Optimal. Leaf size=19 \[ \frac{1}{6} \cos ^3\left (x^2\right )-\frac{\cos \left (x^2\right )}{2} \]
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Rubi [A] time = 0.0137001, antiderivative size = 19, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {3379, 2633} \[ \frac{1}{6} \cos ^3\left (x^2\right )-\frac{\cos \left (x^2\right )}{2} \]
Antiderivative was successfully verified.
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Rule 3379
Rule 2633
Rubi steps
\begin{align*} \int x \sin ^3\left (x^2\right ) \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \sin ^3(x) \, dx,x,x^2\right )\\ &=-\left (\frac{1}{2} \operatorname{Subst}\left (\int \left (1-x^2\right ) \, dx,x,\cos \left (x^2\right )\right )\right )\\ &=-\frac{1}{2} \cos \left (x^2\right )+\frac{1}{6} \cos ^3\left (x^2\right )\\ \end{align*}
Mathematica [A] time = 0.01543, size = 19, normalized size = 1. \[ \frac{1}{24} \cos \left (3 x^2\right )-\frac{3 \cos \left (x^2\right )}{8} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 15, normalized size = 0.8 \begin{align*} -{\frac{ \left ( 2+ \left ( \sin \left ({x}^{2} \right ) \right ) ^{2} \right ) \cos \left ({x}^{2} \right ) }{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.931032, size = 20, normalized size = 1.05 \begin{align*} \frac{1}{24} \, \cos \left (3 \, x^{2}\right ) - \frac{3}{8} \, \cos \left (x^{2}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.06567, size = 42, normalized size = 2.21 \begin{align*} \frac{1}{6} \, \cos \left (x^{2}\right )^{3} - \frac{1}{2} \, \cos \left (x^{2}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.572453, size = 22, normalized size = 1.16 \begin{align*} - \frac{\sin ^{2}{\left (x^{2} \right )} \cos{\left (x^{2} \right )}}{2} - \frac{\cos ^{3}{\left (x^{2} \right )}}{3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.05639, size = 20, normalized size = 1.05 \begin{align*} \frac{1}{6} \, \cos \left (x^{2}\right )^{3} - \frac{1}{2} \, \cos \left (x^{2}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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