Optimal. Leaf size=47 \[ -\frac {2 x}{a^2}-\frac {4 \cos (x)}{3 a^2}-\frac {2 \cos (x)}{a^2 (1+\sin (x))}+\frac {\cos (x) \sin ^2(x)}{3 (a+a \sin (x))^2} \]
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Rubi [A]
time = 0.10, 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 = {2844, 3047,
3102, 12, 2814, 2727} \begin {gather*} -\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} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 2727
Rule 2814
Rule 2844
Rule 3047
Rule 3102
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 = 0.16, size = 84, normalized size = 1.79 \begin {gather*} -\frac {\left (\cos \left (\frac {x}{2}\right )+\sin \left (\frac {x}{2}\right )\right ) \left (6 (-5+6 x) \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 (-9+8 x+4 (1+x) \cos (x)+\cos (2 x)) \sin \left (\frac {x}{2}\right )\right )}{12 a^2 (1+\sin (x))^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.15, size = 56, normalized size = 1.19
method | result | size |
default | \(\frac {\frac {4}{3 \left (\tan \left (\frac {x}{2}\right )+1\right )^{3}}-\frac {2}{\left (\tan \left (\frac {x}{2}\right )+1\right )^{2}}-\frac {4}{\tan \left (\frac {x}{2}\right )+1}-\frac {2}{\tan ^{2}\left (\frac {x}{2}\right )+1}-4 \arctan \left (\tan \left (\frac {x}{2}\right )\right )}{a^{2}}\) | \(56\) |
risch | \(-\frac {2 x}{a^{2}}-\frac {{\mathrm e}^{i x}}{2 a^{2}}-\frac {{\mathrm e}^{-i x}}{2 a^{2}}-\frac {2 \left (15 i {\mathrm e}^{i x}+9 \,{\mathrm e}^{2 i x}-8\right )}{3 a^{2} \left ({\mathrm e}^{i x}+i\right )^{3}}\) | \(60\) |
norman | \(\frac {-\frac {16 \tan \left (\frac {x}{2}\right )}{a}-\frac {4 \left (\tan ^{8}\left (\frac {x}{2}\right )\right )}{a}-\frac {2 x}{a}-\frac {20}{3 a}-\frac {6 x \tan \left (\frac {x}{2}\right )}{a}-\frac {12 x \left (\tan ^{2}\left (\frac {x}{2}\right )\right )}{a}-\frac {20 x \left (\tan ^{3}\left (\frac {x}{2}\right )\right )}{a}-\frac {24 x \left (\tan ^{4}\left (\frac {x}{2}\right )\right )}{a}-\frac {24 x \left (\tan ^{5}\left (\frac {x}{2}\right )\right )}{a}-\frac {20 x \left (\tan ^{6}\left (\frac {x}{2}\right )\right )}{a}-\frac {12 x \left (\tan ^{7}\left (\frac {x}{2}\right )\right )}{a}-\frac {6 x \left (\tan ^{8}\left (\frac {x}{2}\right )\right )}{a}-\frac {2 x \left (\tan ^{9}\left (\frac {x}{2}\right )\right )}{a}-\frac {12 \left (\tan ^{7}\left (\frac {x}{2}\right )\right )}{a}-\frac {28 \left (\tan ^{2}\left (\frac {x}{2}\right )\right )}{a}-\frac {40 \left (\tan ^{4}\left (\frac {x}{2}\right )\right )}{a}-\frac {40 \left (\tan ^{5}\left (\frac {x}{2}\right )\right )}{a}-\frac {44 \left (\tan ^{3}\left (\frac {x}{2}\right )\right )}{a}-\frac {68 \left (\tan ^{6}\left (\frac {x}{2}\right )\right )}{3 a}}{\left (\tan ^{2}\left (\frac {x}{2}\right )+1\right )^{3} a \left (\tan \left (\frac {x}{2}\right )+1\right )^{3}}\) | \(227\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 144 vs.
\(2 (43) = 86\).
time = 0.55, size = 144, normalized size = 3.06 \begin {gather*} -\frac {4 \, {\left (\frac {12 \, \sin \left (x\right )}{\cos \left (x\right ) + 1} + \frac {11 \, \sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} + \frac {9 \, \sin \left (x\right )^{3}}{{\left (\cos \left (x\right ) + 1\right )}^{3}} + \frac {3 \, \sin \left (x\right )^{4}}{{\left (\cos \left (x\right ) + 1\right )}^{4}} + 5\right )}}{3 \, {\left (a^{2} + \frac {3 \, a^{2} \sin \left (x\right )}{\cos \left (x\right ) + 1} + \frac {4 \, a^{2} \sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} + \frac {4 \, a^{2} \sin \left (x\right )^{3}}{{\left (\cos \left (x\right ) + 1\right )}^{3}} + \frac {3 \, a^{2} \sin \left (x\right )^{4}}{{\left (\cos \left (x\right ) + 1\right )}^{4}} + \frac {a^{2} \sin \left (x\right )^{5}}{{\left (\cos \left (x\right ) + 1\right )}^{5}}\right )}} - \frac {4 \, \arctan \left (\frac {\sin \left (x\right )}{\cos \left (x\right ) + 1}\right )}{a^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 95 vs.
\(2 (43) = 86\).
time = 0.34, size = 95, normalized size = 2.02 \begin {gather*} -\frac {{\left (6 \, x - 11\right )} \cos \left (x\right )^{2} + 3 \, \cos \left (x\right )^{3} - {\left (6 \, x + 13\right )} \cos \left (x\right ) - {\left (2 \, {\left (3 \, x + 7\right )} \cos \left (x\right ) + 3 \, \cos \left (x\right )^{2} + 12 \, x + 1\right )} \sin \left (x\right ) - 12 \, x + 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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 779 vs.
\(2 (48) = 96\).
time = 2.61, size = 779, normalized size = 16.57 \begin {gather*} - \frac {6 x \tan ^{5}{\left (\frac {x}{2} \right )}}{3 a^{2} \tan ^{5}{\left (\frac {x}{2} \right )} + 9 a^{2} \tan ^{4}{\left (\frac {x}{2} \right )} + 12 a^{2} \tan ^{3}{\left (\frac {x}{2} \right )} + 12 a^{2} \tan ^{2}{\left (\frac {x}{2} \right )} + 9 a^{2} \tan {\left (\frac {x}{2} \right )} + 3 a^{2}} - \frac {18 x \tan ^{4}{\left (\frac {x}{2} \right )}}{3 a^{2} \tan ^{5}{\left (\frac {x}{2} \right )} + 9 a^{2} \tan ^{4}{\left (\frac {x}{2} \right )} + 12 a^{2} \tan ^{3}{\left (\frac {x}{2} \right )} + 12 a^{2} \tan ^{2}{\left (\frac {x}{2} \right )} + 9 a^{2} \tan {\left (\frac {x}{2} \right )} + 3 a^{2}} - \frac {24 x \tan ^{3}{\left (\frac {x}{2} \right )}}{3 a^{2} \tan ^{5}{\left (\frac {x}{2} \right )} + 9 a^{2} \tan ^{4}{\left (\frac {x}{2} \right )} + 12 a^{2} \tan ^{3}{\left (\frac {x}{2} \right )} + 12 a^{2} \tan ^{2}{\left (\frac {x}{2} \right )} + 9 a^{2} \tan {\left (\frac {x}{2} \right )} + 3 a^{2}} - \frac {24 x \tan ^{2}{\left (\frac {x}{2} \right )}}{3 a^{2} \tan ^{5}{\left (\frac {x}{2} \right )} + 9 a^{2} \tan ^{4}{\left (\frac {x}{2} \right )} + 12 a^{2} \tan ^{3}{\left (\frac {x}{2} \right )} + 12 a^{2} \tan ^{2}{\left (\frac {x}{2} \right )} + 9 a^{2} \tan {\left (\frac {x}{2} \right )} + 3 a^{2}} - \frac {18 x \tan {\left (\frac {x}{2} \right )}}{3 a^{2} \tan ^{5}{\left (\frac {x}{2} \right )} + 9 a^{2} \tan ^{4}{\left (\frac {x}{2} \right )} + 12 a^{2} \tan ^{3}{\left (\frac {x}{2} \right )} + 12 a^{2} \tan ^{2}{\left (\frac {x}{2} \right )} + 9 a^{2} \tan {\left (\frac {x}{2} \right )} + 3 a^{2}} - \frac {6 x}{3 a^{2} \tan ^{5}{\left (\frac {x}{2} \right )} + 9 a^{2} \tan ^{4}{\left (\frac {x}{2} \right )} + 12 a^{2} \tan ^{3}{\left (\frac {x}{2} \right )} + 12 a^{2} \tan ^{2}{\left (\frac {x}{2} \right )} + 9 a^{2} \tan {\left (\frac {x}{2} \right )} + 3 a^{2}} - \frac {12 \tan ^{4}{\left (\frac {x}{2} \right )}}{3 a^{2} \tan ^{5}{\left (\frac {x}{2} \right )} + 9 a^{2} \tan ^{4}{\left (\frac {x}{2} \right )} + 12 a^{2} \tan ^{3}{\left (\frac {x}{2} \right )} + 12 a^{2} \tan ^{2}{\left (\frac {x}{2} \right )} + 9 a^{2} \tan {\left (\frac {x}{2} \right )} + 3 a^{2}} - \frac {36 \tan ^{3}{\left (\frac {x}{2} \right )}}{3 a^{2} \tan ^{5}{\left (\frac {x}{2} \right )} + 9 a^{2} \tan ^{4}{\left (\frac {x}{2} \right )} + 12 a^{2} \tan ^{3}{\left (\frac {x}{2} \right )} + 12 a^{2} \tan ^{2}{\left (\frac {x}{2} \right )} + 9 a^{2} \tan {\left (\frac {x}{2} \right )} + 3 a^{2}} - \frac {44 \tan ^{2}{\left (\frac {x}{2} \right )}}{3 a^{2} \tan ^{5}{\left (\frac {x}{2} \right )} + 9 a^{2} \tan ^{4}{\left (\frac {x}{2} \right )} + 12 a^{2} \tan ^{3}{\left (\frac {x}{2} \right )} + 12 a^{2} \tan ^{2}{\left (\frac {x}{2} \right )} + 9 a^{2} \tan {\left (\frac {x}{2} \right )} + 3 a^{2}} - \frac {48 \tan {\left (\frac {x}{2} \right )}}{3 a^{2} \tan ^{5}{\left (\frac {x}{2} \right )} + 9 a^{2} \tan ^{4}{\left (\frac {x}{2} \right )} + 12 a^{2} \tan ^{3}{\left (\frac {x}{2} \right )} + 12 a^{2} \tan ^{2}{\left (\frac {x}{2} \right )} + 9 a^{2} \tan {\left (\frac {x}{2} \right )} + 3 a^{2}} - \frac {20}{3 a^{2} \tan ^{5}{\left (\frac {x}{2} \right )} + 9 a^{2} \tan ^{4}{\left (\frac {x}{2} \right )} + 12 a^{2} \tan ^{3}{\left (\frac {x}{2} \right )} + 12 a^{2} \tan ^{2}{\left (\frac {x}{2} \right )} + 9 a^{2} \tan {\left (\frac {x}{2} \right )} + 3 a^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.51, size = 51, normalized size = 1.09 \begin {gather*} -\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}} \end {gather*}
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 \begin {gather*} -\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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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