Optimal. Leaf size=12 \[ \frac{2 \cos (x)}{\sqrt{\cos ^3(x)}} \]
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Rubi [A] time = 0.027797, antiderivative size = 12, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273, Rules used = {3207, 2565, 30} \[ \frac{2 \cos (x)}{\sqrt{\cos ^3(x)}} \]
Antiderivative was successfully verified.
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Rule 3207
Rule 2565
Rule 30
Rubi steps
\begin{align*} \int \frac{\sin (x)}{\sqrt{\cos ^3(x)}} \, dx &=\frac{\cos ^{\frac{3}{2}}(x) \int \frac{\sin (x)}{\cos ^{\frac{3}{2}}(x)} \, dx}{\sqrt{\cos ^3(x)}}\\ &=-\frac{\cos ^{\frac{3}{2}}(x) \operatorname{Subst}\left (\int \frac{1}{x^{3/2}} \, dx,x,\cos (x)\right )}{\sqrt{\cos ^3(x)}}\\ &=\frac{2 \cos (x)}{\sqrt{\cos ^3(x)}}\\ \end{align*}
Mathematica [A] time = 0.0129631, size = 12, normalized size = 1. \[ \frac{2 \cos (x)}{\sqrt{\cos ^3(x)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.046, size = 11, normalized size = 0.9 \begin{align*} 2\,{\frac{\cos \left ( x \right ) }{\sqrt{ \left ( \cos \left ( x \right ) \right ) ^{3}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.930996, size = 14, normalized size = 1.17 \begin{align*} \frac{2 \, \cos \left (x\right )}{\sqrt{\cos \left (x\right )^{3}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.456553, size = 36, normalized size = 3. \begin{align*} \frac{2 \, \sqrt{\cos \left (x\right )^{3}}}{\cos \left (x\right )^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.615896, size = 12, normalized size = 1. \begin{align*} \frac{2 \cos{\left (x \right )}}{\sqrt{\cos ^{3}{\left (x \right )}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.09372, size = 8, normalized size = 0.67 \begin{align*} \frac{2}{\sqrt{\cos \left (x\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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