Optimal. Leaf size=41 \[ \frac {x^3}{6}-\frac {1}{2} x^2 \sin (x) \cos (x)-\frac {x}{4}+\frac {1}{2} x \sin ^2(x)+\frac {1}{4} \sin (x) \cos (x) \]
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Rubi [A] time = 0.03, antiderivative size = 41, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {3311, 30, 2635, 8} \[ \frac {x^3}{6}-\frac {1}{2} x^2 \sin (x) \cos (x)-\frac {x}{4}+\frac {1}{2} x \sin ^2(x)+\frac {1}{4} \sin (x) \cos (x) \]
Antiderivative was successfully verified.
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Rule 8
Rule 30
Rule 2635
Rule 3311
Rubi steps
\begin {align*} \int x^2 \sin ^2(x) \, dx &=-\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*}
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Mathematica [A] time = 0.04, size = 29, normalized size = 0.71 \[ \frac {1}{24} \left (4 x^3+\left (3-6 x^2\right ) \sin (2 x)-6 x \cos (2 x)\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.44, size = 29, normalized size = 0.71 \[ \frac {1}{6} \, x^{3} - \frac {1}{2} \, x \cos \relax (x)^{2} - \frac {1}{4} \, {\left (2 \, x^{2} - 1\right )} \cos \relax (x) \sin \relax (x) + \frac {1}{4} \, x \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.01, size = 26, normalized size = 0.63 \[ \frac {1}{6} \, x^{3} - \frac {1}{4} \, x \cos \left (2 \, x\right ) - \frac {1}{8} \, {\left (2 \, x^{2} - 1\right )} \sin \left (2 \, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 37, normalized size = 0.90 \[ -\frac {x^{3}}{3}-\frac {x \left (\cos ^{2}\relax (x )\right )}{2}+\left (-\frac {\cos \relax (x ) \sin \relax (x )}{2}+\frac {x}{2}\right ) x^{2}+\frac {\cos \relax (x ) \sin \relax (x )}{4}+\frac {x}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.50, size = 26, normalized size = 0.63 \[ \frac {1}{6} \, x^{3} - \frac {1}{4} \, x \cos \left (2 \, x\right ) - \frac {1}{8} \, {\left (2 \, x^{2} - 1\right )} \sin \left (2 \, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.06, size = 28, normalized size = 0.68 \[ \frac {\sin \left (2\,x\right )}{8}-\frac {x\,\cos \left (2\,x\right )}{4}-\frac {x^2\,\sin \left (2\,x\right )}{4}+\frac {x^3}{6} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.65, size = 56, normalized size = 1.37 \[ \frac {x^{3} \sin ^{2}{\relax (x )}}{6} + \frac {x^{3} \cos ^{2}{\relax (x )}}{6} - \frac {x^{2} \sin {\relax (x )} \cos {\relax (x )}}{2} + \frac {x \sin ^{2}{\relax (x )}}{4} - \frac {x \cos ^{2}{\relax (x )}}{4} + \frac {\sin {\relax (x )} \cos {\relax (x )}}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
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