Optimal. Leaf size=12 \[ -\frac{\cos (x)}{a \sin (x)+a} \]
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Rubi [A] time = 0.0102863, antiderivative size = 12, 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} \[ -\frac{\cos (x)}{a \sin (x)+a} \]
Antiderivative was successfully verified.
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Rule 2648
Rubi steps
\begin{align*} \int \frac{1}{a+a \sin (x)} \, dx &=-\frac{\cos (x)}{a+a \sin (x)}\\ \end{align*}
Mathematica [B] time = 0.0254837, size = 29, normalized size = 2.42 \[ \frac{2 \sin \left (\frac{x}{2}\right ) \left (\sin \left (\frac{x}{2}\right )+\cos \left (\frac{x}{2}\right )\right )}{a \sin (x)+a} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.017, size = 14, normalized size = 1.2 \begin{align*} -2\,{\frac{1}{a \left ( \tan \left ( x/2 \right ) +1 \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.73839, size = 22, normalized size = 1.83 \begin{align*} -\frac{2}{a + \frac{a \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 = 1.59791, size = 68, normalized size = 5.67 \begin{align*} -\frac{\cos \left (x\right ) - \sin \left (x\right ) + 1}{a \cos \left (x\right ) + a \sin \left (x\right ) + a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.261398, size = 10, normalized size = 0.83 \begin{align*} - \frac{2}{a \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 = 1.66146, size = 18, normalized size = 1.5 \begin{align*} -\frac{2}{a{\left (\tan \left (\frac{1}{2} \, x\right ) + 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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