Optimal. Leaf size=12 \[ x-\frac {\sin (x)}{\cos (x)+1} \]
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Rubi [A] time = 0.05, antiderivative size = 12, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {4392, 2735, 2648} \[ x-\frac {\sin (x)}{\cos (x)+1} \]
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 = 10, normalized size = 0.83 \[ x-\tan \left (\frac {x}{2}\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.59, size = 17, normalized size = 1.42 \[ \frac {x \cos \relax (x) + x - \sin \relax (x)}{\cos \relax (x) + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.22, size = 8, normalized size = 0.67 \[ x - \tan \left (\frac {1}{2} \, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 9, normalized size = 0.75 \[ -\tan \left (\frac {x}{2}\right )+x \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.43, size = 23, normalized size = 1.92 \[ -\frac {\sin \relax (x)}{\cos \relax (x) + 1} + 2 \, \arctan \left (\frac {\sin \relax (x)}{\cos \relax (x) + 1}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.57, size = 8, normalized size = 0.67 \[ x-\mathrm {tan}\left (\frac {x}{2}\right ) \]
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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