Optimal. Leaf size=73 \[ \frac{3}{5} \left (1-\sin \left (\frac{2 x}{3}\right )\right )^{3/2} \cos \left (\frac{2 x}{3}\right )+\frac{8}{5} \sqrt{1-\sin \left (\frac{2 x}{3}\right )} \cos \left (\frac{2 x}{3}\right )+\frac{32 \cos \left (\frac{2 x}{3}\right )}{5 \sqrt{1-\sin \left (\frac{2 x}{3}\right )}} \]
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Rubi [A] time = 0.0325273, antiderivative size = 73, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {2647, 2646} \[ \frac{3}{5} \left (1-\sin \left (\frac{2 x}{3}\right )\right )^{3/2} \cos \left (\frac{2 x}{3}\right )+\frac{8}{5} \sqrt{1-\sin \left (\frac{2 x}{3}\right )} \cos \left (\frac{2 x}{3}\right )+\frac{32 \cos \left (\frac{2 x}{3}\right )}{5 \sqrt{1-\sin \left (\frac{2 x}{3}\right )}} \]
Antiderivative was successfully verified.
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Rule 2647
Rule 2646
Rubi steps
\begin{align*} \int \left (1-\sin \left (\frac{2 x}{3}\right )\right )^{5/2} \, dx &=\frac{3}{5} \cos \left (\frac{2 x}{3}\right ) \left (1-\sin \left (\frac{2 x}{3}\right )\right )^{3/2}+\frac{8}{5} \int \left (1-\sin \left (\frac{2 x}{3}\right )\right )^{3/2} \, dx\\ &=\frac{8}{5} \cos \left (\frac{2 x}{3}\right ) \sqrt{1-\sin \left (\frac{2 x}{3}\right )}+\frac{3}{5} \cos \left (\frac{2 x}{3}\right ) \left (1-\sin \left (\frac{2 x}{3}\right )\right )^{3/2}+\frac{32}{15} \int \sqrt{1-\sin \left (\frac{2 x}{3}\right )} \, dx\\ &=\frac{32 \cos \left (\frac{2 x}{3}\right )}{5 \sqrt{1-\sin \left (\frac{2 x}{3}\right )}}+\frac{8}{5} \cos \left (\frac{2 x}{3}\right ) \sqrt{1-\sin \left (\frac{2 x}{3}\right )}+\frac{3}{5} \cos \left (\frac{2 x}{3}\right ) \left (1-\sin \left (\frac{2 x}{3}\right )\right )^{3/2}\\ \end{align*}
Mathematica [A] time = 0.166133, size = 76, normalized size = 1.04 \[ \frac{\left (1-\sin \left (\frac{2 x}{3}\right )\right )^{5/2} \left (150 \sin \left (\frac{x}{3}\right )-25 \sin (x)-3 \sin \left (\frac{5 x}{3}\right )+150 \cos \left (\frac{x}{3}\right )+25 \cos (x)-3 \cos \left (\frac{5 x}{3}\right )\right )}{20 \left (\cos \left (\frac{x}{3}\right )-\sin \left (\frac{x}{3}\right )\right )^5} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.037, size = 47, normalized size = 0.6 \begin{align*} -{\frac{1}{5} \left ( -1+\sin \left ({\frac{2\,x}{3}} \right ) \right ) \left ( 1+\sin \left ({\frac{2\,x}{3}} \right ) \right ) \left ( 3\, \left ( \sin \left ( 2/3\,x \right ) \right ) ^{2}-14\,\sin \left ( 2/3\,x \right ) +43 \right ) \left ( \cos \left ({\frac{2\,x}{3}} \right ) \right ) ^{-1}{\frac{1}{\sqrt{1-\sin \left ({\frac{2\,x}{3}} \right ) }}}} \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 (-\sin \left (\frac{2}{3} \, x\right ) + 1\right )}^{\frac{5}{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.88725, size = 223, normalized size = 3.05 \begin{align*} -\frac{{\left (3 \, \cos \left (\frac{2}{3} \, x\right )^{3} - 11 \, \cos \left (\frac{2}{3} \, x\right )^{2} +{\left (3 \, \cos \left (\frac{2}{3} \, x\right )^{2} + 14 \, \cos \left (\frac{2}{3} \, x\right ) - 32\right )} \sin \left (\frac{2}{3} \, x\right ) - 46 \, \cos \left (\frac{2}{3} \, x\right ) - 32\right )} \sqrt{-\sin \left (\frac{2}{3} \, x\right ) + 1}}{5 \,{\left (\cos \left (\frac{2}{3} \, x\right ) - \sin \left (\frac{2}{3} \, x\right ) + 1\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(-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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