Optimal. Leaf size=50 \[ \frac {\cos (x)}{5 (a+a \sin (x))^3}-\frac {\cos (x)}{5 a (a+a \sin (x))^2}-\frac {\cos (x)}{5 \left (a^3+a^3 \sin (x)\right )} \]
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Rubi [A]
time = 0.03, antiderivative size = 50, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {2829, 2729,
2727} \begin {gather*} -\frac {\cos (x)}{5 \left (a^3 \sin (x)+a^3\right )}-\frac {\cos (x)}{5 a (a \sin (x)+a)^2}+\frac {\cos (x)}{5 (a \sin (x)+a)^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 2727
Rule 2729
Rule 2829
Rubi steps
\begin {align*} \int \frac {\sin (x)}{(a+a \sin (x))^3} \, dx &=\frac {\cos (x)}{5 (a+a \sin (x))^3}+\frac {3 \int \frac {1}{(a+a \sin (x))^2} \, dx}{5 a}\\ &=\frac {\cos (x)}{5 (a+a \sin (x))^3}-\frac {\cos (x)}{5 a (a+a \sin (x))^2}+\frac {\int \frac {1}{a+a \sin (x)} \, dx}{5 a^2}\\ &=\frac {\cos (x)}{5 (a+a \sin (x))^3}-\frac {\cos (x)}{5 a (a+a \sin (x))^2}-\frac {\cos (x)}{5 \left (a^3+a^3 \sin (x)\right )}\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 41, normalized size = 0.82 \begin {gather*} \frac {\sin ^2\left (\frac {x}{2}\right ) (7+4 \cos (x)-\cos (2 x)+8 \sin (x)+\sin (2 x))}{5 a^3 (1+\sin (x))^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.14, size = 45, normalized size = 0.90
method | result | size |
risch | \(-\frac {2 i \left (5 i {\mathrm e}^{2 i x}+5 \,{\mathrm e}^{3 i x}-i-5 \,{\mathrm e}^{i x}\right )}{5 \left ({\mathrm e}^{i x}+i\right )^{5} a^{3}}\) | \(42\) |
default | \(\frac {\frac {8}{5 \left (\tan \left (\frac {x}{2}\right )+1\right )^{5}}-\frac {4}{\left (\tan \left (\frac {x}{2}\right )+1\right )^{4}}-\frac {2}{\left (\tan \left (\frac {x}{2}\right )+1\right )^{2}}+\frac {4}{\left (\tan \left (\frac {x}{2}\right )+1\right )^{3}}}{a^{3}}\) | \(45\) |
norman | \(\frac {-\frac {2 \tan \left (\frac {x}{2}\right )}{a}-\frac {2 \left (\tan ^{5}\left (\frac {x}{2}\right )\right )}{a}-\frac {4 \left (\tan ^{3}\left (\frac {x}{2}\right )\right )}{a}-\frac {2 \left (\tan ^{4}\left (\frac {x}{2}\right )\right )}{a}-\frac {2}{5 a}-\frac {12 \left (\tan ^{2}\left (\frac {x}{2}\right )\right )}{5 a}}{\left (\tan ^{2}\left (\frac {x}{2}\right )+1\right ) a^{2} \left (\tan \left (\frac {x}{2}\right )+1\right )^{5}}\) | \(82\) |
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. 116 vs.
\(2 (44) = 88\).
time = 0.29, size = 116, normalized size = 2.32 \begin {gather*} -\frac {2 \, {\left (\frac {5 \, \sin \left (x\right )}{\cos \left (x\right ) + 1} + \frac {5 \, \sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} + \frac {5 \, \sin \left (x\right )^{3}}{{\left (\cos \left (x\right ) + 1\right )}^{3}} + 1\right )}}{5 \, {\left (a^{3} + \frac {5 \, a^{3} \sin \left (x\right )}{\cos \left (x\right ) + 1} + \frac {10 \, a^{3} \sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} + \frac {10 \, a^{3} \sin \left (x\right )^{3}}{{\left (\cos \left (x\right ) + 1\right )}^{3}} + \frac {5 \, a^{3} \sin \left (x\right )^{4}}{{\left (\cos \left (x\right ) + 1\right )}^{4}} + \frac {a^{3} \sin \left (x\right )^{5}}{{\left (\cos \left (x\right ) + 1\right )}^{5}}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.34, size = 88, normalized size = 1.76 \begin {gather*} -\frac {\cos \left (x\right )^{3} - 2 \, \cos \left (x\right )^{2} - {\left (\cos \left (x\right )^{2} + 3 \, \cos \left (x\right ) + 1\right )} \sin \left (x\right ) - 2 \, \cos \left (x\right ) + 1}{5 \, {\left (a^{3} \cos \left (x\right )^{3} + 3 \, a^{3} \cos \left (x\right )^{2} - 2 \, a^{3} \cos \left (x\right ) - 4 \, a^{3} + {\left (a^{3} \cos \left (x\right )^{2} - 2 \, a^{3} \cos \left (x\right ) - 4 \, a^{3}\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. 277 vs.
\(2 (44) = 88\).
time = 1.52, size = 277, normalized size = 5.54 \begin {gather*} - \frac {10 \tan ^{3}{\left (\frac {x}{2} \right )}}{5 a^{3} \tan ^{5}{\left (\frac {x}{2} \right )} + 25 a^{3} \tan ^{4}{\left (\frac {x}{2} \right )} + 50 a^{3} \tan ^{3}{\left (\frac {x}{2} \right )} + 50 a^{3} \tan ^{2}{\left (\frac {x}{2} \right )} + 25 a^{3} \tan {\left (\frac {x}{2} \right )} + 5 a^{3}} - \frac {10 \tan ^{2}{\left (\frac {x}{2} \right )}}{5 a^{3} \tan ^{5}{\left (\frac {x}{2} \right )} + 25 a^{3} \tan ^{4}{\left (\frac {x}{2} \right )} + 50 a^{3} \tan ^{3}{\left (\frac {x}{2} \right )} + 50 a^{3} \tan ^{2}{\left (\frac {x}{2} \right )} + 25 a^{3} \tan {\left (\frac {x}{2} \right )} + 5 a^{3}} - \frac {10 \tan {\left (\frac {x}{2} \right )}}{5 a^{3} \tan ^{5}{\left (\frac {x}{2} \right )} + 25 a^{3} \tan ^{4}{\left (\frac {x}{2} \right )} + 50 a^{3} \tan ^{3}{\left (\frac {x}{2} \right )} + 50 a^{3} \tan ^{2}{\left (\frac {x}{2} \right )} + 25 a^{3} \tan {\left (\frac {x}{2} \right )} + 5 a^{3}} - \frac {2}{5 a^{3} \tan ^{5}{\left (\frac {x}{2} \right )} + 25 a^{3} \tan ^{4}{\left (\frac {x}{2} \right )} + 50 a^{3} \tan ^{3}{\left (\frac {x}{2} \right )} + 50 a^{3} \tan ^{2}{\left (\frac {x}{2} \right )} + 25 a^{3} \tan {\left (\frac {x}{2} \right )} + 5 a^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.53, size = 37, normalized size = 0.74 \begin {gather*} -\frac {2 \, {\left (5 \, \tan \left (\frac {1}{2} \, x\right )^{3} + 5 \, \tan \left (\frac {1}{2} \, x\right )^{2} + 5 \, \tan \left (\frac {1}{2} \, x\right ) + 1\right )}}{5 \, a^{3} {\left (\tan \left (\frac {1}{2} \, x\right ) + 1\right )}^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 6.85, size = 37, normalized size = 0.74 \begin {gather*} -\frac {2\,\left (5\,{\mathrm {tan}\left (\frac {x}{2}\right )}^3+5\,{\mathrm {tan}\left (\frac {x}{2}\right )}^2+5\,\mathrm {tan}\left (\frac {x}{2}\right )+1\right )}{5\,a^3\,{\left (\mathrm {tan}\left (\frac {x}{2}\right )+1\right )}^5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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