Optimal. Leaf size=69 \[ -\frac{3}{2} x^{2/3} \sin \left (\sqrt [3]{x}\right ) \cos \left (\sqrt [3]{x}\right )+\frac{x}{2}-\frac{3 \sqrt [3]{x}}{4}+\frac{3}{2} \sqrt [3]{x} \sin ^2\left (\sqrt [3]{x}\right )+\frac{3}{4} \sin \left (\sqrt [3]{x}\right ) \cos \left (\sqrt [3]{x}\right ) \]
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Rubi [A] time = 0.043028, antiderivative size = 69, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.625, Rules used = {3361, 3311, 30, 2635, 8} \[ -\frac{3}{2} x^{2/3} \sin \left (\sqrt [3]{x}\right ) \cos \left (\sqrt [3]{x}\right )+\frac{x}{2}-\frac{3 \sqrt [3]{x}}{4}+\frac{3}{2} \sqrt [3]{x} \sin ^2\left (\sqrt [3]{x}\right )+\frac{3}{4} \sin \left (\sqrt [3]{x}\right ) \cos \left (\sqrt [3]{x}\right ) \]
Antiderivative was successfully verified.
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Rule 3361
Rule 3311
Rule 30
Rule 2635
Rule 8
Rubi steps
\begin{align*} \int \sin ^2\left (\sqrt [3]{x}\right ) \, dx &=3 \operatorname{Subst}\left (\int x^2 \sin ^2(x) \, dx,x,\sqrt [3]{x}\right )\\ &=-\frac{3}{2} x^{2/3} \cos \left (\sqrt [3]{x}\right ) \sin \left (\sqrt [3]{x}\right )+\frac{3}{2} \sqrt [3]{x} \sin ^2\left (\sqrt [3]{x}\right )+\frac{3}{2} \operatorname{Subst}\left (\int x^2 \, dx,x,\sqrt [3]{x}\right )-\frac{3}{2} \operatorname{Subst}\left (\int \sin ^2(x) \, dx,x,\sqrt [3]{x}\right )\\ &=\frac{x}{2}+\frac{3}{4} \cos \left (\sqrt [3]{x}\right ) \sin \left (\sqrt [3]{x}\right )-\frac{3}{2} x^{2/3} \cos \left (\sqrt [3]{x}\right ) \sin \left (\sqrt [3]{x}\right )+\frac{3}{2} \sqrt [3]{x} \sin ^2\left (\sqrt [3]{x}\right )-\frac{3}{4} \operatorname{Subst}\left (\int 1 \, dx,x,\sqrt [3]{x}\right )\\ &=-\frac{3 \sqrt [3]{x}}{4}+\frac{x}{2}+\frac{3}{4} \cos \left (\sqrt [3]{x}\right ) \sin \left (\sqrt [3]{x}\right )-\frac{3}{2} x^{2/3} \cos \left (\sqrt [3]{x}\right ) \sin \left (\sqrt [3]{x}\right )+\frac{3}{2} \sqrt [3]{x} \sin ^2\left (\sqrt [3]{x}\right )\\ \end{align*}
Mathematica [A] time = 0.049048, size = 41, normalized size = 0.59 \[ \frac{1}{8} \left (\left (3-6 x^{2/3}\right ) \sin \left (2 \sqrt [3]{x}\right )+4 x-6 \sqrt [3]{x} \cos \left (2 \sqrt [3]{x}\right )\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.01, size = 52, normalized size = 0.8 \begin{align*} 3\,{x}^{2/3} \left ( -1/2\,\cos \left ( \sqrt [3]{x} \right ) \sin \left ( \sqrt [3]{x} \right ) +1/2\,\sqrt [3]{x} \right ) -{\frac{3}{2}\sqrt [3]{x} \left ( \cos \left ( \sqrt [3]{x} \right ) \right ) ^{2}}+{\frac{3}{4}\cos \left ( \sqrt [3]{x} \right ) \sin \left ( \sqrt [3]{x} \right ) }+{\frac{3}{4}\sqrt [3]{x}}-x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.950069, size = 41, normalized size = 0.59 \begin{align*} -\frac{3}{8} \,{\left (2 \, x^{\frac{2}{3}} - 1\right )} \sin \left (2 \, x^{\frac{1}{3}}\right ) - \frac{3}{4} \, x^{\frac{1}{3}} \cos \left (2 \, x^{\frac{1}{3}}\right ) + \frac{1}{2} \, x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.62251, size = 134, normalized size = 1.94 \begin{align*} -\frac{3}{4} \,{\left (2 \, x^{\frac{2}{3}} - 1\right )} \cos \left (x^{\frac{1}{3}}\right ) \sin \left (x^{\frac{1}{3}}\right ) - \frac{3}{2} \, x^{\frac{1}{3}} \cos \left (x^{\frac{1}{3}}\right )^{2} + \frac{1}{2} \, x + \frac{3}{4} \, x^{\frac{1}{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 1.51308, size = 379, normalized size = 5.49 \begin{align*} \frac{12 x^{\frac{2}{3}} \tan ^{3}{\left (\frac{\sqrt [3]{x}}{2} \right )}}{4 \tan ^{4}{\left (\frac{\sqrt [3]{x}}{2} \right )} + 8 \tan ^{2}{\left (\frac{\sqrt [3]{x}}{2} \right )} + 4} - \frac{12 x^{\frac{2}{3}} \tan{\left (\frac{\sqrt [3]{x}}{2} \right )}}{4 \tan ^{4}{\left (\frac{\sqrt [3]{x}}{2} \right )} + 8 \tan ^{2}{\left (\frac{\sqrt [3]{x}}{2} \right )} + 4} - \frac{3 \sqrt [3]{x} \tan ^{4}{\left (\frac{\sqrt [3]{x}}{2} \right )}}{4 \tan ^{4}{\left (\frac{\sqrt [3]{x}}{2} \right )} + 8 \tan ^{2}{\left (\frac{\sqrt [3]{x}}{2} \right )} + 4} + \frac{18 \sqrt [3]{x} \tan ^{2}{\left (\frac{\sqrt [3]{x}}{2} \right )}}{4 \tan ^{4}{\left (\frac{\sqrt [3]{x}}{2} \right )} + 8 \tan ^{2}{\left (\frac{\sqrt [3]{x}}{2} \right )} + 4} - \frac{3 \sqrt [3]{x}}{4 \tan ^{4}{\left (\frac{\sqrt [3]{x}}{2} \right )} + 8 \tan ^{2}{\left (\frac{\sqrt [3]{x}}{2} \right )} + 4} + \frac{2 x \tan ^{4}{\left (\frac{\sqrt [3]{x}}{2} \right )}}{4 \tan ^{4}{\left (\frac{\sqrt [3]{x}}{2} \right )} + 8 \tan ^{2}{\left (\frac{\sqrt [3]{x}}{2} \right )} + 4} + \frac{4 x \tan ^{2}{\left (\frac{\sqrt [3]{x}}{2} \right )}}{4 \tan ^{4}{\left (\frac{\sqrt [3]{x}}{2} \right )} + 8 \tan ^{2}{\left (\frac{\sqrt [3]{x}}{2} \right )} + 4} + \frac{2 x}{4 \tan ^{4}{\left (\frac{\sqrt [3]{x}}{2} \right )} + 8 \tan ^{2}{\left (\frac{\sqrt [3]{x}}{2} \right )} + 4} - \frac{6 \tan ^{3}{\left (\frac{\sqrt [3]{x}}{2} \right )}}{4 \tan ^{4}{\left (\frac{\sqrt [3]{x}}{2} \right )} + 8 \tan ^{2}{\left (\frac{\sqrt [3]{x}}{2} \right )} + 4} + \frac{6 \tan{\left (\frac{\sqrt [3]{x}}{2} \right )}}{4 \tan ^{4}{\left (\frac{\sqrt [3]{x}}{2} \right )} + 8 \tan ^{2}{\left (\frac{\sqrt [3]{x}}{2} \right )} + 4} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.09981, size = 41, normalized size = 0.59 \begin{align*} -\frac{3}{8} \,{\left (2 \, x^{\frac{2}{3}} - 1\right )} \sin \left (2 \, x^{\frac{1}{3}}\right ) - \frac{3}{4} \, x^{\frac{1}{3}} \cos \left (2 \, x^{\frac{1}{3}}\right ) + \frac{1}{2} \, x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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