Optimal. Leaf size=20 \[ -\frac {1}{4} \cot ^4(x)+\frac {\cot ^2(x)}{2}+\log (\sin (x)) \]
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Rubi [A] time = 0.02, antiderivative size = 20, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {3473, 3475} \[ -\frac {1}{4} \cot ^4(x)+\frac {\cot ^2(x)}{2}+\log (\sin (x)) \]
Antiderivative was successfully verified.
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Rule 3473
Rule 3475
Rubi steps
\begin {align*} \int \cot ^5(x) \, dx &=-\frac {1}{4} \cot ^4(x)-\int \cot ^3(x) \, dx\\ &=\frac {\cot ^2(x)}{2}-\frac {\cot ^4(x)}{4}+\int \cot (x) \, dx\\ &=\frac {\cot ^2(x)}{2}-\frac {\cot ^4(x)}{4}+\log (\sin (x))\\ \end {align*}
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Mathematica [A] time = 0.00, size = 16, normalized size = 0.80 \[ -\frac {1}{4} \csc ^4(x)+\csc ^2(x)+\log (\sin (x)) \]
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \cot ^5(x) \, dx \]
Verification is Not applicable to the result.
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fricas [B] time = 0.94, size = 40, normalized size = 2.00 \[ \frac {2 \, \log \left (\frac {\tan \relax (x)^{2}}{\tan \relax (x)^{2} + 1}\right ) \tan \relax (x)^{4} + 3 \, \tan \relax (x)^{4} + 2 \, \tan \relax (x)^{2} - 1}{4 \, \tan \relax (x)^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.83, size = 37, normalized size = 1.85 \[ -\frac {3 \, \tan \relax (x)^{4} - 2 \, \tan \relax (x)^{2} + 1}{4 \, \tan \relax (x)^{4}} - \frac {1}{2} \, \log \left (\tan \relax (x)^{2} + 1\right ) + \frac {1}{2} \, \log \left (\tan \relax (x)^{2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 26, normalized size = 1.30
method | result | size |
derivativedivides | \(-\frac {\ln \left (1+\tan ^{2}\relax (x )\right )}{2}-\frac {1}{4 \tan \relax (x )^{4}}+\ln \left (\tan \relax (x )\right )+\frac {1}{2 \tan \relax (x )^{2}}\) | \(26\) |
default | \(-\frac {\ln \left (1+\tan ^{2}\relax (x )\right )}{2}-\frac {1}{4 \tan \relax (x )^{4}}+\ln \left (\tan \relax (x )\right )+\frac {1}{2 \tan \relax (x )^{2}}\) | \(26\) |
norman | \(\frac {-\frac {1}{4}+\frac {\left (\tan ^{2}\relax (x )\right )}{2}}{\tan \relax (x )^{4}}-\frac {\ln \left (1+\tan ^{2}\relax (x )\right )}{2}+\ln \left (\tan \relax (x )\right )\) | \(27\) |
risch | \(-i x -\frac {4 \left ({\mathrm e}^{6 i x}-{\mathrm e}^{4 i x}+{\mathrm e}^{2 i x}\right )}{\left ({\mathrm e}^{2 i x}-1\right )^{4}}+\ln \left ({\mathrm e}^{2 i x}-1\right )\) | \(43\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.61, size = 22, normalized size = 1.10 \[ \frac {4 \, \sin \relax (x)^{2} - 1}{4 \, \sin \relax (x)^{4}} + \frac {1}{2} \, \log \left (\sin \relax (x)^{2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.27, size = 26, normalized size = 1.30 \[ \ln \left (\mathrm {tan}\relax (x)\right )-\frac {\ln \left ({\mathrm {tan}\relax (x)}^2+1\right )}{2}+\frac {\frac {{\mathrm {tan}\relax (x)}^2}{2}-\frac {1}{4}}{{\mathrm {tan}\relax (x)}^4} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.11, size = 19, normalized size = 0.95 \[ \frac {4 \sin ^{2}{\relax (x )} - 1}{4 \sin ^{4}{\relax (x )}} + \log {\left (\sin {\relax (x )} \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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