Integrand size = 16, antiderivative size = 19 \[ \int x^2 \cos \left (4 x^3\right ) \cos \left (5 x^3\right ) \, dx=\frac {\sin \left (x^3\right )}{6}+\frac {1}{54} \sin \left (9 x^3\right ) \]
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Time = 0.05 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {4668, 3461, 2717} \[ \int x^2 \cos \left (4 x^3\right ) \cos \left (5 x^3\right ) \, dx=\frac {\sin \left (x^3\right )}{6}+\frac {1}{54} \sin \left (9 x^3\right ) \]
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Rule 2717
Rule 3461
Rule 4668
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {1}{2} x^2 \cos \left (x^3\right )+\frac {1}{2} x^2 \cos \left (9 x^3\right )\right ) \, dx \\ & = \frac {1}{2} \int x^2 \cos \left (x^3\right ) \, dx+\frac {1}{2} \int x^2 \cos \left (9 x^3\right ) \, dx \\ & = \frac {1}{6} \text {Subst}\left (\int \cos (x) \, dx,x,x^3\right )+\frac {1}{6} \text {Subst}\left (\int \cos (9 x) \, dx,x,x^3\right ) \\ & = \frac {\sin \left (x^3\right )}{6}+\frac {1}{54} \sin \left (9 x^3\right ) \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.00 \[ \int x^2 \cos \left (4 x^3\right ) \cos \left (5 x^3\right ) \, dx=\frac {\sin \left (x^3\right )}{6}+\frac {1}{54} \sin \left (9 x^3\right ) \]
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Time = 0.45 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.84
method | result | size |
default | \(\frac {\sin \left (x^{3}\right )}{6}+\frac {\sin \left (9 x^{3}\right )}{54}\) | \(16\) |
risch | \(\frac {\sin \left (x^{3}\right )}{6}+\frac {\sin \left (9 x^{3}\right )}{54}\) | \(16\) |
parallelrisch | \(\frac {\sin \left (x^{3}\right )}{6}+\frac {\sin \left (9 x^{3}\right )}{54}\) | \(16\) |
norman | \(\frac {\frac {8 \tan \left (2 x^{3}\right ) \tan \left (\frac {5 x^{3}}{2}\right )^{2}}{27}-\frac {10 \tan \left (2 x^{3}\right )^{2} \tan \left (\frac {5 x^{3}}{2}\right )}{27}-\frac {8 \tan \left (2 x^{3}\right )}{27}+\frac {10 \tan \left (\frac {5 x^{3}}{2}\right )}{27}}{\left (1+\tan \left (2 x^{3}\right )^{2}\right ) \left (1+\tan \left (\frac {5 x^{3}}{2}\right )^{2}\right )}\) | \(75\) |
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Leaf count of result is larger than twice the leaf count of optimal. 40 vs. \(2 (15) = 30\).
Time = 0.26 (sec) , antiderivative size = 40, normalized size of antiderivative = 2.11 \[ \int x^2 \cos \left (4 x^3\right ) \cos \left (5 x^3\right ) \, dx=\frac {1}{27} \, {\left (128 \, \cos \left (x^{3}\right )^{8} - 224 \, \cos \left (x^{3}\right )^{6} + 120 \, \cos \left (x^{3}\right )^{4} - 20 \, \cos \left (x^{3}\right )^{2} + 5\right )} \sin \left (x^{3}\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. 32 vs. \(2 (14) = 28\).
Time = 0.60 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.68 \[ \int x^2 \cos \left (4 x^3\right ) \cos \left (5 x^3\right ) \, dx=- \frac {4 \sin {\left (4 x^{3} \right )} \cos {\left (5 x^{3} \right )}}{27} + \frac {5 \sin {\left (5 x^{3} \right )} \cos {\left (4 x^{3} \right )}}{27} \]
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none
Time = 0.19 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.79 \[ \int x^2 \cos \left (4 x^3\right ) \cos \left (5 x^3\right ) \, dx=\frac {1}{54} \, \sin \left (9 \, x^{3}\right ) + \frac {1}{6} \, \sin \left (x^{3}\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. 39 vs. \(2 (15) = 30\).
Time = 0.28 (sec) , antiderivative size = 39, normalized size of antiderivative = 2.05 \[ \int x^2 \cos \left (4 x^3\right ) \cos \left (5 x^3\right ) \, dx=\frac {128}{27} \, \sin \left (x^{3}\right )^{9} - \frac {32}{3} \, \sin \left (x^{3}\right )^{7} + 8 \, \sin \left (x^{3}\right )^{5} - \frac {20}{9} \, \sin \left (x^{3}\right )^{3} + \frac {1}{3} \, \sin \left (x^{3}\right ) \]
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Time = 26.65 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.79 \[ \int x^2 \cos \left (4 x^3\right ) \cos \left (5 x^3\right ) \, dx=\frac {\sin \left (x^3\right )}{6}+\frac {\sin \left (9\,x^3\right )}{54} \]
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