Optimal. Leaf size=73 \[ -\frac{2}{9} a \sin ^2(x) \cos (x) \sqrt{a \sin ^3(x)}-\frac{14}{45} a \cos (x) \sqrt{a \sin ^3(x)}-\frac{14 a E\left (\left .\frac{\pi }{4}-\frac{x}{2}\right |2\right ) \sqrt{a \sin ^3(x)}}{15 \sin ^{\frac{3}{2}}(x)} \]
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Rubi [A] time = 0.0247842, antiderivative size = 73, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, 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 3207
Rule 2635
Rule 2639
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*}
Mathematica [A] time = 0.100251, 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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Maple [C] time = 0.213, size = 337, normalized size = 4.6 \begin{align*} -{\frac{1}{45\, \left ( \sin \left ( x \right ) \right ) ^{5}} \left ( 42\,\sqrt{2}\cos \left ( x \right ) \sqrt{{\frac{-i\cos \left ( x \right ) +\sin \left ( x \right ) +i}{\sin \left ( x \right ) }}}{\it EllipticE} \left ( \sqrt{{\frac{i\cos \left ( x \right ) +\sin \left ( x \right ) -i}{\sin \left ( x \right ) }}},1/2\,\sqrt{2} \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 ) }}}-21\,\sqrt{2}\cos \left ( x \right ) \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 ) }}}{\it EllipticF} \left ( \sqrt{{\frac{i\cos \left ( x \right ) +\sin \left ( x \right ) -i}{\sin \left ( x \right ) }}},1/2\,\sqrt{2} \right ) +42\,\sqrt{2}\sqrt{{\frac{-i\cos \left ( x \right ) +\sin \left ( x \right ) +i}{\sin \left ( x \right ) }}}{\it EllipticE} \left ( \sqrt{{\frac{i\cos \left ( x \right ) +\sin \left ( x \right ) -i}{\sin \left ( x \right ) }}},1/2\,\sqrt{2} \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 ) }}}-21\,\sqrt{2}\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 ) }}}{\it EllipticF} \left ( \sqrt{{\frac{i\cos \left ( x \right ) +\sin \left ( x \right ) -i}{\sin \left ( x \right ) }}},1/2\,\sqrt{2} \right ) +10\, \left ( \cos \left ( x \right ) \right ) ^{5}-34\, \left ( \cos \left ( x \right ) \right ) ^{3}+66\,\cos \left ( x \right ) -42 \right ) \left ( a \left ( \sin \left ( x \right ) \right ) ^{3} \right ) ^{{\frac{3}{2}}}} \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 \left (a \sin \left (x\right )^{3}\right )^{\frac{3}{2}}\,{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 (-{\left (a \cos \left (x\right )^{2} - a\right )} \sqrt{-{\left (a \cos \left (x\right )^{2} - a\right )} \sin \left (x\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(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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 \left (a \sin \left (x\right )^{3}\right )^{\frac{3}{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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