Optimal. Leaf size=17 \[ \frac{x}{a}+\frac{\cos (x)}{a \sin (x)+a} \]
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Rubi [A] time = 0.0285508, antiderivative size = 17, 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 = {2735, 2648} \[ \frac{x}{a}+\frac{\cos (x)}{a \sin (x)+a} \]
Antiderivative was successfully verified.
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Rule 2735
Rule 2648
Rubi steps
\begin{align*} \int \frac{\sin (x)}{a+a \sin (x)} \, dx &=\frac{x}{a}-\int \frac{1}{a+a \sin (x)} \, dx\\ &=\frac{x}{a}+\frac{\cos (x)}{a+a \sin (x)}\\ \end{align*}
Mathematica [B] time = 0.0366239, size = 42, normalized size = 2.47 \[ \frac{\left (\sin \left (\frac{x}{2}\right )+\cos \left (\frac{x}{2}\right )\right ) \left ((x-2) \sin \left (\frac{x}{2}\right )+x \cos \left (\frac{x}{2}\right )\right )}{a (\sin (x)+1)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.022, size = 25, normalized size = 1.5 \begin{align*} 2\,{\frac{\arctan \left ( \tan \left ( x/2 \right ) \right ) }{a}}+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 = 2.36792, size = 43, normalized size = 2.53 \begin{align*} \frac{2 \, \arctan \left (\frac{\sin \left (x\right )}{\cos \left (x\right ) + 1}\right )}{a} + \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 = 1.57898, size = 93, normalized size = 5.47 \begin{align*} \frac{{\left (x + 1\right )} \cos \left (x\right ) +{\left (x - 1\right )} \sin \left (x\right ) + x + 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 [B] time = 0.631777, size = 39, normalized size = 2.29 \begin{align*} \frac{x \tan{\left (\frac{x}{2} \right )}}{a \tan{\left (\frac{x}{2} \right )} + a} + \frac{x}{a \tan{\left (\frac{x}{2} \right )} + a} - \frac{2 \tan{\left (\frac{x}{2} \right )}}{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 = 2.36122, size = 26, normalized size = 1.53 \begin{align*} \frac{x}{a} + \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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