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 = 0.03, antiderivative size = 33, normalized size of antiderivative = 1.00, 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 2648
Rule 2750
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*}
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Mathematica [A] time = 0.04, 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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fricas [B] time = 0.45, size = 60, normalized size = 1.82 \[ \frac {2 \, \cos \relax (x)^{2} + {\left (2 \, \cos \relax (x) + 1\right )} \sin \relax (x) + \cos \relax (x) - 1}{3 \, {\left (a^{2} \cos \relax (x)^{2} - a^{2} \cos \relax (x) - 2 \, a^{2} - {\left (a^{2} \cos \relax (x) + 2 \, a^{2}\right )} \sin \relax (x)\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.95, size = 21, normalized size = 0.64 \[ -\frac {2 \, {\left (3 \, \tan \left (\frac {1}{2} \, x\right ) + 1\right )}}{3 \, a^{2} {\left (\tan \left (\frac {1}{2} \, x\right ) + 1\right )}^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.09, size = 27, normalized size = 0.82 \[ \frac {\frac {4}{3 \left (\tan \left (\frac {x}{2}\right )+1\right )^{3}}-\frac {2}{\left (\tan \left (\frac {x}{2}\right )+1\right )^{2}}}{a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.60, size = 62, normalized size = 1.88 \[ -\frac {2 \, {\left (\frac {3 \, \sin \relax (x)}{\cos \relax (x) + 1} + 1\right )}}{3 \, {\left (a^{2} + \frac {3 \, a^{2} \sin \relax (x)}{\cos \relax (x) + 1} + \frac {3 \, a^{2} \sin \relax (x)^{2}}{{\left (\cos \relax (x) + 1\right )}^{2}} + \frac {a^{2} \sin \relax (x)^{3}}{{\left (\cos \relax (x) + 1\right )}^{3}}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 6.30, size = 21, normalized size = 0.64 \[ -\frac {2\,\left (3\,\mathrm {tan}\left (\frac {x}{2}\right )+1\right )}{3\,a^2\,{\left (\mathrm {tan}\left (\frac {x}{2}\right )+1\right )}^3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 2.13, size = 87, normalized size = 2.64 \[ - \frac {6 \tan {\left (\frac {x}{2} \right )}}{3 a^{2} \tan ^{3}{\left (\frac {x}{2} \right )} + 9 a^{2} \tan ^{2}{\left (\frac {x}{2} \right )} + 9 a^{2} \tan {\left (\frac {x}{2} \right )} + 3 a^{2}} - \frac {2}{3 a^{2} \tan ^{3}{\left (\frac {x}{2} \right )} + 9 a^{2} \tan ^{2}{\left (\frac {x}{2} \right )} + 9 a^{2} \tan {\left (\frac {x}{2} \right )} + 3 a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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