Optimal. Leaf size=20 \[ \frac{\sin \left (x^2\right )}{2}-\frac{1}{2} x^2 \cos \left (x^2\right ) \]
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Rubi [A] time = 0.0189159, 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 = {3379, 3296, 2637} \[ \frac{\sin \left (x^2\right )}{2}-\frac{1}{2} x^2 \cos \left (x^2\right ) \]
Antiderivative was successfully verified.
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Rule 3379
Rule 3296
Rule 2637
Rubi steps
\begin{align*} \int x^3 \sin \left (x^2\right ) \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int x \sin (x) \, dx,x,x^2\right )\\ &=-\frac{1}{2} x^2 \cos \left (x^2\right )+\frac{1}{2} \operatorname{Subst}\left (\int \cos (x) \, dx,x,x^2\right )\\ &=-\frac{1}{2} x^2 \cos \left (x^2\right )+\frac{\sin \left (x^2\right )}{2}\\ \end{align*}
Mathematica [A] time = 0.0017644, size = 20, normalized size = 1. \[ \frac{\sin \left (x^2\right )}{2}-\frac{1}{2} x^2 \cos \left (x^2\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0., size = 17, normalized size = 0.9 \begin{align*} -{\frac{{x}^{2}\cos \left ({x}^{2} \right ) }{2}}+{\frac{\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 = 0.93255, size = 22, normalized size = 1.1 \begin{align*} -\frac{1}{2} \, x^{2} \cos \left (x^{2}\right ) + \frac{1}{2} \, \sin \left (x^{2}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.98983, size = 46, normalized size = 2.3 \begin{align*} -\frac{1}{2} \, x^{2} \cos \left (x^{2}\right ) + \frac{1}{2} \, \sin \left (x^{2}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.548402, size = 15, normalized size = 0.75 \begin{align*} - \frac{x^{2} \cos{\left (x^{2} \right )}}{2} + \frac{\sin{\left (x^{2} \right )}}{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.07265, size = 22, normalized size = 1.1 \begin{align*} -\frac{1}{2} \, x^{2} \cos \left (x^{2}\right ) + \frac{1}{2} \, \sin \left (x^{2}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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