Optimal. Leaf size=16 \[ -x-\frac {\sin (x)}{1-\cos (x)} \]
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Rubi [A]
time = 0.02, antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {2814, 2727}
\begin {gather*} -x-\frac {\sin (x)}{1-\cos (x)} \end {gather*}
Antiderivative was successfully verified.
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Rule 2727
Rule 2814
Rubi steps
\begin {align*} \int \frac {\cos (x)}{1-\cos (x)} \, dx &=-x+\int \frac {1}{1-\cos (x)} \, dx\\ &=-x-\frac {\sin (x)}{1-\cos (x)}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 21, normalized size = 1.31 \begin {gather*} \frac {2 x \sin ^2\left (\frac {x}{2}\right )+\sin (x)}{-1+\cos (x)} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.02, size = 17, normalized size = 1.06
method | result | size |
default | \(-2 \arctan \left (\tan \left (\frac {x}{2}\right )\right )-\frac {1}{\tan \left (\frac {x}{2}\right )}\) | \(17\) |
risch | \(-x -\frac {2 i}{{\mathrm e}^{i x}-1}\) | \(17\) |
norman | \(\frac {-1-\left (\tan ^{2}\left (\frac {x}{2}\right )\right )-x \tan \left (\frac {x}{2}\right )-x \left (\tan ^{3}\left (\frac {x}{2}\right )\right )}{\left (1+\tan ^{2}\left (\frac {x}{2}\right )\right ) \tan \left (\frac {x}{2}\right )}\) | \(44\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 2.28, size = 23, normalized size = 1.44 \begin {gather*} -\frac {\cos \left (x\right ) + 1}{\sin \left (x\right )} - 2 \, \arctan \left (\frac {\sin \left (x\right )}{\cos \left (x\right ) + 1}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.55, size = 14, normalized size = 0.88 \begin {gather*} -\frac {x \sin \left (x\right ) + \cos \left (x\right ) + 1}{\sin \left (x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.20, size = 8, normalized size = 0.50 \begin {gather*} - x - \frac {1}{\tan {\left (\frac {x}{2} \right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.11, size = 12, normalized size = 0.75 \begin {gather*} -x - \frac {1}{\tan \left (\frac {1}{2} \, x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.17, size = 10, normalized size = 0.62 \begin {gather*} -x-\mathrm {cot}\left (\frac {x}{2}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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