Optimal. Leaf size=47 \[ -\frac {2 x}{a^2}-\frac {4 \cos (x)}{3 a^2}-\frac {2 \cos (x)}{a^2 (\sin (x)+1)}+\frac {\sin ^2(x) \cos (x)}{3 (a \sin (x)+a)^2} \]
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Rubi [A] time = 0.14, antiderivative size = 47, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.462, Rules used = {2765, 2968, 3023, 12, 2735, 2648} \[ -\frac {2 x}{a^2}-\frac {4 \cos (x)}{3 a^2}-\frac {2 \cos (x)}{a^2 (\sin (x)+1)}+\frac {\sin ^2(x) \cos (x)}{3 (a \sin (x)+a)^2} \]
Antiderivative was successfully verified.
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Rule 12
Rule 2648
Rule 2735
Rule 2765
Rule 2968
Rule 3023
Rubi steps
\begin {align*} \int \frac {\sin ^3(x)}{(a+a \sin (x))^2} \, dx &=\frac {\cos (x) \sin ^2(x)}{3 (a+a \sin (x))^2}-\frac {\int \frac {\sin (x) (2 a-4 a \sin (x))}{a+a \sin (x)} \, dx}{3 a^2}\\ &=\frac {\cos (x) \sin ^2(x)}{3 (a+a \sin (x))^2}-\frac {\int \frac {2 a \sin (x)-4 a \sin ^2(x)}{a+a \sin (x)} \, dx}{3 a^2}\\ &=-\frac {4 \cos (x)}{3 a^2}+\frac {\cos (x) \sin ^2(x)}{3 (a+a \sin (x))^2}-\frac {\int \frac {6 a^2 \sin (x)}{a+a \sin (x)} \, dx}{3 a^3}\\ &=-\frac {4 \cos (x)}{3 a^2}+\frac {\cos (x) \sin ^2(x)}{3 (a+a \sin (x))^2}-\frac {2 \int \frac {\sin (x)}{a+a \sin (x)} \, dx}{a}\\ &=-\frac {2 x}{a^2}-\frac {4 \cos (x)}{3 a^2}+\frac {\cos (x) \sin ^2(x)}{3 (a+a \sin (x))^2}+\frac {2 \int \frac {1}{a+a \sin (x)} \, dx}{a}\\ &=-\frac {2 x}{a^2}-\frac {4 \cos (x)}{3 a^2}+\frac {\cos (x) \sin ^2(x)}{3 (a+a \sin (x))^2}-\frac {2 \cos (x)}{a^2+a^2 \sin (x)}\\ \end {align*}
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Mathematica [A] time = 0.24, size = 84, normalized size = 1.79 \[ -\frac {\left (\sin \left (\frac {x}{2}\right )+\cos \left (\frac {x}{2}\right )\right ) \left (6 (6 x-5) \cos \left (\frac {x}{2}\right )+(41-12 x) \cos \left (\frac {3 x}{2}\right )-3 \cos \left (\frac {5 x}{2}\right )+6 \sin \left (\frac {x}{2}\right ) (8 x+4 (x+1) \cos (x)+\cos (2 x)-9)\right )}{12 a^2 (\sin (x)+1)^2} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.46, size = 95, normalized size = 2.02 \[ -\frac {{\left (6 \, x - 11\right )} \cos \relax (x)^{2} + 3 \, \cos \relax (x)^{3} - {\left (6 \, x + 13\right )} \cos \relax (x) - {\left (2 \, {\left (3 \, x + 7\right )} \cos \relax (x) + 3 \, \cos \relax (x)^{2} + 12 \, x + 1\right )} \sin \relax (x) - 12 \, x + 1}{3 \, {\left (a^{2} \cos \relax (x)^{2} - a^{2} \cos \relax (x) - 2 \, a^{2} - {\left (a^{2} \cos \relax (x) + 2 \, a^{2}\right )} \sin \relax (x)\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.38, size = 51, normalized size = 1.09 \[ -\frac {2 \, x}{a^{2}} - \frac {2}{{\left (\tan \left (\frac {1}{2} \, x\right )^{2} + 1\right )} a^{2}} - \frac {2 \, {\left (6 \, \tan \left (\frac {1}{2} \, x\right )^{2} + 15 \, \tan \left (\frac {1}{2} \, x\right ) + 7\right )}}{3 \, a^{2} {\left (\tan \left (\frac {1}{2} \, x\right ) + 1\right )}^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.10, size = 66, normalized size = 1.40 \[ -\frac {2}{a^{2} \left (\tan ^{2}\left (\frac {x}{2}\right )+1\right )}-\frac {4 \arctan \left (\tan \left (\frac {x}{2}\right )\right )}{a^{2}}+\frac {4}{3 a^{2} \left (\tan \left (\frac {x}{2}\right )+1\right )^{3}}-\frac {2}{a^{2} \left (\tan \left (\frac {x}{2}\right )+1\right )^{2}}-\frac {4}{a^{2} \left (\tan \left (\frac {x}{2}\right )+1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.87, size = 144, normalized size = 3.06 \[ -\frac {4 \, {\left (\frac {12 \, \sin \relax (x)}{\cos \relax (x) + 1} + \frac {11 \, \sin \relax (x)^{2}}{{\left (\cos \relax (x) + 1\right )}^{2}} + \frac {9 \, \sin \relax (x)^{3}}{{\left (\cos \relax (x) + 1\right )}^{3}} + \frac {3 \, \sin \relax (x)^{4}}{{\left (\cos \relax (x) + 1\right )}^{4}} + 5\right )}}{3 \, {\left (a^{2} + \frac {3 \, a^{2} \sin \relax (x)}{\cos \relax (x) + 1} + \frac {4 \, a^{2} \sin \relax (x)^{2}}{{\left (\cos \relax (x) + 1\right )}^{2}} + \frac {4 \, a^{2} \sin \relax (x)^{3}}{{\left (\cos \relax (x) + 1\right )}^{3}} + \frac {3 \, a^{2} \sin \relax (x)^{4}}{{\left (\cos \relax (x) + 1\right )}^{4}} + \frac {a^{2} \sin \relax (x)^{5}}{{\left (\cos \relax (x) + 1\right )}^{5}}\right )}} - \frac {4 \, \arctan \left (\frac {\sin \relax (x)}{\cos \relax (x) + 1}\right )}{a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 6.46, size = 62, normalized size = 1.32 \[ -\frac {2\,x}{a^2}-\frac {4\,{\mathrm {tan}\left (\frac {x}{2}\right )}^4+12\,{\mathrm {tan}\left (\frac {x}{2}\right )}^3+\frac {44\,{\mathrm {tan}\left (\frac {x}{2}\right )}^2}{3}+16\,\mathrm {tan}\left (\frac {x}{2}\right )+\frac {20}{3}}{a^2\,\left ({\mathrm {tan}\left (\frac {x}{2}\right )}^2+1\right )\,{\left (\mathrm {tan}\left (\frac {x}{2}\right )+1\right )}^3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 6.79, size = 779, normalized size = 16.57 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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