Optimal. Leaf size=10 \[ -\frac{\log (\cos (x)+1)}{a} \]
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Rubi [A] time = 0.0235585, antiderivative size = 10, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {2667, 31} \[ -\frac{\log (\cos (x)+1)}{a} \]
Antiderivative was successfully verified.
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Rule 2667
Rule 31
Rubi steps
\begin{align*} \int \frac{\sin (x)}{a+a \cos (x)} \, dx &=-\frac{\operatorname{Subst}\left (\int \frac{1}{a+x} \, dx,x,a \cos (x)\right )}{a}\\ &=-\frac{\log (1+\cos (x))}{a}\\ \end{align*}
Mathematica [A] time = 0.0058053, size = 12, normalized size = 1.2 \[ -\frac{2 \log \left (\cos \left (\frac{x}{2}\right )\right )}{a} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.031, size = 13, normalized size = 1.3 \begin{align*} -{\frac{\ln \left ( a+a\cos \left ( x \right ) \right ) }{a}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.11335, size = 16, normalized size = 1.6 \begin{align*} -\frac{\log \left (a \cos \left (x\right ) + a\right )}{a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.61993, size = 35, normalized size = 3.5 \begin{align*} -\frac{\log \left (\frac{1}{2} \, \cos \left (x\right ) + \frac{1}{2}\right )}{a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.146093, size = 8, normalized size = 0.8 \begin{align*} - \frac{\log{\left (\cos{\left (x \right )} + 1 \right )}}{a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.10571, size = 14, normalized size = 1.4 \begin{align*} -\frac{\log \left (\cos \left (x\right ) + 1\right )}{a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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