Optimal. Leaf size=21 \[ \frac{2}{3} \cos ^3\left (\sqrt{x}\right )-2 \cos \left (\sqrt{x}\right ) \]
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Rubi [A] time = 0.0204478, antiderivative size = 21, 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 = {3379, 2633} \[ \frac{2}{3} \cos ^3\left (\sqrt{x}\right )-2 \cos \left (\sqrt{x}\right ) \]
Antiderivative was successfully verified.
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Rule 3379
Rule 2633
Rubi steps
\begin{align*} \int \frac{\sin ^3\left (\sqrt{x}\right )}{\sqrt{x}} \, dx &=2 \operatorname{Subst}\left (\int \sin ^3(x) \, dx,x,\sqrt{x}\right )\\ &=-\left (2 \operatorname{Subst}\left (\int \left (1-x^2\right ) \, dx,x,\cos \left (\sqrt{x}\right )\right )\right )\\ &=-2 \cos \left (\sqrt{x}\right )+\frac{2}{3} \cos ^3\left (\sqrt{x}\right )\\ \end{align*}
Mathematica [A] time = 0.0231795, size = 23, normalized size = 1.1 \[ \frac{1}{6} \cos \left (3 \sqrt{x}\right )-\frac{3 \cos \left (\sqrt{x}\right )}{2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.009, size = 15, normalized size = 0.7 \begin{align*} -{\frac{2}{3} \left ( 2+ \left ( \sin \left ( \sqrt{x} \right ) \right ) ^{2} \right ) \cos \left ( \sqrt{x} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.940897, size = 20, normalized size = 0.95 \begin{align*} \frac{2}{3} \, \cos \left (\sqrt{x}\right )^{3} - 2 \, \cos \left (\sqrt{x}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.65846, size = 50, normalized size = 2.38 \begin{align*} \frac{2}{3} \, \cos \left (\sqrt{x}\right )^{3} - 2 \, \cos \left (\sqrt{x}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.902545, size = 29, normalized size = 1.38 \begin{align*} - 2 \sin ^{2}{\left (\sqrt{x} \right )} \cos{\left (\sqrt{x} \right )} - \frac{4 \cos ^{3}{\left (\sqrt{x} \right )}}{3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.09923, size = 20, normalized size = 0.95 \begin{align*} \frac{2}{3} \, \cos \left (\sqrt{x}\right )^{3} - 2 \, \cos \left (\sqrt{x}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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