Optimal. Leaf size=83 \[ -\frac {14}{45} a \cosh (x) \sqrt {a \sinh ^3(x)}+\frac {2}{9} a \sinh ^2(x) \cosh (x) \sqrt {a \sinh ^3(x)}+\frac {14 i a \text {csch}(x) E\left (\left .\frac {\pi }{4}-\frac {i x}{2}\right |2\right ) \sqrt {a \sinh ^3(x)}}{15 \sqrt {i \sinh (x)}} \]
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Rubi [A] time = 0.04, antiderivative size = 83, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {3207, 2635, 2640, 2639} \[ \frac {2}{9} a \sinh ^2(x) \cosh (x) \sqrt {a \sinh ^3(x)}-\frac {14}{45} a \cosh (x) \sqrt {a \sinh ^3(x)}+\frac {14 i a \text {csch}(x) E\left (\left .\frac {\pi }{4}-\frac {i x}{2}\right |2\right ) \sqrt {a \sinh ^3(x)}}{15 \sqrt {i \sinh (x)}} \]
Antiderivative was successfully verified.
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Rule 2635
Rule 2639
Rule 2640
Rule 3207
Rubi steps
\begin {align*} \int \left (a \sinh ^3(x)\right )^{3/2} \, dx &=\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*}
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Mathematica [A] time = 0.06, size = 57, normalized size = 0.69 \[ \frac {1}{180} a \text {csch}(x) \sqrt {a \sinh ^3(x)} \left (-38 \sinh (2 x)+5 \sinh (4 x)+168 \sqrt {i \sinh (x)} \text {csch}(x) E\left (\left .\frac {1}{4} (\pi -2 i x)\right |2\right )\right ) \]
Antiderivative was successfully verified.
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fricas [F] time = 0.47, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\sqrt {a \sinh \relax (x)^{3}} a \sinh \relax (x)^{3}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (a \sinh \relax (x)^{3}\right )^{\frac {3}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.09, size = 0, normalized size = 0.00 \[ \int \left (a \left (\sinh ^{3}\relax (x )\right )\right )^{\frac {3}{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (a \sinh \relax (x)^{3}\right )^{\frac {3}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int {\left (a\,{\mathrm {sinh}\relax (x)}^3\right )}^{3/2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (a \sinh ^{3}{\relax (x )}\right )^{\frac {3}{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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