Optimal. Leaf size=50 \[ -\frac {2}{3} \cot (x) \sqrt {a \sin ^3(x)}-\frac {2 F\left (\left .\frac {\pi }{4}-\frac {x}{2}\right |2\right ) \sqrt {a \sin ^3(x)}}{3 \sin ^{\frac {3}{2}}(x)} \]
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Rubi [A] time = 0.02, antiderivative size = 50, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {3207, 2635, 2641} \[ -\frac {2}{3} \cot (x) \sqrt {a \sin ^3(x)}-\frac {2 F\left (\left .\frac {\pi }{4}-\frac {x}{2}\right |2\right ) \sqrt {a \sin ^3(x)}}{3 \sin ^{\frac {3}{2}}(x)} \]
Antiderivative was successfully verified.
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Rule 2635
Rule 2641
Rule 3207
Rubi steps
\begin {align*} \int \sqrt {a \sin ^3(x)} \, dx &=\frac {\sqrt {a \sin ^3(x)} \int \sin ^{\frac {3}{2}}(x) \, dx}{\sin ^{\frac {3}{2}}(x)}\\ &=-\frac {2}{3} \cot (x) \sqrt {a \sin ^3(x)}+\frac {\sqrt {a \sin ^3(x)} \int \frac {1}{\sqrt {\sin (x)}} \, dx}{3 \sin ^{\frac {3}{2}}(x)}\\ &=-\frac {2}{3} \cot (x) \sqrt {a \sin ^3(x)}-\frac {2 F\left (\left .\frac {\pi }{4}-\frac {x}{2}\right |2\right ) \sqrt {a \sin ^3(x)}}{3 \sin ^{\frac {3}{2}}(x)}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 41, normalized size = 0.82 \[ -\frac {2 \sqrt {a \sin ^3(x)} \left (F\left (\left .\frac {1}{4} (\pi -2 x)\right |2\right )+\sqrt {\sin (x)} \cos (x)\right )}{3 \sin ^{\frac {3}{2}}(x)} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.44, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\sqrt {-{\left (a \cos \relax (x)^{2} - a\right )} \sin \relax (x)}, x\right ) \]
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 \[ \int \sqrt {a \sin \relax (x)^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.48, size = 124, normalized size = 2.48 \[ -\frac {\left (i \sqrt {-\frac {i \left (-1+\cos \relax (x )\right )}{\sin \relax (x )}}\, \sin \relax (x ) \sqrt {\frac {i \cos \relax (x )+\sin \relax (x )-i}{\sin \relax (x )}}\, \sqrt {-\frac {i \cos \relax (x )-\sin \relax (x )-i}{\sin \relax (x )}}\, \EllipticF \left (\sqrt {\frac {i \cos \relax (x )+\sin \relax (x )-i}{\sin \relax (x )}}, \frac {\sqrt {2}}{2}\right )+\left (\cos ^{2}\relax (x )\right ) \sqrt {2}-\cos \relax (x ) \sqrt {2}\right ) \sqrt {a \left (1-\left (\cos ^{2}\relax (x )\right )\right ) \sin \relax (x )}\, \sqrt {8}}{6 \sin \relax (x ) \left (-1+\cos \relax (x )\right )} \]
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 \[ \int \sqrt {a \sin \relax (x)^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \sqrt {a\,{\sin \relax (x)}^3} \,d x \]
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 \[ \int \sqrt {a \sin ^{3}{\relax (x )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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