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 = 0.02, antiderivative size = 29, normalized size of antiderivative = 1.00, 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 2638
Rule 3176
Rule 3203
Rule 3207
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*}
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Mathematica [A] time = 0.03, 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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fricas [A] time = 0.68, size = 11, normalized size = 0.38 \[ \frac {1}{3} \, \cos \relax (x)^{3} - \cos \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.23, size = 45, normalized size = 1.55 \[ -\frac {4 \, {\left (3 \, \mathrm {sgn}\left (\tan \left (\frac {1}{2} \, x\right )^{3} + \tan \left (\frac {1}{2} \, x\right )\right ) \tan \left (\frac {1}{2} \, x\right )^{2} + \mathrm {sgn}\left (\tan \left (\frac {1}{2} \, x\right )^{3} + \tan \left (\frac {1}{2} \, x\right )\right )\right )}}{3 \, {\left (\tan \left (\frac {1}{2} \, x\right )^{2} + 1\right )}^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 1.43, size = 19, normalized size = 0.66 \[ \frac {2 \sin \relax (x ) \cos \relax (x ) \left (\cos ^{2}\relax (x )-3\right )}{3 \sqrt {2-2 \cos \left (2 x \right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.98, size = 11, normalized size = 0.38 \[ -\frac {1}{12} \, \cos \left (3 \, x\right ) + \frac {3}{4} \, \cos \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.03 \[ \int {\left (1-{\cos \relax (x)}^2\right )}^{3/2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (1 - \cos ^{2}{\relax (x )}\right )^{\frac {3}{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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