Optimal. Leaf size=16 \[ -\frac{\cos (3 x)}{3 \cos ^{\frac{3}{2}}(2 x)} \]
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Rubi [A] time = 0.0134062, antiderivative size = 16, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {4331} \[ -\frac{\cos (3 x)}{3 \cos ^{\frac{3}{2}}(2 x)} \]
Antiderivative was successfully verified.
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Rule 4331
Rubi steps
\begin{align*} \int \frac{\sin (x)}{\cos ^{\frac{5}{2}}(2 x)} \, dx &=-\frac{\cos (3 x)}{3 \cos ^{\frac{3}{2}}(2 x)}\\ \end{align*}
Mathematica [A] time = 0.0357061, size = 16, normalized size = 1. \[ -\frac{\cos (3 x)}{3 \cos ^{\frac{3}{2}}(2 x)} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.075, size = 39, normalized size = 2.4 \begin{align*}{\frac{\cos \left ( x \right ) \left ( 4\, \left ( \sin \left ( x \right ) \right ) ^{2}-1 \right ) }{12\, \left ( \sin \left ( x \right ) \right ) ^{4}-12\, \left ( \sin \left ( x \right ) \right ) ^{2}+3}\sqrt{-2\, \left ( \sin \left ( x \right ) \right ) ^{2}+1}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.51037, size = 122, normalized size = 7.62 \begin{align*} -\frac{\sqrt{2} \sin \left (\frac{3}{2} \, \arctan \left (\sin \left (4 \, x\right ), \cos \left (4 \, x\right ) + 1\right )\right ) \sin \left (\frac{3}{2} \, \arctan \left (\sin \left (4 \, x\right ), \cos \left (4 \, x\right )\right )\right ) +{\left (\sqrt{2} \cos \left (\frac{3}{2} \, \arctan \left (\sin \left (4 \, x\right ), \cos \left (4 \, x\right )\right )\right ) + \sqrt{2}\right )} \cos \left (\frac{3}{2} \, \arctan \left (\sin \left (4 \, x\right ), \cos \left (4 \, x\right ) + 1\right )\right )}{3 \,{\left (\cos \left (4 \, x\right )^{2} + \sin \left (4 \, x\right )^{2} + 2 \, \cos \left (4 \, x\right ) + 1\right )}^{\frac{3}{4}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 3.198, size = 109, normalized size = 6.81 \begin{align*} -\frac{{\left (4 \, \cos \left (x\right )^{3} - 3 \, \cos \left (x\right )\right )} \sqrt{2 \, \cos \left (x\right )^{2} - 1}}{3 \,{\left (4 \, \cos \left (x\right )^{4} - 4 \, \cos \left (x\right )^{2} + 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 [B] time = 1.27446, size = 62, normalized size = 3.88 \begin{align*} \frac{{\left ({\left (\tan \left (\frac{1}{2} \, x\right )^{2} - 15\right )} \tan \left (\frac{1}{2} \, x\right )^{2} + 15\right )} \tan \left (\frac{1}{2} \, x\right )^{2} - 1}{3 \,{\left (\tan \left (\frac{1}{2} \, x\right )^{4} - 6 \, \tan \left (\frac{1}{2} \, x\right )^{2} + 1\right )}^{\frac{3}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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