Optimal. Leaf size=12 \[ -\frac {\sin (x)}{1-\cos (x)} \]
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Rubi [A] time = 0.01, antiderivative size = 12, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {2648} \[ -\frac {\sin (x)}{1-\cos (x)} \]
Antiderivative was successfully verified.
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Rule 2648
Rubi steps
\begin {align*} \int \frac {1}{1-\cos (x)} \, dx &=-\frac {\sin (x)}{1-\cos (x)}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 8, normalized size = 0.67 \[ -\cot \left (\frac {x}{2}\right ) \]
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{1-\cos (x)} \, dx \]
Verification is Not applicable to the result.
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fricas [A] time = 1.05, size = 10, normalized size = 0.83 \[ -\frac {\cos \relax (x) + 1}{\sin \relax (x)} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.08, size = 8, normalized size = 0.67 \[ -\frac {1}{\tan \left (\frac {1}{2} \, x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 9, normalized size = 0.75
method | result | size |
default | \(-\frac {1}{\tan \left (\frac {x}{2}\right )}\) | \(9\) |
norman | \(-\frac {1}{\tan \left (\frac {x}{2}\right )}\) | \(9\) |
risch | \(-\frac {2 i}{{\mathrm e}^{i x}-1}\) | \(13\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.42, size = 10, normalized size = 0.83 \[ -\frac {\cos \relax (x) + 1}{\sin \relax (x)} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.00, size = 6, normalized size = 0.50 \[ -\mathrm {cot}\left (\frac {x}{2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.34, size = 7, normalized size = 0.58 \[ - \frac {1}{\tan {\left (\frac {x}{2} \right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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