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 = 0.02, antiderivative size = 20, normalized size of antiderivative = 1.00, 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 2637
Rule 3296
Rule 3379
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*}
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Mathematica [A] time = 0.00, size = 20, normalized size = 1.00 \[ \frac {\sin \left (x^2\right )}{2}-\frac {1}{2} x^2 \cos \left (x^2\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.44, size = 16, normalized size = 0.80 \[ -\frac {1}{2} \, x^{2} \cos \left (x^{2}\right ) + \frac {1}{2} \, \sin \left (x^{2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.04, size = 16, normalized size = 0.80 \[ -\frac {1}{2} \, x^{2} \cos \left (x^{2}\right ) + \frac {1}{2} \, \sin \left (x^{2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 17, normalized size = 0.85 \[ -\frac {x^{2} \cos \left (x^{2}\right )}{2}+\frac {\sin \left (x^{2}\right )}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.42, size = 16, normalized size = 0.80 \[ -\frac {1}{2} \, x^{2} \cos \left (x^{2}\right ) + \frac {1}{2} \, \sin \left (x^{2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.16, size = 16, normalized size = 0.80 \[ \frac {\sin \left (x^2\right )}{2}-\frac {x^2\,\cos \left (x^2\right )}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.57, size = 15, normalized size = 0.75 \[ - \frac {x^{2} \cos {\left (x^{2} \right )}}{2} + \frac {\sin {\left (x^{2} \right )}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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