Optimal. Leaf size=20 \[ \frac{1}{2} x^2 \sin \left (x^2\right )+\frac{\cos \left (x^2\right )}{2} \]
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Rubi [A] time = 0.0168783, antiderivative size = 20, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.375, Rules used = {3380, 3296, 2638} \[ \frac{1}{2} x^2 \sin \left (x^2\right )+\frac{\cos \left (x^2\right )}{2} \]
Antiderivative was successfully verified.
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Rule 3380
Rule 3296
Rule 2638
Rubi steps
\begin{align*} \int x^3 \cos \left (x^2\right ) \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int x \cos (x) \, dx,x,x^2\right )\\ &=\frac{1}{2} x^2 \sin \left (x^2\right )-\frac{1}{2} \operatorname{Subst}\left (\int \sin (x) \, dx,x,x^2\right )\\ &=\frac{\cos \left (x^2\right )}{2}+\frac{1}{2} x^2 \sin \left (x^2\right )\\ \end{align*}
Mathematica [A] time = 0.0075433, size = 20, normalized size = 1. \[ \frac{1}{2} x^2 \sin \left (x^2\right )+\frac{\cos \left (x^2\right )}{2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 17, normalized size = 0.9 \begin{align*}{\frac{\cos \left ({x}^{2} \right ) }{2}}+{\frac{{x}^{2}\sin \left ({x}^{2} \right ) }{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.974607, size = 22, normalized size = 1.1 \begin{align*} \frac{1}{2} \, x^{2} \sin \left (x^{2}\right ) + \frac{1}{2} \, \cos \left (x^{2}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.96105, size = 45, normalized size = 2.25 \begin{align*} \frac{1}{2} \, x^{2} \sin \left (x^{2}\right ) + \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 = 0.549555, size = 15, normalized size = 0.75 \begin{align*} \frac{x^{2} \sin{\left (x^{2} \right )}}{2} + \frac{\cos{\left (x^{2} \right )}}{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.09492, size = 22, normalized size = 1.1 \begin{align*} \frac{1}{2} \, x^{2} \sin \left (x^{2}\right ) + \frac{1}{2} \, \cos \left (x^{2}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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