Optimal. Leaf size=3 \[ \log (\sin (x)) \]
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Rubi [A]
time = 0.00, antiderivative size = 3, normalized size of antiderivative = 1.00, number of steps
used = 1, number of rules used = 1, integrand size = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {3556}
\begin {gather*} \log (\sin (x)) \end {gather*}
Antiderivative was successfully verified.
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Rule 3556
Rubi steps
\begin {align*} \int \cot (x) \, dx &=\log (\sin (x))\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 3, normalized size = 1.00 \begin {gather*} \log (\sin (x)) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.01, size = 4, normalized size = 1.33
method | result | size |
lookup | \(\ln \left (\sin \left (x \right )\right )\) | \(4\) |
default | \(\ln \left (\sin \left (x \right )\right )\) | \(4\) |
derivativedivides | \(-\frac {\ln \left (\cot ^{2}\left (x \right )+1\right )}{2}\) | \(10\) |
norman | \(-\frac {\ln \left (1+\tan ^{2}\left (x \right )\right )}{2}+\ln \left (\tan \left (x \right )\right )\) | \(14\) |
risch | \(-i x +\ln \left ({\mathrm e}^{2 i x}-1\right )\) | \(14\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 3.59, size = 3, normalized size = 1.00 \begin {gather*} \log \left (\sin \left (x\right )\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. 11 vs.
\(2 (3) = 6\).
time = 1.26, size = 11, normalized size = 3.67 \begin {gather*} \frac {1}{2} \, \log \left (-\frac {1}{2} \, \cos \left (2 \, x\right ) + \frac {1}{2}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.03, size = 3, normalized size = 1.00 \begin {gather*} \log {\left (\sin {\left (x \right )} \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.44, size = 4, normalized size = 1.33 \begin {gather*} \log \left ({\left | \sin \left (x\right ) \right |}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.02, size = 3, normalized size = 1.00 \begin {gather*} \ln \left (\sin \left (x\right )\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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