Integrand size = 8, antiderivative size = 41 \[ \int x^2 \sin ^2(x) \, dx=-\frac {x}{4}+\frac {x^3}{6}+\frac {1}{4} \cos (x) \sin (x)-\frac {1}{2} x^2 \cos (x) \sin (x)+\frac {1}{2} x \sin ^2(x) \]
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Time = 0.03 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {3392, 30, 2715, 8} \[ \int x^2 \sin ^2(x) \, dx=\frac {x^3}{6}-\frac {1}{2} x^2 \sin (x) \cos (x)-\frac {x}{4}+\frac {1}{2} x \sin ^2(x)+\frac {1}{4} \sin (x) \cos (x) \]
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Rule 8
Rule 30
Rule 2715
Rule 3392
Rubi steps \begin{align*} \text {integral}& = -\frac {1}{2} x^2 \cos (x) \sin (x)+\frac {1}{2} x \sin ^2(x)+\frac {\int x^2 \, dx}{2}-\frac {1}{2} \int \sin ^2(x) \, dx \\ & = \frac {x^3}{6}+\frac {1}{4} \cos (x) \sin (x)-\frac {1}{2} x^2 \cos (x) \sin (x)+\frac {1}{2} x \sin ^2(x)-\frac {\int 1 \, dx}{4} \\ & = -\frac {x}{4}+\frac {x^3}{6}+\frac {1}{4} \cos (x) \sin (x)-\frac {1}{2} x^2 \cos (x) \sin (x)+\frac {1}{2} x \sin ^2(x) \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.71 \[ \int x^2 \sin ^2(x) \, dx=\frac {1}{24} \left (4 x^3-6 x \cos (2 x)+\left (3-6 x^2\right ) \sin (2 x)\right ) \]
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Result contains higher order function than in optimal. Order 5 vs. order 3.
Time = 0.07 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.46
method | result | size |
meijerg | \(\frac {x^{5} {}_{2}^{}{\moversetsp {}{\mundersetsp {}{F_{3}^{}}}}\left (1,\frac {5}{2};\frac {3}{2},2,\frac {7}{2};-x^{2}\right )}{5}\) | \(19\) |
risch | \(\frac {x^{3}}{6}-\frac {x \cos \left (2 x \right )}{4}-\frac {\left (2 x^{2}-1\right ) \sin \left (2 x \right )}{8}\) | \(27\) |
default | \(x^{2} \left (\frac {x}{2}-\frac {\cos \left (x \right ) \sin \left (x \right )}{2}\right )-\frac {\left (\cos ^{2}\left (x \right )\right ) x}{2}+\frac {\cos \left (x \right ) \sin \left (x \right )}{4}+\frac {x}{4}-\frac {x^{3}}{3}\) | \(37\) |
norman | \(\frac {x^{2} \left (\tan ^{3}\left (\frac {x}{2}\right )\right )-\frac {x}{4}+\frac {x^{3}}{6}-\frac {\left (\tan ^{3}\left (\frac {x}{2}\right )\right )}{2}+\frac {3 x \left (\tan ^{2}\left (\frac {x}{2}\right )\right )}{2}-\frac {x \left (\tan ^{4}\left (\frac {x}{2}\right )\right )}{4}-x^{2} \tan \left (\frac {x}{2}\right )+\frac {x^{3} \left (\tan ^{2}\left (\frac {x}{2}\right )\right )}{3}+\frac {x^{3} \left (\tan ^{4}\left (\frac {x}{2}\right )\right )}{6}+\frac {\tan \left (\frac {x}{2}\right )}{2}}{\left (1+\tan ^{2}\left (\frac {x}{2}\right )\right )^{2}}\) | \(94\) |
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Time = 0.24 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.71 \[ \int x^2 \sin ^2(x) \, dx=\frac {1}{6} \, x^{3} - \frac {1}{2} \, x \cos \left (x\right )^{2} - \frac {1}{4} \, {\left (2 \, x^{2} - 1\right )} \cos \left (x\right ) \sin \left (x\right ) + \frac {1}{4} \, x \]
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Time = 0.18 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.37 \[ \int x^2 \sin ^2(x) \, dx=\frac {x^{3} \sin ^{2}{\left (x \right )}}{6} + \frac {x^{3} \cos ^{2}{\left (x \right )}}{6} - \frac {x^{2} \sin {\left (x \right )} \cos {\left (x \right )}}{2} + \frac {x \sin ^{2}{\left (x \right )}}{4} - \frac {x \cos ^{2}{\left (x \right )}}{4} + \frac {\sin {\left (x \right )} \cos {\left (x \right )}}{4} \]
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Time = 0.19 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.63 \[ \int x^2 \sin ^2(x) \, dx=\frac {1}{6} \, x^{3} - \frac {1}{4} \, x \cos \left (2 \, x\right ) - \frac {1}{8} \, {\left (2 \, x^{2} - 1\right )} \sin \left (2 \, x\right ) \]
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Time = 0.26 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.63 \[ \int x^2 \sin ^2(x) \, dx=\frac {1}{6} \, x^{3} - \frac {1}{4} \, x \cos \left (2 \, x\right ) - \frac {1}{8} \, {\left (2 \, x^{2} - 1\right )} \sin \left (2 \, x\right ) \]
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Time = 0.07 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.68 \[ \int x^2 \sin ^2(x) \, dx=\frac {\sin \left (2\,x\right )}{8}-\frac {x\,\cos \left (2\,x\right )}{4}-\frac {x^2\,\sin \left (2\,x\right )}{4}+\frac {x^3}{6} \]
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