Optimal. Leaf size=3 \[ \log (\sin (x)) \]
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Rubi [B] Leaf count is larger than twice the leaf count of optimal. \(13\) vs. \(2(3)=6\).
time = 0.02, antiderivative size = 13, normalized size of antiderivative = 4.33, number of steps
used = 2, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {2746, 31}
\begin {gather*} \frac {1}{2} \log (1-\cos (2 x)) \end {gather*}
Antiderivative was successfully verified.
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Rule 31
Rule 2746
Rubi steps
\begin {align*} \int \frac {\sin (2 x)}{1-\cos (2 x)} \, dx &=\frac {1}{2} \text {Subst}\left (\int \frac {1}{1+x} \, dx,x,-\cos (2 x)\right )\\ &=\frac {1}{2} \log (1-\cos (2 x))\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 3, normalized size = 1.00 \begin {gather*} \log (\sin (x)) \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(11\) vs.
\(2(3)=6\).
time = 0.04, size = 12, normalized size = 4.00
method | result | size |
derivativedivides | \(\frac {\ln \left (1-\cos \left (2 x \right )\right )}{2}\) | \(12\) |
default | \(\frac {\ln \left (1-\cos \left (2 x \right )\right )}{2}\) | \(12\) |
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 [B] Leaf count of result is larger than twice the leaf count of optimal. 9 vs.
\(2 (3) = 6\).
time = 0.27, size = 9, normalized size = 3.00 \begin {gather*} \frac {1}{2} \, \log \left (\cos \left (2 \, x\right ) - 1\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 = 0.37, 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 [B] Leaf count of result is larger than twice the leaf count of optimal. 8 vs.
\(2 (3) = 6\).
time = 0.04, size = 8, normalized size = 2.67 \begin {gather*} \frac {\log {\left (\cos {\left (2 x \right )} - 1 \right )}}{2} \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. 11 vs.
\(2 (3) = 6\).
time = 0.42, size = 11, normalized size = 3.67 \begin {gather*} \frac {1}{2} \, \log \left (-\cos \left (2 \, x\right ) + 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.08, size = 9, normalized size = 3.00 \begin {gather*} \frac {\ln \left (-{\sin \left (x\right )}^2\right )}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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