Optimal. Leaf size=29 \[ -\frac{1}{3} \sin ^2(x)^{3/2} \cot (x)-\frac{2}{3} \sqrt{\sin ^2(x)} \cot (x) \]
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Rubi [A] time = 0.022335, antiderivative size = 29, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {3176, 3203, 3207, 2638} \[ -\frac{1}{3} \sin ^2(x)^{3/2} \cot (x)-\frac{2}{3} \sqrt{\sin ^2(x)} \cot (x) \]
Antiderivative was successfully verified.
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Rule 3176
Rule 3203
Rule 3207
Rule 2638
Rubi steps
\begin{align*} \int \left (1-\cos ^2(x)\right )^{3/2} \, dx &=\int \sin ^2(x)^{3/2} \, dx\\ &=-\frac{1}{3} \cot (x) \sin ^2(x)^{3/2}+\frac{2}{3} \int \sqrt{\sin ^2(x)} \, dx\\ &=-\frac{1}{3} \cot (x) \sin ^2(x)^{3/2}+\frac{1}{3} \left (2 \csc (x) \sqrt{\sin ^2(x)}\right ) \int \sin (x) \, dx\\ &=-\frac{2}{3} \cot (x) \sqrt{\sin ^2(x)}-\frac{1}{3} \cot (x) \sin ^2(x)^{3/2}\\ \end{align*}
Mathematica [A] time = 0.0263914, size = 23, normalized size = 0.79 \[ \frac{1}{12} \sqrt{\sin ^2(x)} (\cos (3 x)-9 \cos (x)) \csc (x) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.756, size = 19, normalized size = 0.7 \begin{align*}{\frac{\cos \left ( x \right ) \sin \left ( x \right ) \left ( \left ( \cos \left ( x \right ) \right ) ^{2}-3 \right ) }{3}{\frac{1}{\sqrt{ \left ( \sin \left ( x \right ) \right ) ^{2}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.63589, size = 15, normalized size = 0.52 \begin{align*} -\frac{1}{12} \, \cos \left (3 \, x\right ) + \frac{3}{4} \, \cos \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.72423, size = 31, normalized size = 1.07 \begin{align*} \frac{1}{3} \, \cos \left (x\right )^{3} - \cos \left (x\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 = 1.21325, size = 61, normalized size = 2.1 \begin{align*} -\frac{4 \,{\left (3 \, \mathrm{sgn}\left (\tan \left (\frac{1}{2} \, x\right )^{3} + \tan \left (\frac{1}{2} \, x\right )\right ) \tan \left (\frac{1}{2} \, x\right )^{2} + \mathrm{sgn}\left (\tan \left (\frac{1}{2} \, x\right )^{3} + \tan \left (\frac{1}{2} \, x\right )\right )\right )}}{3 \,{\left (\tan \left (\frac{1}{2} \, x\right )^{2} + 1\right )}^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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