Optimal. Leaf size=21 \[ \frac{2}{7} \sin ^{\frac{7}{2}}(x)-\frac{2}{11} \sin ^{\frac{11}{2}}(x) \]
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Rubi [A] time = 0.0249217, antiderivative size = 21, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {2564, 14} \[ \frac{2}{7} \sin ^{\frac{7}{2}}(x)-\frac{2}{11} \sin ^{\frac{11}{2}}(x) \]
Antiderivative was successfully verified.
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Rule 2564
Rule 14
Rubi steps
\begin{align*} \int \cos ^3(x) \sin ^{\frac{5}{2}}(x) \, dx &=\operatorname{Subst}\left (\int x^{5/2} \left (1-x^2\right ) \, dx,x,\sin (x)\right )\\ &=\operatorname{Subst}\left (\int \left (x^{5/2}-x^{9/2}\right ) \, dx,x,\sin (x)\right )\\ &=\frac{2}{7} \sin ^{\frac{7}{2}}(x)-\frac{2}{11} \sin ^{\frac{11}{2}}(x)\\ \end{align*}
Mathematica [A] time = 0.0115088, size = 18, normalized size = 0.86 \[ \frac{1}{77} \sin ^{\frac{7}{2}}(x) (7 \cos (2 x)+15) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.036, size = 14, normalized size = 0.7 \begin{align*}{\frac{2}{7} \left ( \sin \left ( x \right ) \right ) ^{{\frac{7}{2}}}}-{\frac{2}{11} \left ( \sin \left ( x \right ) \right ) ^{{\frac{11}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.963436, size = 18, normalized size = 0.86 \begin{align*} -\frac{2}{11} \, \sin \left (x\right )^{\frac{11}{2}} + \frac{2}{7} \, \sin \left (x\right )^{\frac{7}{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.34417, size = 68, normalized size = 3.24 \begin{align*} -\frac{2}{77} \,{\left (7 \, \cos \left (x\right )^{4} - 3 \, \cos \left (x\right )^{2} - 4\right )} \sin \left (x\right )^{\frac{3}{2}} \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 [A] time = 1.09783, size = 18, normalized size = 0.86 \begin{align*} -\frac{2}{11} \, \sin \left (x\right )^{\frac{11}{2}} + \frac{2}{7} \, \sin \left (x\right )^{\frac{7}{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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