Optimal. Leaf size=33 \[ \frac {2}{3} \cot (x) \sqrt {-\sin ^2(x)}-\frac {1}{3} \cot (x) \left (-\sin ^2(x)\right )^{3/2} \]
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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 = {3255, 3282,
3286, 2718} \begin {gather*} \frac {2}{3} \sqrt {-\sin ^2(x)} \cot (x)-\frac {1}{3} \left (-\sin ^2(x)\right )^{3/2} \cot (x) \end {gather*}
Antiderivative was successfully verified.
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Rule 2718
Rule 3255
Rule 3282
Rule 3286
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 \begin {gather*} -\frac {1}{12} (-9 \cos (x)+\cos (3 x)) \csc (x) \sqrt {-\sin ^2(x)} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.28, size = 21, normalized size = 0.64
method | result | size |
default | \(-\frac {\sin \left (x \right ) \cos \left (x \right ) \left (\sin ^{2}\left (x \right )+2\right )}{3 \sqrt {-\left (\sin ^{2}\left (x \right )\right )}}\) | \(21\) |
risch | \(-\frac {i {\mathrm e}^{4 i x} \sqrt {\left ({\mathrm e}^{2 i x}-1\right )^{2} {\mathrm e}^{-2 i x}}}{24 \left ({\mathrm e}^{2 i x}-1\right )}+\frac {3 i \sqrt {\left ({\mathrm e}^{2 i x}-1\right )^{2} {\mathrm e}^{-2 i x}}\, {\mathrm e}^{2 i x}}{8 \left ({\mathrm e}^{2 i x}-1\right )}+\frac {3 i \sqrt {\left ({\mathrm e}^{2 i x}-1\right )^{2} {\mathrm e}^{-2 i x}}}{8 \left ({\mathrm e}^{2 i x}-1\right )}-\frac {i {\mathrm e}^{-2 i x} \sqrt {\left ({\mathrm e}^{2 i x}-1\right )^{2} {\mathrm e}^{-2 i x}}}{24 \left ({\mathrm e}^{2 i x}-1\right )}\) | \(133\) |
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.39, size = 1, normalized size = 0.03 \begin {gather*} 0 \end {gather*}
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 \begin {gather*} \int \left (\cos ^{2}{\left (x \right )} - 1\right )^{\frac {3}{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [C] Result contains complex when optimal does not.
time = 0.42, size = 55, normalized size = 1.67 \begin {gather*} -\frac {4 \, {\left (3 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} + i \, \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 {gather*}
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 \begin {gather*} \int {\left ({\cos \left (x\right )}^2-1\right )}^{3/2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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