Optimal. Leaf size=9 \[ \frac{\sin (x)}{\cos (x)+1} \]
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Rubi [A] time = 0.0075139, antiderivative size = 9, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 6, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {2648} \[ \frac{\sin (x)}{\cos (x)+1} \]
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*}
Mathematica [A] time = 0.0036081, size = 6, normalized size = 0.67 \[ \tan \left (\frac{x}{2}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0., size = 5, normalized size = 0.6 \begin{align*} \tan \left ({\frac{x}{2}} \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.918008, size = 12, normalized size = 1.33 \begin{align*} \frac{\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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Fricas [A] time = 1.94506, size = 28, normalized size = 3.11 \begin{align*} \frac{\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 [A] time = 0.179115, size = 3, normalized size = 0.33 \begin{align*} \tan{\left (\frac{x}{2} \right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.10891, size = 41, normalized size = 4.56 \begin{align*} -\frac{2 \, \tan \left (\frac{1}{2} \, x\right )}{{\left (x^{2} + 1\right )}{\left (\frac{x^{2} - 1}{x^{2} + 1} - 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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