Optimal. Leaf size=16 \[ -x-\frac {2 \sin (x)}{1-\cos (x)} \]
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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 = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {2680, 8} \[ -x-\frac {2 \sin (x)}{1-\cos (x)} \]
Antiderivative was successfully verified.
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Rule 8
Rule 2680
Rubi steps
\begin {align*} \int \frac {\sin ^2(x)}{(1-\cos (x))^2} \, dx &=-\frac {2 \sin (x)}{1-\cos (x)}-\int 1 \, dx\\ &=-x-\frac {2 \sin (x)}{1-\cos (x)}\\ \end {align*}
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Mathematica [C] time = 0.01, size = 26, normalized size = 1.62 \[ -2 \cot \left (\frac {x}{2}\right ) \, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};-\tan ^2\left (\frac {x}{2}\right )\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 1.41, size = 16, normalized size = 1.00 \[ -\frac {x \sin \relax (x) + 2 \, \cos \relax (x) + 2}{\sin \relax (x)} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.37, size = 12, normalized size = 0.75 \[ -x - \frac {2}{\tan \left (\frac {1}{2} \, x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.07, size = 13, normalized size = 0.81 \[ -\frac {2}{\tan \left (\frac {x}{2}\right )}-x \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.14, size = 23, normalized size = 1.44 \[ -\frac {2 \, {\left (\cos \relax (x) + 1\right )}}{\sin \relax (x)} - 2 \, \arctan \left (\frac {\sin \relax (x)}{\cos \relax (x) + 1}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.31, size = 10, normalized size = 0.62 \[ -x-2\,\mathrm {cot}\left (\frac {x}{2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.88, size = 8, normalized size = 0.50 \[ - x - \frac {2}{\tan {\left (\frac {x}{2} \right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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