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 = 0.0313716, antiderivative size = 41, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, 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 3311
Rule 30
Rule 2635
Rule 8
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*}
Mathematica [A] time = 0.0322167, 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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Maple [A] time = 0.02, size = 37, normalized size = 0.9 \begin{align*}{x}^{2} \left ({\frac{x}{2}}-{\frac{\cos \left ( x \right ) \sin \left ( x \right ) }{2}} \right ) -{\frac{x \left ( \cos \left ( x \right ) \right ) ^{2}}{2}}+{\frac{\cos \left ( x \right ) \sin \left ( x \right ) }{4}}+{\frac{x}{4}}-{\frac{{x}^{3}}{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.965939, size = 35, normalized size = 0.85 \begin{align*} \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 ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.455595, size = 89, normalized size = 2.17 \begin{align*} \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 \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.665146, size = 56, normalized size = 1.37 \begin{align*} \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.07779, size = 35, normalized size = 0.85 \begin{align*} \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 ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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