Integrand size = 12, antiderivative size = 76 \[ \int \frac {1}{(3+3 \sin (e+f x))^3} \, dx=-\frac {\cos (e+f x)}{5 f (3+3 \sin (e+f x))^3}-\frac {2 \cos (e+f x)}{45 f (3+3 \sin (e+f x))^2}-\frac {2 \cos (e+f x)}{15 f (27+27 \sin (e+f x))} \]
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Time = 0.03 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.09, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {2729, 2727} \[ \int \frac {1}{(3+3 \sin (e+f x))^3} \, dx=-\frac {2 \cos (e+f x)}{15 f \left (a^3 \sin (e+f x)+a^3\right )}-\frac {2 \cos (e+f x)}{15 a f (a \sin (e+f x)+a)^2}-\frac {\cos (e+f x)}{5 f (a \sin (e+f x)+a)^3} \]
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Rule 2727
Rule 2729
Rubi steps \begin{align*} \text {integral}& = -\frac {\cos (e+f x)}{5 f (a+a \sin (e+f x))^3}+\frac {2 \int \frac {1}{(a+a \sin (e+f x))^2} \, dx}{5 a} \\ & = -\frac {\cos (e+f x)}{5 f (a+a \sin (e+f x))^3}-\frac {2 \cos (e+f x)}{15 a f (a+a \sin (e+f x))^2}+\frac {2 \int \frac {1}{a+a \sin (e+f x)} \, dx}{15 a^2} \\ & = -\frac {\cos (e+f x)}{5 f (a+a \sin (e+f x))^3}-\frac {2 \cos (e+f x)}{15 a f (a+a \sin (e+f x))^2}-\frac {2 \cos (e+f x)}{15 f \left (a^3+a^3 \sin (e+f x)\right )} \\ \end{align*}
Time = 0.19 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.96 \[ \int \frac {1}{(3+3 \sin (e+f x))^3} \, dx=\frac {10-15 \cos (e+f x)-6 \cos (2 (e+f x))+\cos (3 (e+f x))+15 \sin (e+f x)-6 \sin (2 (e+f x))-\sin (3 (e+f x))}{810 f (1+\sin (e+f x))^3} \]
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Result contains complex when optimal does not.
Time = 0.58 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.63
method | result | size |
risch | \(\frac {-\frac {4}{15}+\frac {8 \,{\mathrm e}^{2 i \left (f x +e \right )}}{3}+\frac {4 i {\mathrm e}^{i \left (f x +e \right )}}{3}}{f \,a^{3} \left ({\mathrm e}^{i \left (f x +e \right )}+i\right )^{5}}\) | \(48\) |
parallelrisch | \(\frac {-\frac {14}{15}-2 \left (\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )-4 \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )-\frac {16 \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{3}-\frac {8 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{3}}{f \,a^{3} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{5}}\) | \(74\) |
derivativedivides | \(\frac {-\frac {16}{3 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{3}}-\frac {2}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1}-\frac {8}{5 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{5}}+\frac {4}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{4}}+\frac {4}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{2}}}{a^{3} f}\) | \(85\) |
default | \(\frac {-\frac {16}{3 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{3}}-\frac {2}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1}-\frac {8}{5 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{5}}+\frac {4}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{4}}+\frac {4}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{2}}}{a^{3} f}\) | \(85\) |
norman | \(\frac {-\frac {2 \left (\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{f a}-\frac {14}{15 a f}-\frac {4 \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{f a}-\frac {8 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{3 f a}-\frac {16 \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{3 f a}}{a^{2} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{5}}\) | \(101\) |
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Time = 0.26 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.93 \[ \int \frac {1}{(3+3 \sin (e+f x))^3} \, dx=-\frac {2 \, \cos \left (f x + e\right )^{3} - 4 \, \cos \left (f x + e\right )^{2} - {\left (2 \, \cos \left (f x + e\right )^{2} + 6 \, \cos \left (f x + e\right ) - 3\right )} \sin \left (f x + e\right ) - 9 \, \cos \left (f x + e\right ) - 3}{15 \, {\left (a^{3} f \cos \left (f x + e\right )^{3} + 3 \, a^{3} f \cos \left (f x + e\right )^{2} - 2 \, a^{3} f \cos \left (f x + e\right ) - 4 \, a^{3} f + {\left (a^{3} f \cos \left (f x + e\right )^{2} - 2 \, a^{3} f \cos \left (f x + e\right ) - 4 \, a^{3} f\right )} \sin \left (f x + e\right )\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 558 vs. \(2 (73) = 146\).
Time = 1.28 (sec) , antiderivative size = 558, normalized size of antiderivative = 7.34 \[ \int \frac {1}{(3+3 \sin (e+f x))^3} \, dx=\begin {cases} - \frac {30 \tan ^{4}{\left (\frac {e}{2} + \frac {f x}{2} \right )}}{15 a^{3} f \tan ^{5}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 75 a^{3} f \tan ^{4}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 150 a^{3} f \tan ^{3}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 150 a^{3} f \tan ^{2}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 75 a^{3} f \tan {\left (\frac {e}{2} + \frac {f x}{2} \right )} + 15 a^{3} f} - \frac {60 \tan ^{3}{\left (\frac {e}{2} + \frac {f x}{2} \right )}}{15 a^{3} f \tan ^{5}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 75 a^{3} f \tan ^{4}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 150 a^{3} f \tan ^{3}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 150 a^{3} f \tan ^{2}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 75 a^{3} f \tan {\left (\frac {e}{2} + \frac {f x}{2} \right )} + 15 a^{3} f} - \frac {80 \tan ^{2}{\left (\frac {e}{2} + \frac {f x}{2} \right )}}{15 a^{3} f \tan ^{5}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 75 a^{3} f \tan ^{4}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 150 a^{3} f \tan ^{3}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 150 a^{3} f \tan ^{2}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 75 a^{3} f \tan {\left (\frac {e}{2} + \frac {f x}{2} \right )} + 15 a^{3} f} - \frac {40 \tan {\left (\frac {e}{2} + \frac {f x}{2} \right )}}{15 a^{3} f \tan ^{5}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 75 a^{3} f \tan ^{4}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 150 a^{3} f \tan ^{3}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 150 a^{3} f \tan ^{2}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 75 a^{3} f \tan {\left (\frac {e}{2} + \frac {f x}{2} \right )} + 15 a^{3} f} - \frac {14}{15 a^{3} f \tan ^{5}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 75 a^{3} f \tan ^{4}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 150 a^{3} f \tan ^{3}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 150 a^{3} f \tan ^{2}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 75 a^{3} f \tan {\left (\frac {e}{2} + \frac {f x}{2} \right )} + 15 a^{3} f} & \text {for}\: f \neq 0 \\\frac {x}{\left (a \sin {\left (e \right )} + a\right )^{3}} & \text {otherwise} \end {cases} \]
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Leaf count of result is larger than twice the leaf count of optimal. 203 vs. \(2 (77) = 154\).
Time = 0.20 (sec) , antiderivative size = 203, normalized size of antiderivative = 2.67 \[ \int \frac {1}{(3+3 \sin (e+f x))^3} \, dx=-\frac {2 \, {\left (\frac {20 \, \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {40 \, \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {30 \, \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} + \frac {15 \, \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} + 7\right )}}{15 \, {\left (a^{3} + \frac {5 \, a^{3} \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {10 \, a^{3} \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {10 \, a^{3} \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} + \frac {5 \, a^{3} \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} + \frac {a^{3} \sin \left (f x + e\right )^{5}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{5}}\right )} f} \]
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Time = 0.31 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.96 \[ \int \frac {1}{(3+3 \sin (e+f x))^3} \, dx=-\frac {2 \, {\left (15 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} + 30 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 40 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 20 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 7\right )}}{15 \, a^{3} f {\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1\right )}^{5}} \]
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Time = 6.90 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.75 \[ \int \frac {1}{(3+3 \sin (e+f x))^3} \, dx=-\frac {2\,\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\left (7\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4+20\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3\,\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )+40\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2\,{\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2+30\,\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )\,{\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3+15\,{\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4\right )}{15\,a^3\,f\,{\left (\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )+\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )\right )}^5} \]
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