Optimal. Leaf size=34 \[ \frac {\cos \left (a+b x^2\right )}{2 b^2}+\frac {x^2 \sin \left (a+b x^2\right )}{2 b} \]
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Rubi [A] time = 0.03, antiderivative size = 34, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3380, 3296, 2638} \[ \frac {\cos \left (a+b x^2\right )}{2 b^2}+\frac {x^2 \sin \left (a+b x^2\right )}{2 b} \]
Antiderivative was successfully verified.
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Rule 2638
Rule 3296
Rule 3380
Rubi steps
\begin {align*} \int x^3 \cos \left (a+b x^2\right ) \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int x \cos (a+b x) \, dx,x,x^2\right )\\ &=\frac {x^2 \sin \left (a+b x^2\right )}{2 b}-\frac {\operatorname {Subst}\left (\int \sin (a+b x) \, dx,x,x^2\right )}{2 b}\\ &=\frac {\cos \left (a+b x^2\right )}{2 b^2}+\frac {x^2 \sin \left (a+b x^2\right )}{2 b}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 29, normalized size = 0.85 \[ \frac {b x^2 \sin \left (a+b x^2\right )+\cos \left (a+b x^2\right )}{2 b^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.57, size = 27, normalized size = 0.79 \[ \frac {b x^{2} \sin \left (b x^{2} + a\right ) + \cos \left (b x^{2} + a\right )}{2 \, b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.38, size = 27, normalized size = 0.79 \[ \frac {b x^{2} \sin \left (b x^{2} + a\right ) + \cos \left (b x^{2} + a\right )}{2 \, b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 31, normalized size = 0.91 \[ \frac {\cos \left (b \,x^{2}+a \right )}{2 b^{2}}+\frac {x^{2} \sin \left (b \,x^{2}+a \right )}{2 b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.11, size = 27, normalized size = 0.79 \[ \frac {b x^{2} \sin \left (b x^{2} + a\right ) + \cos \left (b x^{2} + a\right )}{2 \, b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.10, size = 27, normalized size = 0.79 \[ \frac {\cos \left (b\,x^2+a\right )+b\,x^2\,\sin \left (b\,x^2+a\right )}{2\,b^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.77, size = 36, normalized size = 1.06 \[ \begin {cases} \frac {x^{2} \sin {\left (a + b x^{2} \right )}}{2 b} + \frac {\cos {\left (a + b x^{2} \right )}}{2 b^{2}} & \text {for}\: b \neq 0 \\\frac {x^{4} \cos {\relax (a )}}{4} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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