Optimal. Leaf size=73 \[ -\frac {14}{45} a \cos (x) \sqrt {a \sin ^3(x)}-\frac {14 a E\left (\left .\frac {\pi }{4}-\frac {x}{2}\right |2\right ) \sqrt {a \sin ^3(x)}}{15 \sin ^{\frac {3}{2}}(x)}-\frac {2}{9} a \cos (x) \sin ^2(x) \sqrt {a \sin ^3(x)} \]
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Rubi [A]
time = 0.02, antiderivative size = 73, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {3286, 2715,
2719} \begin {gather*} -\frac {14}{45} a \cos (x) \sqrt {a \sin ^3(x)}-\frac {2}{9} a \sin ^2(x) \cos (x) \sqrt {a \sin ^3(x)}-\frac {14 a E\left (\left .\frac {\pi }{4}-\frac {x}{2}\right |2\right ) \sqrt {a \sin ^3(x)}}{15 \sin ^{\frac {3}{2}}(x)} \end {gather*}
Antiderivative was successfully verified.
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Rule 2715
Rule 2719
Rule 3286
Rubi steps
\begin {align*} \int \left (a \sin ^3(x)\right )^{3/2} \, dx &=\frac {\left (a \sqrt {a \sin ^3(x)}\right ) \int \sin ^{\frac {9}{2}}(x) \, dx}{\sin ^{\frac {3}{2}}(x)}\\ &=-\frac {2}{9} a \cos (x) \sin ^2(x) \sqrt {a \sin ^3(x)}+\frac {\left (7 a \sqrt {a \sin ^3(x)}\right ) \int \sin ^{\frac {5}{2}}(x) \, dx}{9 \sin ^{\frac {3}{2}}(x)}\\ &=-\frac {14}{45} a \cos (x) \sqrt {a \sin ^3(x)}-\frac {2}{9} a \cos (x) \sin ^2(x) \sqrt {a \sin ^3(x)}+\frac {\left (7 a \sqrt {a \sin ^3(x)}\right ) \int \sqrt {\sin (x)} \, dx}{15 \sin ^{\frac {3}{2}}(x)}\\ &=-\frac {14}{45} a \cos (x) \sqrt {a \sin ^3(x)}-\frac {14 a E\left (\left .\frac {\pi }{4}-\frac {x}{2}\right |2\right ) \sqrt {a \sin ^3(x)}}{15 \sin ^{\frac {3}{2}}(x)}-\frac {2}{9} a \cos (x) \sin ^2(x) \sqrt {a \sin ^3(x)}\\ \end {align*}
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Mathematica [A]
time = 0.07, size = 54, normalized size = 0.74 \begin {gather*} \frac {\left (a \sin ^3(x)\right )^{3/2} \left (-168 E\left (\left .\frac {1}{4} (\pi -2 x)\right |2\right )+\sqrt {\sin (x)} (-38 \sin (2 x)+5 \sin (4 x))\right )}{180 \sin ^{\frac {9}{2}}(x)} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains complex when optimal does not.
time = 0.34, size = 337, normalized size = 4.62
method | result | size |
default | \(-\frac {\left (42 \sqrt {\frac {i \cos \left (x \right )+\sin \left (x \right )-i}{\sin \left (x \right )}}\, \sqrt {\frac {-i \cos \left (x \right )+\sin \left (x \right )+i}{\sin \left (x \right )}}\, \EllipticE \left (\sqrt {\frac {i \cos \left (x \right )+\sin \left (x \right )-i}{\sin \left (x \right )}}, \frac {\sqrt {2}}{2}\right ) \sqrt {-\frac {i \left (-1+\cos \left (x \right )\right )}{\sin \left (x \right )}}\, \sqrt {2}\, \cos \left (x \right )-21 \sqrt {\frac {i \cos \left (x \right )+\sin \left (x \right )-i}{\sin \left (x \right )}}\, \sqrt {\frac {-i \cos \left (x \right )+\sin \left (x \right )+i}{\sin \left (x \right )}}\, \sqrt {-\frac {i \left (-1+\cos \left (x \right )\right )}{\sin \left (x \right )}}\, \EllipticF \left (\sqrt {\frac {i \cos \left (x \right )+\sin \left (x \right )-i}{\sin \left (x \right )}}, \frac {\sqrt {2}}{2}\right ) \sqrt {2}\, \cos \left (x \right )+42 \sqrt {\frac {i \cos \left (x \right )+\sin \left (x \right )-i}{\sin \left (x \right )}}\, \sqrt {\frac {-i \cos \left (x \right )+\sin \left (x \right )+i}{\sin \left (x \right )}}\, \EllipticE \left (\sqrt {\frac {i \cos \left (x \right )+\sin \left (x \right )-i}{\sin \left (x \right )}}, \frac {\sqrt {2}}{2}\right ) \sqrt {-\frac {i \left (-1+\cos \left (x \right )\right )}{\sin \left (x \right )}}\, \sqrt {2}-21 \sqrt {\frac {i \cos \left (x \right )+\sin \left (x \right )-i}{\sin \left (x \right )}}\, \sqrt {\frac {-i \cos \left (x \right )+\sin \left (x \right )+i}{\sin \left (x \right )}}\, \sqrt {-\frac {i \left (-1+\cos \left (x \right )\right )}{\sin \left (x \right )}}\, \EllipticF \left (\sqrt {\frac {i \cos \left (x \right )+\sin \left (x \right )-i}{\sin \left (x \right )}}, \frac {\sqrt {2}}{2}\right ) \sqrt {2}+10 \left (\cos ^{5}\left (x \right )\right )-34 \left (\cos ^{3}\left (x \right )\right )+66 \cos \left (x \right )-42\right ) \left (a \left (\sin ^{3}\left (x \right )\right )\right )^{\frac {3}{2}}}{45 \sin \left (x \right )^{5}}\) | \(337\) |
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order
4.
time = 0.10, size = 80, normalized size = 1.10 \begin {gather*} \frac {7}{15} i \, \sqrt {2} \sqrt {-i \, a} a {\rm weierstrassZeta}\left (4, 0, {\rm weierstrassPInverse}\left (4, 0, \cos \left (x\right ) + i \, \sin \left (x\right )\right )\right ) - \frac {7}{15} i \, \sqrt {2} \sqrt {i \, a} a {\rm weierstrassZeta}\left (4, 0, {\rm weierstrassPInverse}\left (4, 0, \cos \left (x\right ) - i \, \sin \left (x\right )\right )\right ) + \frac {2}{45} \, {\left (5 \, a \cos \left (x\right )^{3} - 12 \, a \cos \left (x\right )\right )} \sqrt {-{\left (a \cos \left (x\right )^{2} - a\right )} \sin \left (x\right )} \end {gather*}
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 \begin {gather*} \int \left (a \sin ^{3}{\left (x \right )}\right )^{\frac {3}{2}}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int {\left (a\,{\sin \left (x\right )}^3\right )}^{3/2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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