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 = 0.0176481, antiderivative size = 50, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, 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 3207
Rule 2635
Rule 2641
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*}
Mathematica [A] time = 0.0305087, 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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Maple [C] time = 0.271, size = 118, normalized size = 2.4 \begin{align*} -{\frac{\sqrt{8}}{6\,\sin \left ( x \right ) \left ( -1+\cos \left ( x \right ) \right ) } \left ( i\sqrt{{\frac{-i \left ( -1+\cos \left ( x \right ) \right ) }{\sin \left ( x \right ) }}}\sin \left ( x \right ) \sqrt{-{\frac{i\cos \left ( x \right ) -\sin \left ( x \right ) -i}{\sin \left ( x \right ) }}}{\it EllipticF} \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\cos \left ( x \right ) +\sin \left ( x \right ) -i}{\sin \left ( x \right ) }}}+ \left ( \cos \left ( x \right ) \right ) ^{2}\sqrt{2}-\cos \left ( x \right ) \sqrt{2} \right ) \sqrt{a \left ( \sin \left ( x \right ) \right ) ^{3}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{a \sin \left (x\right )^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\sqrt{-{\left (a \cos \left (x\right )^{2} - a\right )} \sin \left (x\right )}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{a \sin ^{3}{\left (x \right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{a \sin \left (x\right )^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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