Optimal. Leaf size=73 \[ -\frac {14}{45} a \cos (x) \sqrt {a \sin ^3(x)}-\frac {2}{9} a \sin ^2(x) \cos (x) \sqrt {a \sin ^3(x)}-\frac {14 a E\left (\left .\frac {\pi }{4}-\frac {x}{2}\right |2\right ) \sqrt {a \sin ^3(x)}}{15 \sin ^{\frac {3}{2}}(x)} \]
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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 = {3207, 2635, 2639} \[ -\frac {2}{9} a \sin ^2(x) \cos (x) \sqrt {a \sin ^3(x)}-\frac {14}{45} a \cos (x) \sqrt {a \sin ^3(x)}-\frac {14 a E\left (\left .\frac {\pi }{4}-\frac {x}{2}\right |2\right ) \sqrt {a \sin ^3(x)}}{15 \sin ^{\frac {3}{2}}(x)} \]
Antiderivative was successfully verified.
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Rule 2635
Rule 2639
Rule 3207
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 = 0.10, size = 54, normalized size = 0.74 \[ \frac {\left (a \sin ^3(x)\right )^{3/2} \left (\sqrt {\sin (x)} (5 \sin (4 x)-38 \sin (2 x))-168 E\left (\left .\frac {1}{4} (\pi -2 x)\right |2\right )\right )}{180 \sin ^{\frac {9}{2}}(x)} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.48, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-{\left (a \cos \relax (x)^{2} - a\right )} \sqrt {-{\left (a \cos \relax (x)^{2} - a\right )} \sin \relax (x)} \sin \relax (x), x\right ) \]
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 \[ \int \left (a \sin \relax (x)^{3}\right )^{\frac {3}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.51, size = 337, normalized size = 4.62 \[ \frac {\left (21 \sqrt {\frac {i \cos \relax (x )+\sin \relax (x )-i}{\sin \relax (x )}}\, \sqrt {\frac {-i \cos \relax (x )+\sin \relax (x )+i}{\sin \relax (x )}}\, \sqrt {-\frac {i \left (-1+\cos \relax (x )\right )}{\sin \relax (x )}}\, \EllipticF \left (\sqrt {\frac {i \cos \relax (x )+\sin \relax (x )-i}{\sin \relax (x )}}, \frac {\sqrt {2}}{2}\right ) \sqrt {2}\, \cos \relax (x )-42 \sqrt {\frac {i \cos \relax (x )+\sin \relax (x )-i}{\sin \relax (x )}}\, \sqrt {\frac {-i \cos \relax (x )+\sin \relax (x )+i}{\sin \relax (x )}}\, \sqrt {-\frac {i \left (-1+\cos \relax (x )\right )}{\sin \relax (x )}}\, \EllipticE \left (\sqrt {\frac {i \cos \relax (x )+\sin \relax (x )-i}{\sin \relax (x )}}, \frac {\sqrt {2}}{2}\right ) \sqrt {2}\, \cos \relax (x )+21 \sqrt {\frac {i \cos \relax (x )+\sin \relax (x )-i}{\sin \relax (x )}}\, \sqrt {\frac {-i \cos \relax (x )+\sin \relax (x )+i}{\sin \relax (x )}}\, \sqrt {-\frac {i \left (-1+\cos \relax (x )\right )}{\sin \relax (x )}}\, \EllipticF \left (\sqrt {\frac {i \cos \relax (x )+\sin \relax (x )-i}{\sin \relax (x )}}, \frac {\sqrt {2}}{2}\right ) \sqrt {2}-42 \sqrt {\frac {i \cos \relax (x )+\sin \relax (x )-i}{\sin \relax (x )}}\, \sqrt {\frac {-i \cos \relax (x )+\sin \relax (x )+i}{\sin \relax (x )}}\, \sqrt {-\frac {i \left (-1+\cos \relax (x )\right )}{\sin \relax (x )}}\, \EllipticE \left (\sqrt {\frac {i \cos \relax (x )+\sin \relax (x )-i}{\sin \relax (x )}}, \frac {\sqrt {2}}{2}\right ) \sqrt {2}-10 \left (\cos ^{5}\relax (x )\right )+34 \left (\cos ^{3}\relax (x )\right )-66 \cos \relax (x )+42\right ) \left (a \left (\sin ^{3}\relax (x )\right )\right )^{\frac {3}{2}}}{45 \sin \relax (x )^{5}} \]
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 \[ \int \left (a \sin \relax (x)^{3}\right )^{\frac {3}{2}}\,{d x} \]
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 \[ \int {\left (a\,{\sin \relax (x)}^3\right )}^{3/2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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