Optimal. Leaf size=16 \[ -x-\frac {\sin (x)}{1-\cos (x)} \]
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Rubi [A] time = 0.06, antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {4392, 2735, 2648} \[ -x-\frac {\sin (x)}{1-\cos (x)} \]
Antiderivative was successfully verified.
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Rule 2648
Rule 2735
Rule 4392
Rubi steps
\begin {align*} \int \frac {\cot (x)}{-\cot (x)+\csc (x)} \, dx &=\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 = 16, normalized size = 1.00 \[ \frac {1}{2} \left (-2 x-2 \cot \left (\frac {x}{2}\right )\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.69, 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 = 0.16, 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 = 0.07, size = 13, normalized size = 0.81 \[ -\frac {1}{\tan \left (\frac {x}{2}\right )}-x \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.45, 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 = 0.58, 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 [F] time = 0.00, size = 0, normalized size = 0.00 \[ - \int \frac {\cot {\relax (x )}}{\cot {\relax (x )} - \csc {\relax (x )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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