Optimal. Leaf size=50 \[ -\frac {7 \cos (x)}{15 \left (a^3 \sin (x)+a^3\right )}+\frac {8 \cos (x)}{15 a (a \sin (x)+a)^2}-\frac {\cos (x)}{5 (a \sin (x)+a)^3} \]
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Rubi [A] time = 0.08, antiderivative size = 50, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {2758, 2750, 2648} \[ -\frac {7 \cos (x)}{15 \left (a^3 \sin (x)+a^3\right )}+\frac {8 \cos (x)}{15 a (a \sin (x)+a)^2}-\frac {\cos (x)}{5 (a \sin (x)+a)^3} \]
Antiderivative was successfully verified.
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Rule 2648
Rule 2750
Rule 2758
Rubi steps
\begin {align*} \int \frac {\sin ^2(x)}{(a+a \sin (x))^3} \, dx &=-\frac {\cos (x)}{5 (a+a \sin (x))^3}+\frac {\int \frac {-3 a+5 a \sin (x)}{(a+a \sin (x))^2} \, dx}{5 a^2}\\ &=-\frac {\cos (x)}{5 (a+a \sin (x))^3}+\frac {8 \cos (x)}{15 a (a+a \sin (x))^2}+\frac {7 \int \frac {1}{a+a \sin (x)} \, dx}{15 a^2}\\ &=-\frac {\cos (x)}{5 (a+a \sin (x))^3}+\frac {8 \cos (x)}{15 a (a+a \sin (x))^2}-\frac {7 \cos (x)}{15 \left (a^3+a^3 \sin (x)\right )}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 47, normalized size = 0.94 \[ \frac {105 \sin (x)-12 \sin (2 x)-7 \sin (3 x)-15 \cos (x)-42 \cos (2 x)+7 \cos (3 x)+70}{60 a^3 (\sin (x)+1)^3} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.42, size = 90, normalized size = 1.80 \[ -\frac {7 \, \cos \relax (x)^{3} + \cos \relax (x)^{2} - {\left (7 \, \cos \relax (x)^{2} + 6 \, \cos \relax (x) - 3\right )} \sin \relax (x) - 9 \, \cos \relax (x) - 3}{15 \, {\left (a^{3} \cos \relax (x)^{3} + 3 \, a^{3} \cos \relax (x)^{2} - 2 \, a^{3} \cos \relax (x) - 4 \, a^{3} + {\left (a^{3} \cos \relax (x)^{2} - 2 \, a^{3} \cos \relax (x) - 4 \, a^{3}\right )} \sin \relax (x)\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.27, size = 29, normalized size = 0.58 \[ -\frac {4 \, {\left (10 \, \tan \left (\frac {1}{2} \, x\right )^{2} + 5 \, \tan \left (\frac {1}{2} \, x\right ) + 1\right )}}{15 \, a^{3} {\left (\tan \left (\frac {1}{2} \, x\right ) + 1\right )}^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.09, size = 37, normalized size = 0.74 \[ \frac {-\frac {8}{5 \left (\tan \left (\frac {x}{2}\right )+1\right )^{5}}+\frac {4}{\left (\tan \left (\frac {x}{2}\right )+1\right )^{4}}-\frac {8}{3 \left (\tan \left (\frac {x}{2}\right )+1\right )^{3}}}{a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.64, size = 104, normalized size = 2.08 \[ -\frac {4 \, {\left (\frac {5 \, \sin \relax (x)}{\cos \relax (x) + 1} + \frac {10 \, \sin \relax (x)^{2}}{{\left (\cos \relax (x) + 1\right )}^{2}} + 1\right )}}{15 \, {\left (a^{3} + \frac {5 \, a^{3} \sin \relax (x)}{\cos \relax (x) + 1} + \frac {10 \, a^{3} \sin \relax (x)^{2}}{{\left (\cos \relax (x) + 1\right )}^{2}} + \frac {10 \, a^{3} \sin \relax (x)^{3}}{{\left (\cos \relax (x) + 1\right )}^{3}} + \frac {5 \, a^{3} \sin \relax (x)^{4}}{{\left (\cos \relax (x) + 1\right )}^{4}} + \frac {a^{3} \sin \relax (x)^{5}}{{\left (\cos \relax (x) + 1\right )}^{5}}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 6.64, size = 29, normalized size = 0.58 \[ -\frac {4\,\left (10\,{\mathrm {tan}\left (\frac {x}{2}\right )}^2+5\,\mathrm {tan}\left (\frac {x}{2}\right )+1\right )}{15\,a^3\,{\left (\mathrm {tan}\left (\frac {x}{2}\right )+1\right )}^5} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 6.92, size = 206, normalized size = 4.12 \[ - \frac {40 \tan ^{2}{\left (\frac {x}{2} \right )}}{15 a^{3} \tan ^{5}{\left (\frac {x}{2} \right )} + 75 a^{3} \tan ^{4}{\left (\frac {x}{2} \right )} + 150 a^{3} \tan ^{3}{\left (\frac {x}{2} \right )} + 150 a^{3} \tan ^{2}{\left (\frac {x}{2} \right )} + 75 a^{3} \tan {\left (\frac {x}{2} \right )} + 15 a^{3}} - \frac {20 \tan {\left (\frac {x}{2} \right )}}{15 a^{3} \tan ^{5}{\left (\frac {x}{2} \right )} + 75 a^{3} \tan ^{4}{\left (\frac {x}{2} \right )} + 150 a^{3} \tan ^{3}{\left (\frac {x}{2} \right )} + 150 a^{3} \tan ^{2}{\left (\frac {x}{2} \right )} + 75 a^{3} \tan {\left (\frac {x}{2} \right )} + 15 a^{3}} - \frac {4}{15 a^{3} \tan ^{5}{\left (\frac {x}{2} \right )} + 75 a^{3} \tan ^{4}{\left (\frac {x}{2} \right )} + 150 a^{3} \tan ^{3}{\left (\frac {x}{2} \right )} + 150 a^{3} \tan ^{2}{\left (\frac {x}{2} \right )} + 75 a^{3} \tan {\left (\frac {x}{2} \right )} + 15 a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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