Optimal. Leaf size=33 \[ \frac{\cos (x)}{3 (a \sin (x)+a)^2}-\frac{2 \cos (x)}{3 \left (a^2 \sin (x)+a^2\right )} \]
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Rubi [A] time = 0.0314165, antiderivative size = 33, 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 = {2750, 2648} \[ \frac{\cos (x)}{3 (a \sin (x)+a)^2}-\frac{2 \cos (x)}{3 \left (a^2 \sin (x)+a^2\right )} \]
Antiderivative was successfully verified.
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Rule 2750
Rule 2648
Rubi steps
\begin{align*} \int \frac{\sin (x)}{(a+a \sin (x))^2} \, dx &=\frac{\cos (x)}{3 (a+a \sin (x))^2}+\frac{2 \int \frac{1}{a+a \sin (x)} \, dx}{3 a}\\ &=\frac{\cos (x)}{3 (a+a \sin (x))^2}-\frac{2 \cos (x)}{3 \left (a^2+a^2 \sin (x)\right )}\\ \end{align*}
Mathematica [A] time = 0.0421223, size = 29, normalized size = 0.88 \[ -\frac{-4 \sin (x)+\sin (2 x)+\cos (x)+\cos (2 x)-3}{3 a^2 (\sin (x)+1)^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.032, size = 27, normalized size = 0.8 \begin{align*} 4\,{\frac{1}{{a}^{2}} \left ( 1/3\, \left ( \tan \left ( x/2 \right ) +1 \right ) ^{-3}-1/2\, \left ( \tan \left ( x/2 \right ) +1 \right ) ^{-2} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.07915, size = 84, normalized size = 2.55 \begin{align*} -\frac{2 \,{\left (\frac{3 \, \sin \left (x\right )}{\cos \left (x\right ) + 1} + 1\right )}}{3 \,{\left (a^{2} + \frac{3 \, a^{2} \sin \left (x\right )}{\cos \left (x\right ) + 1} + \frac{3 \, a^{2} \sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} + \frac{a^{2} \sin \left (x\right )^{3}}{{\left (\cos \left (x\right ) + 1\right )}^{3}}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.39024, size = 163, normalized size = 4.94 \begin{align*} \frac{2 \, \cos \left (x\right )^{2} +{\left (2 \, \cos \left (x\right ) + 1\right )} \sin \left (x\right ) + \cos \left (x\right ) - 1}{3 \,{\left (a^{2} \cos \left (x\right )^{2} - a^{2} \cos \left (x\right ) - 2 \, a^{2} -{\left (a^{2} \cos \left (x\right ) + 2 \, a^{2}\right )} \sin \left (x\right )\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 3.41913, size = 178, normalized size = 5.39 \begin{align*} \frac{\tan ^{3}{\left (\frac{x}{2} \right )}}{6 a^{2} \tan ^{3}{\left (\frac{x}{2} \right )} + 18 a^{2} \tan ^{2}{\left (\frac{x}{2} \right )} + 18 a^{2} \tan{\left (\frac{x}{2} \right )} + 6 a^{2}} + \frac{3 \tan ^{2}{\left (\frac{x}{2} \right )}}{6 a^{2} \tan ^{3}{\left (\frac{x}{2} \right )} + 18 a^{2} \tan ^{2}{\left (\frac{x}{2} \right )} + 18 a^{2} \tan{\left (\frac{x}{2} \right )} + 6 a^{2}} - \frac{9 \tan{\left (\frac{x}{2} \right )}}{6 a^{2} \tan ^{3}{\left (\frac{x}{2} \right )} + 18 a^{2} \tan ^{2}{\left (\frac{x}{2} \right )} + 18 a^{2} \tan{\left (\frac{x}{2} \right )} + 6 a^{2}} - \frac{3}{6 a^{2} \tan ^{3}{\left (\frac{x}{2} \right )} + 18 a^{2} \tan ^{2}{\left (\frac{x}{2} \right )} + 18 a^{2} \tan{\left (\frac{x}{2} \right )} + 6 a^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.88682, size = 28, normalized size = 0.85 \begin{align*} -\frac{2 \,{\left (3 \, \tan \left (\frac{1}{2} \, x\right ) + 1\right )}}{3 \, a^{2}{\left (\tan \left (\frac{1}{2} \, x\right ) + 1\right )}^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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