Optimal. Leaf size=14 \[ -\frac {1}{2} \cot ^2(x)-\log (\sin (x)) \]
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Rubi [A]
time = 0.01, antiderivative size = 14, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {3554, 3556}
\begin {gather*} -\frac {1}{2} \cot ^2(x)-\log (\sin (x)) \end {gather*}
Antiderivative was successfully verified.
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Rule 3554
Rule 3556
Rubi steps
\begin {align*} \int \cot ^3(x) \, dx &=-\frac {1}{2} \cot ^2(x)-\int \cot (x) \, dx\\ &=-\frac {1}{2} \cot ^2(x)-\log (\sin (x))\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 14, normalized size = 1.00 \begin {gather*} -\frac {1}{2} \csc ^2(x)-\log (\sin (x)) \end {gather*}
Antiderivative was successfully verified.
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Mathics [A]
time = 1.90, size = 12, normalized size = 0.86 \begin {gather*} -\text {Log}\left [\text {Sin}\left [x\right ]\right ]-\frac {1}{2 \text {Sin}\left [x\right ]^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.02, size = 22, normalized size = 1.57
method | result | size |
derivativedivides | \(\frac {\ln \left (1+\tan ^{2}\left (x \right )\right )}{2}-\frac {1}{2 \tan \left (x \right )^{2}}-\ln \left (\tan \left (x \right )\right )\) | \(22\) |
default | \(\frac {\ln \left (1+\tan ^{2}\left (x \right )\right )}{2}-\frac {1}{2 \tan \left (x \right )^{2}}-\ln \left (\tan \left (x \right )\right )\) | \(22\) |
norman | \(\frac {\ln \left (1+\tan ^{2}\left (x \right )\right )}{2}-\frac {1}{2 \tan \left (x \right )^{2}}-\ln \left (\tan \left (x \right )\right )\) | \(22\) |
risch | \(i x +\frac {2 \,{\mathrm e}^{2 i x}}{\left ({\mathrm e}^{2 i x}-1\right )^{2}}-\ln \left ({\mathrm e}^{2 i x}-1\right )\) | \(32\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 14, normalized size = 1.00 \begin {gather*} -\frac {1}{2 \, \sin \left (x\right )^{2}} - \frac {1}{2} \, \log \left (\sin \left (x\right )^{2}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 31 vs.
\(2 (12) = 24\).
time = 0.31, size = 31, normalized size = 2.21 \begin {gather*} -\frac {\log \left (\frac {\tan \left (x\right )^{2}}{\tan \left (x\right )^{2} + 1}\right ) \tan \left (x\right )^{2} + \tan \left (x\right )^{2} + 1}{2 \, \tan \left (x\right )^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.05, size = 14, normalized size = 1.00 \begin {gather*} - \log {\left (\sin {\left (x \right )} \right )} - \frac {1}{2 \sin ^{2}{\left (x \right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 29 vs.
\(2 (12) = 24\).
time = 0.00, size = 34, normalized size = 2.43 \begin {gather*} \frac {\ln \left (\tan ^{2}x+1\right )}{2}-\frac {\ln \left (\tan ^{2}x\right )}{2}+\frac {\tan ^{2}x-1}{2 \tan ^{2}x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.17, size = 21, normalized size = 1.50 \begin {gather*} \frac {\ln \left ({\mathrm {tan}\left (x\right )}^2+1\right )}{2}-\ln \left (\mathrm {tan}\left (x\right )\right )-\frac {1}{2\,{\mathrm {tan}\left (x\right )}^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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