Optimal. Leaf size=11 \[ \frac{\sin (x)}{a \cos (x)+a} \]
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Rubi [A] time = 0.0103396, antiderivative size = 11, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {2648} \[ \frac{\sin (x)}{a \cos (x)+a} \]
Antiderivative was successfully verified.
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Rule 2648
Rubi steps
\begin{align*} \int \frac{1}{a+a \cos (x)} \, dx &=\frac{\sin (x)}{a+a \cos (x)}\\ \end{align*}
Mathematica [A] time = 0.0046866, size = 10, normalized size = 0.91 \[ \frac{\tan \left (\frac{x}{2}\right )}{a} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.025, size = 9, normalized size = 0.8 \begin{align*}{\frac{1}{a}\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 = 1.10538, size = 16, normalized size = 1.45 \begin{align*} \frac{\sin \left (x\right )}{a{\left (\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 = 1.51081, size = 31, normalized size = 2.82 \begin{align*} \frac{\sin \left (x\right )}{a \cos \left (x\right ) + a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.221091, size = 5, normalized size = 0.45 \begin{align*} \frac{\tan{\left (\frac{x}{2} \right )}}{a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.15774, size = 11, normalized size = 1. \begin{align*} \frac{\tan \left (\frac{1}{2} \, x\right )}{a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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