Optimal. Leaf size=73 \[ \frac{1}{3} x^2 \sin ^3(x)+2 x^2 \sin (x)-\frac{2}{3} x^3 \cos (x)-\frac{1}{3} x^3 \sin ^2(x) \cos (x)-\frac{2 \sin ^3(x)}{27}-\frac{40 \sin (x)}{9}+\frac{40}{9} x \cos (x)+\frac{2}{9} x \sin ^2(x) \cos (x) \]
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Rubi [A] time = 0.0842397, antiderivative size = 73, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 4, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {3311, 3296, 2637, 3310} \[ \frac{1}{3} x^2 \sin ^3(x)+2 x^2 \sin (x)-\frac{2}{3} x^3 \cos (x)-\frac{1}{3} x^3 \sin ^2(x) \cos (x)-\frac{2 \sin ^3(x)}{27}-\frac{40 \sin (x)}{9}+\frac{40}{9} x \cos (x)+\frac{2}{9} x \sin ^2(x) \cos (x) \]
Antiderivative was successfully verified.
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Rule 3311
Rule 3296
Rule 2637
Rule 3310
Rubi steps
\begin{align*} \int x^3 \sin ^3(x) \, dx &=-\frac{1}{3} x^3 \cos (x) \sin ^2(x)+\frac{1}{3} x^2 \sin ^3(x)+\frac{2}{3} \int x^3 \sin (x) \, dx-\frac{2}{3} \int x \sin ^3(x) \, dx\\ &=-\frac{2}{3} x^3 \cos (x)+\frac{2}{9} x \cos (x) \sin ^2(x)-\frac{1}{3} x^3 \cos (x) \sin ^2(x)-\frac{2 \sin ^3(x)}{27}+\frac{1}{3} x^2 \sin ^3(x)-\frac{4}{9} \int x \sin (x) \, dx+2 \int x^2 \cos (x) \, dx\\ &=\frac{4}{9} x \cos (x)-\frac{2}{3} x^3 \cos (x)+2 x^2 \sin (x)+\frac{2}{9} x \cos (x) \sin ^2(x)-\frac{1}{3} x^3 \cos (x) \sin ^2(x)-\frac{2 \sin ^3(x)}{27}+\frac{1}{3} x^2 \sin ^3(x)-\frac{4}{9} \int \cos (x) \, dx-4 \int x \sin (x) \, dx\\ &=\frac{40}{9} x \cos (x)-\frac{2}{3} x^3 \cos (x)-\frac{4 \sin (x)}{9}+2 x^2 \sin (x)+\frac{2}{9} x \cos (x) \sin ^2(x)-\frac{1}{3} x^3 \cos (x) \sin ^2(x)-\frac{2 \sin ^3(x)}{27}+\frac{1}{3} x^2 \sin ^3(x)-4 \int \cos (x) \, dx\\ &=\frac{40}{9} x \cos (x)-\frac{2}{3} x^3 \cos (x)-\frac{40 \sin (x)}{9}+2 x^2 \sin (x)+\frac{2}{9} x \cos (x) \sin ^2(x)-\frac{1}{3} x^3 \cos (x) \sin ^2(x)-\frac{2 \sin ^3(x)}{27}+\frac{1}{3} x^2 \sin ^3(x)\\ \end{align*}
Mathematica [A] time = 0.0958343, size = 51, normalized size = 0.7 \[ \frac{1}{108} \left (243 \left (x^2-2\right ) \sin (x)-\left (9 x^2-2\right ) \sin (3 x)-81 x \left (x^2-6\right ) \cos (x)+3 x \left (3 x^2-2\right ) \cos (3 x)\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.023, size = 57, normalized size = 0.8 \begin{align*} -{\frac{{x}^{3} \left ( 2+ \left ( \sin \left ( x \right ) \right ) ^{2} \right ) \cos \left ( x \right ) }{3}}+2\,{x}^{2}\sin \left ( x \right ) -{\frac{40\,\sin \left ( x \right ) }{9}}+4\,x\cos \left ( x \right ) +{\frac{{x}^{2} \left ( \sin \left ( x \right ) \right ) ^{3}}{3}}+{\frac{2\,x \left ( 2+ \left ( \sin \left ( x \right ) \right ) ^{2} \right ) \cos \left ( x \right ) }{9}}-{\frac{2\, \left ( \sin \left ( x \right ) \right ) ^{3}}{27}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.976939, size = 66, normalized size = 0.9 \begin{align*} \frac{1}{36} \,{\left (3 \, x^{3} - 2 \, x\right )} \cos \left (3 \, x\right ) - \frac{3}{4} \,{\left (x^{3} - 6 \, x\right )} \cos \left (x\right ) - \frac{1}{108} \,{\left (9 \, x^{2} - 2\right )} \sin \left (3 \, x\right ) + \frac{9}{4} \,{\left (x^{2} - 2\right )} \sin \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.24924, size = 146, normalized size = 2. \begin{align*} \frac{1}{9} \,{\left (3 \, x^{3} - 2 \, x\right )} \cos \left (x\right )^{3} - \frac{1}{3} \,{\left (3 \, x^{3} - 14 \, x\right )} \cos \left (x\right ) - \frac{1}{27} \,{\left ({\left (9 \, x^{2} - 2\right )} \cos \left (x\right )^{2} - 63 \, x^{2} + 122\right )} \sin \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.95342, size = 92, normalized size = 1.26 \begin{align*} - x^{3} \sin ^{2}{\left (x \right )} \cos{\left (x \right )} - \frac{2 x^{3} \cos ^{3}{\left (x \right )}}{3} + \frac{7 x^{2} \sin ^{3}{\left (x \right )}}{3} + 2 x^{2} \sin{\left (x \right )} \cos ^{2}{\left (x \right )} + \frac{14 x \sin ^{2}{\left (x \right )} \cos{\left (x \right )}}{3} + \frac{40 x \cos ^{3}{\left (x \right )}}{9} - \frac{122 \sin ^{3}{\left (x \right )}}{27} - \frac{40 \sin{\left (x \right )} \cos ^{2}{\left (x \right )}}{9} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.07071, size = 66, normalized size = 0.9 \begin{align*} \frac{1}{36} \,{\left (3 \, x^{3} - 2 \, x\right )} \cos \left (3 \, x\right ) - \frac{3}{4} \,{\left (x^{3} - 6 \, x\right )} \cos \left (x\right ) - \frac{1}{108} \,{\left (9 \, x^{2} - 2\right )} \sin \left (3 \, x\right ) + \frac{9}{4} \,{\left (x^{2} - 2\right )} \sin \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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