Integrand size = 8, antiderivative size = 50 \[ \int \frac {1}{(a+a \sin (x))^3} \, dx=-\frac {\cos (x)}{5 (a+a \sin (x))^3}-\frac {2 \cos (x)}{15 a (a+a \sin (x))^2}-\frac {2 \cos (x)}{15 \left (a^3+a^3 \sin (x)\right )} \]
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Time = 0.02 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {2729, 2727} \[ \int \frac {1}{(a+a \sin (x))^3} \, dx=-\frac {2 \cos (x)}{15 \left (a^3 \sin (x)+a^3\right )}-\frac {2 \cos (x)}{15 a (a \sin (x)+a)^2}-\frac {\cos (x)}{5 (a \sin (x)+a)^3} \]
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Rule 2727
Rule 2729
Rubi steps \begin{align*} \text {integral}& = -\frac {\cos (x)}{5 (a+a \sin (x))^3}+\frac {2 \int \frac {1}{(a+a \sin (x))^2} \, dx}{5 a} \\ & = -\frac {\cos (x)}{5 (a+a \sin (x))^3}-\frac {2 \cos (x)}{15 a (a+a \sin (x))^2}+\frac {2 \int \frac {1}{a+a \sin (x)} \, dx}{15 a^2} \\ & = -\frac {\cos (x)}{5 (a+a \sin (x))^3}-\frac {2 \cos (x)}{15 a (a+a \sin (x))^2}-\frac {2 \cos (x)}{15 \left (a^3+a^3 \sin (x)\right )} \\ \end{align*}
Time = 0.11 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.90 \[ \int \frac {1}{(a+a \sin (x))^3} \, dx=-\frac {5 \cos \left (\frac {3 x}{2}\right )-10 \sin \left (\frac {x}{2}\right )+\sin \left (\frac {5 x}{2}\right )}{15 a^3 \left (\cos \left (\frac {x}{2}\right )+\sin \left (\frac {x}{2}\right )\right )^5} \]
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Result contains complex when optimal does not.
Time = 0.33 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.66
method | result | size |
risch | \(\frac {-\frac {4}{15}+\frac {8 \,{\mathrm e}^{2 i x}}{3}+\frac {4 i {\mathrm e}^{i x}}{3}}{\left ({\mathrm e}^{i x}+i\right )^{5} a^{3}}\) | \(33\) |
parallelrisch | \(\frac {12 \tan \left (x \right ) \left (\sec ^{4}\left (x \right )\right )-12 \left (\sec ^{5}\left (x \right )\right )+\tan \left (x \right ) \left (\sec ^{2}\left (x \right )\right )+5 \left (\sec ^{3}\left (x \right )\right )+2 \tan \left (x \right )-7}{15 a^{3}}\) | \(39\) |
default | \(\frac {-\frac {8}{5 \left (\tan \left (\frac {x}{2}\right )+1\right )^{5}}-\frac {16}{3 \left (\tan \left (\frac {x}{2}\right )+1\right )^{3}}+\frac {4}{\left (\tan \left (\frac {x}{2}\right )+1\right )^{2}}+\frac {4}{\left (\tan \left (\frac {x}{2}\right )+1\right )^{4}}-\frac {2}{\tan \left (\frac {x}{2}\right )+1}}{a^{3}}\) | \(57\) |
norman | \(\frac {-\frac {2 \left (\tan ^{4}\left (\frac {x}{2}\right )\right )}{a}-\frac {14}{15 a}-\frac {4 \left (\tan ^{3}\left (\frac {x}{2}\right )\right )}{a}-\frac {8 \tan \left (\frac {x}{2}\right )}{3 a}-\frac {16 \left (\tan ^{2}\left (\frac {x}{2}\right )\right )}{3 a}}{a^{2} \left (\tan \left (\frac {x}{2}\right )+1\right )^{5}}\) | \(61\) |
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Leaf count of result is larger than twice the leaf count of optimal. 92 vs. \(2 (44) = 88\).
Time = 0.26 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.84 \[ \int \frac {1}{(a+a \sin (x))^3} \, dx=-\frac {2 \, \cos \left (x\right )^{3} - 4 \, \cos \left (x\right )^{2} - {\left (2 \, \cos \left (x\right )^{2} + 6 \, \cos \left (x\right ) - 3\right )} \sin \left (x\right ) - 9 \, \cos \left (x\right ) - 3}{15 \, {\left (a^{3} \cos \left (x\right )^{3} + 3 \, a^{3} \cos \left (x\right )^{2} - 2 \, a^{3} \cos \left (x\right ) - 4 \, a^{3} + {\left (a^{3} \cos \left (x\right )^{2} - 2 \, a^{3} \cos \left (x\right ) - 4 \, a^{3}\right )} \sin \left (x\right )\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 348 vs. \(2 (49) = 98\).
Time = 0.60 (sec) , antiderivative size = 348, normalized size of antiderivative = 6.96 \[ \int \frac {1}{(a+a \sin (x))^3} \, dx=- \frac {30 \tan ^{4}{\left (\frac {x}{2} \right )}}{15 a^{3} \tan ^{5}{\left (\frac {x}{2} \right )} + 75 a^{3} \tan ^{4}{\left (\frac {x}{2} \right )} + 150 a^{3} \tan ^{3}{\left (\frac {x}{2} \right )} + 150 a^{3} \tan ^{2}{\left (\frac {x}{2} \right )} + 75 a^{3} \tan {\left (\frac {x}{2} \right )} + 15 a^{3}} - \frac {60 \tan ^{3}{\left (\frac {x}{2} \right )}}{15 a^{3} \tan ^{5}{\left (\frac {x}{2} \right )} + 75 a^{3} \tan ^{4}{\left (\frac {x}{2} \right )} + 150 a^{3} \tan ^{3}{\left (\frac {x}{2} \right )} + 150 a^{3} \tan ^{2}{\left (\frac {x}{2} \right )} + 75 a^{3} \tan {\left (\frac {x}{2} \right )} + 15 a^{3}} - \frac {80 \tan ^{2}{\left (\frac {x}{2} \right )}}{15 a^{3} \tan ^{5}{\left (\frac {x}{2} \right )} + 75 a^{3} \tan ^{4}{\left (\frac {x}{2} \right )} + 150 a^{3} \tan ^{3}{\left (\frac {x}{2} \right )} + 150 a^{3} \tan ^{2}{\left (\frac {x}{2} \right )} + 75 a^{3} \tan {\left (\frac {x}{2} \right )} + 15 a^{3}} - \frac {40 \tan {\left (\frac {x}{2} \right )}}{15 a^{3} \tan ^{5}{\left (\frac {x}{2} \right )} + 75 a^{3} \tan ^{4}{\left (\frac {x}{2} \right )} + 150 a^{3} \tan ^{3}{\left (\frac {x}{2} \right )} + 150 a^{3} \tan ^{2}{\left (\frac {x}{2} \right )} + 75 a^{3} \tan {\left (\frac {x}{2} \right )} + 15 a^{3}} - \frac {14}{15 a^{3} \tan ^{5}{\left (\frac {x}{2} \right )} + 75 a^{3} \tan ^{4}{\left (\frac {x}{2} \right )} + 150 a^{3} \tan ^{3}{\left (\frac {x}{2} \right )} + 150 a^{3} \tan ^{2}{\left (\frac {x}{2} \right )} + 75 a^{3} \tan {\left (\frac {x}{2} \right )} + 15 a^{3}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 128 vs. \(2 (44) = 88\).
Time = 0.20 (sec) , antiderivative size = 128, normalized size of antiderivative = 2.56 \[ \int \frac {1}{(a+a \sin (x))^3} \, dx=-\frac {2 \, {\left (\frac {20 \, \sin \left (x\right )}{\cos \left (x\right ) + 1} + \frac {40 \, \sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} + \frac {30 \, \sin \left (x\right )^{3}}{{\left (\cos \left (x\right ) + 1\right )}^{3}} + \frac {15 \, \sin \left (x\right )^{4}}{{\left (\cos \left (x\right ) + 1\right )}^{4}} + 7\right )}}{15 \, {\left (a^{3} + \frac {5 \, a^{3} \sin \left (x\right )}{\cos \left (x\right ) + 1} + \frac {10 \, a^{3} \sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} + \frac {10 \, a^{3} \sin \left (x\right )^{3}}{{\left (\cos \left (x\right ) + 1\right )}^{3}} + \frac {5 \, a^{3} \sin \left (x\right )^{4}}{{\left (\cos \left (x\right ) + 1\right )}^{4}} + \frac {a^{3} \sin \left (x\right )^{5}}{{\left (\cos \left (x\right ) + 1\right )}^{5}}\right )}} \]
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none
Time = 0.29 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.90 \[ \int \frac {1}{(a+a \sin (x))^3} \, dx=-\frac {2 \, {\left (15 \, \tan \left (\frac {1}{2} \, x\right )^{4} + 30 \, \tan \left (\frac {1}{2} \, x\right )^{3} + 40 \, \tan \left (\frac {1}{2} \, x\right )^{2} + 20 \, \tan \left (\frac {1}{2} \, x\right ) + 7\right )}}{15 \, a^{3} {\left (\tan \left (\frac {1}{2} \, x\right ) + 1\right )}^{5}} \]
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Time = 5.95 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.90 \[ \int \frac {1}{(a+a \sin (x))^3} \, dx=-\frac {2\,\left (15\,{\mathrm {tan}\left (\frac {x}{2}\right )}^4+30\,{\mathrm {tan}\left (\frac {x}{2}\right )}^3+40\,{\mathrm {tan}\left (\frac {x}{2}\right )}^2+20\,\mathrm {tan}\left (\frac {x}{2}\right )+7\right )}{15\,a^3\,{\left (\mathrm {tan}\left (\frac {x}{2}\right )+1\right )}^5} \]
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