Optimal. Leaf size=33 \[ \frac {2}{3} \sqrt {-\sin ^2(x)} \cot (x)-\frac {1}{3} \left (-\sin ^2(x)\right )^{3/2} \cot (x) \]
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Rubi [A] time = 0.02, antiderivative size = 33, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {3176, 3203, 3207, 2638} \[ \frac {2}{3} \sqrt {-\sin ^2(x)} \cot (x)-\frac {1}{3} \left (-\sin ^2(x)\right )^{3/2} \cot (x) \]
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 \left (-\sin ^2(x)\right )^{3/2} \, dx\\ &=-\frac {1}{3} \cot (x) \left (-\sin ^2(x)\right )^{3/2}-\frac {2}{3} \int \sqrt {-\sin ^2(x)} \, dx\\ &=-\frac {1}{3} \cot (x) \left (-\sin ^2(x)\right )^{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) \left (-\sin ^2(x)\right )^{3/2}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 25, normalized size = 0.76 \[ -\frac {1}{12} \sqrt {-\sin ^2(x)} (\cos (3 x)-9 \cos (x)) \csc (x) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 1, normalized size = 0.03 \[ 0 \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [C] time = 0.65, size = 55, normalized size = 1.67 \[ -\frac {12 i \, \mathrm {sgn}\left (-\tan \left (\frac {1}{2} \, x\right )^{3} - \tan \left (\frac {1}{2} \, x\right )\right ) \tan \left (\frac {1}{2} \, x\right )^{2} + 4 i \, \mathrm {sgn}\left (-\tan \left (\frac {1}{2} \, x\right )^{3} - \tan \left (\frac {1}{2} \, x\right )\right )}{3 \, {\left (\tan \left (\frac {1}{2} \, x\right )^{2} + 1\right )}^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.99, size = 21, normalized size = 0.64 \[ -\frac {\sin \relax (x ) \cos \relax (x ) \left (\sin ^{2}\relax (x )+2\right )}{3 \sqrt {-\left (\sin ^{2}\relax (x )\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (\cos \relax (x)^{2} - 1\right )}^{\frac {3}{2}}\,{d x} \]
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 \[ \int {\left ({\cos \relax (x)}^2-1\right )}^{3/2} \,d x \]
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 \[ \int \left (\cos ^{2}{\relax (x )} - 1\right )^{\frac {3}{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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