) \sqrt {a \sinh ^3(x)} \\ \end{align*}
Time = 0.16 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.50 \[ \int \left (a \sinh ^3(x)\right )^{5/2} \, dx=\frac {a^2 \text {csch}(x) \left (-15465 \cosh (x)+3657 \cosh (3 x)-749 \cosh (5 x)+77 \cosh (7 x)-\frac {12480 \operatorname {EllipticF}\left (\frac {1}{4} (\pi -2 i x),2\right )}{\sqrt {i \sinh (x)}}\right ) \sqrt {a \sinh ^3(x)}}{36960} \]
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\[\int \left (a \sinh \left (x \right )^{3}\right )^{\frac {5}{2}}d x\]
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.11 (sec) , antiderivative size = 823, normalized size of antiderivative = 6.10 \[ \int \left (a \sinh ^3(x)\right )^{5/2} \, dx=\text {Too large to display} \]
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\[ \int \left (a \sinh ^3(x)\right )^{5/2} \, dx=\int \left (a \sinh ^{3}{\left (x \right )}\right )^{\frac {5}{2}}\, dx \]
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\[ \int \left (a \sinh ^3(x)\right )^{5/2} \, dx=\int { \left (a \sinh \left (x\right )^{3}\right )^{\frac {5}{2}} \,d x } \]
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\[ \int \left (a \sinh ^3(x)\right )^{5/2} \, dx=\int { \left (a \sinh \left (x\right )^{3}\right )^{\frac {5}{2}} \,d x } \]
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Timed out. \[ \int \left (a \sinh ^3(x)\right )^{5/2} \, dx=\int {\left (a\,{\mathrm {sinh}\left (x\right )}^3\right )}^{5/2} \,d x \]
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