Optimal. Leaf size=16 \[ -x-\frac {\sin (x)}{1-\cos (x)} \]
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Rubi [A] time = 0.03, 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 = {2735, 2648} \[ -x-\frac {\sin (x)}{1-\cos (x)} \]
Antiderivative was successfully verified.
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Rule 2648
Rule 2735
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 \[ \frac {2 x \sin ^2\left (\frac {x}{2}\right )+\sin (x)}{\cos (x)-1} \]
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\cos (x)}{1-\cos (x)} \, dx \]
Verification is Not applicable to the result.
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fricas [A] time = 0.91, size = 14, normalized size = 0.88 \[ -\frac {x \sin \relax (x) + \cos \relax (x) + 1}{\sin \relax (x)} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.01, size = 12, normalized size = 0.75 \[ -x - \frac {1}{\tan \left (\frac {1}{2} \, x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 17, normalized size = 1.06
method | result | size |
default | \(-\frac {1}{\tan \left (\frac {x}{2}\right )}-2 \arctan \left (\tan \left (\frac {x}{2}\right )\right )\) | \(17\) |
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 = 0.96, size = 23, normalized size = 1.44 \[ -\frac {\cos \relax (x) + 1}{\sin \relax (x)} - 2 \, \arctan \left (\frac {\sin \relax (x)}{\cos \relax (x) + 1}\right ) \]
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 \[ -x-\mathrm {cot}\left (\frac {x}{2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.49, size = 8, normalized size = 0.50 \[ - x - \frac {1}{\tan {\left (\frac {x}{2} \right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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