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 = 0.01, antiderivative size = 19, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, 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 2633
Rule 3379
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*}
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Mathematica [A] time = 0.02, size = 19, normalized size = 1.00 \[ \frac {1}{24} \cos \left (3 x^2\right )-\frac {3 \cos \left (x^2\right )}{8} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.44, size = 15, normalized size = 0.79 \[ \frac {1}{6} \, \cos \left (x^{2}\right )^{3} - \frac {1}{2} \, \cos \left (x^{2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.92, size = 15, normalized size = 0.79 \[ \frac {1}{6} \, \cos \left (x^{2}\right )^{3} - \frac {1}{2} \, \cos \left (x^{2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 15, normalized size = 0.79 \[ -\frac {\left (\sin ^{2}\left (x^{2}\right )+2\right ) \cos \left (x^{2}\right )}{6} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.44, size = 15, normalized size = 0.79 \[ \frac {1}{24} \, \cos \left (3 \, x^{2}\right ) - \frac {3}{8} \, \cos \left (x^{2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.18, size = 14, normalized size = 0.74 \[ \frac {\cos \left (x^2\right )\,\left ({\cos \left (x^2\right )}^2-3\right )}{6} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.62, size = 22, normalized size = 1.16 \[ - \frac {\sin ^{2}{\left (x^{2} \right )} \cos {\left (x^{2} \right )}}{2} - \frac {\cos ^{3}{\left (x^{2} \right )}}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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