Optimal. Leaf size=12 \[ x-\frac{\sin (x)}{\cos (x)+1} \]
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Rubi [A] time = 0.0535888, antiderivative size = 12, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {4392, 2735, 2648} \[ x-\frac{\sin (x)}{\cos (x)+1} \]
Antiderivative was successfully verified.
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Rule 4392
Rule 2735
Rule 2648
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*}
Mathematica [A] time = 0.017545, size = 10, normalized size = 0.83 \[ x-\tan \left (\frac{x}{2}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.057, size = 9, normalized size = 0.8 \begin{align*} -\tan \left ({\frac{x}{2}} \right ) +x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.66572, size = 31, normalized size = 2.58 \begin{align*} -\frac{\sin \left (x\right )}{\cos \left (x\right ) + 1} + 2 \, \arctan \left (\frac{\sin \left (x\right )}{\cos \left (x\right ) + 1}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.48709, size = 51, normalized size = 4.25 \begin{align*} \frac{x \cos \left (x\right ) + x - \sin \left (x\right )}{\cos \left (x\right ) + 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\cot{\left (x \right )}}{\cot{\left (x \right )} + \csc{\left (x \right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.13233, size = 11, normalized size = 0.92 \begin{align*} x - \tan \left (\frac{1}{2} \, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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