Integrand size = 8, antiderivative size = 20 \[ \int x^3 \sin \left (x^2\right ) \, dx=-\frac {1}{2} x^2 \cos \left (x^2\right )+\frac {\sin \left (x^2\right )}{2} \]
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Time = 0.02 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {3460, 3377, 2717} \[ \int x^3 \sin \left (x^2\right ) \, dx=\frac {\sin \left (x^2\right )}{2}-\frac {1}{2} x^2 \cos \left (x^2\right ) \]
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Rule 2717
Rule 3377
Rule 3460
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \text {Subst}\left (\int x \sin (x) \, dx,x,x^2\right ) \\ & = -\frac {1}{2} x^2 \cos \left (x^2\right )+\frac {1}{2} \text {Subst}\left (\int \cos (x) \, dx,x,x^2\right ) \\ & = -\frac {1}{2} x^2 \cos \left (x^2\right )+\frac {\sin \left (x^2\right )}{2} \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00 \[ \int x^3 \sin \left (x^2\right ) \, dx=-\frac {1}{2} x^2 \cos \left (x^2\right )+\frac {\sin \left (x^2\right )}{2} \]
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Time = 0.56 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.85
method | result | size |
derivativedivides | \(-\frac {x^{2} \cos \left (x^{2}\right )}{2}+\frac {\sin \left (x^{2}\right )}{2}\) | \(17\) |
default | \(-\frac {x^{2} \cos \left (x^{2}\right )}{2}+\frac {\sin \left (x^{2}\right )}{2}\) | \(17\) |
risch | \(-\frac {x^{2} \cos \left (x^{2}\right )}{2}+\frac {\sin \left (x^{2}\right )}{2}\) | \(17\) |
parallelrisch | \(-\frac {x^{2} \cos \left (x^{2}\right )}{2}+\frac {\sin \left (x^{2}\right )}{2}\) | \(17\) |
meijerg | \(\sqrt {\pi }\, \left (-\frac {x^{2} \cos \left (x^{2}\right )}{2 \sqrt {\pi }}+\frac {\sin \left (x^{2}\right )}{2 \sqrt {\pi }}\right )\) | \(27\) |
norman | \(\frac {-\frac {x^{2}}{2}+\frac {x^{2} \tan \left (\frac {x^{2}}{2}\right )^{2}}{2}+\tan \left (\frac {x^{2}}{2}\right )}{1+\tan \left (\frac {x^{2}}{2}\right )^{2}}\) | \(39\) |
parts | \(\frac {\sqrt {2}\, \sqrt {\pi }\, \operatorname {FresnelS}\left (\frac {\sqrt {2}\, x}{\sqrt {\pi }}\right ) x^{3}}{2}-\frac {3 \pi ^{2} \left (\frac {2 \,\operatorname {FresnelS}\left (\frac {\sqrt {2}\, x}{\sqrt {\pi }}\right ) \sqrt {2}\, x^{3}}{3 \pi ^{\frac {3}{2}}}+\frac {2 x^{2} \cos \left (x^{2}\right )}{3 \pi ^{2}}-\frac {2 \sin \left (x^{2}\right )}{3 \pi ^{2}}\right )}{4}\) | \(69\) |
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Time = 0.24 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.80 \[ \int x^3 \sin \left (x^2\right ) \, dx=-\frac {1}{2} \, x^{2} \cos \left (x^{2}\right ) + \frac {1}{2} \, \sin \left (x^{2}\right ) \]
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Time = 0.12 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.75 \[ \int x^3 \sin \left (x^2\right ) \, dx=- \frac {x^{2} \cos {\left (x^{2} \right )}}{2} + \frac {\sin {\left (x^{2} \right )}}{2} \]
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Time = 0.20 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.80 \[ \int x^3 \sin \left (x^2\right ) \, dx=-\frac {1}{2} \, x^{2} \cos \left (x^{2}\right ) + \frac {1}{2} \, \sin \left (x^{2}\right ) \]
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Time = 0.28 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.80 \[ \int x^3 \sin \left (x^2\right ) \, dx=-\frac {1}{2} \, x^{2} \cos \left (x^{2}\right ) + \frac {1}{2} \, \sin \left (x^{2}\right ) \]
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Time = 26.82 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.80 \[ \int x^3 \sin \left (x^2\right ) \, dx=\frac {\sin \left (x^2\right )}{2}-\frac {x^2\,\cos \left (x^2\right )}{2} \]
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