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 = 0.0317191, antiderivative size = 34, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, 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 3380
Rule 3296
Rule 2638
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*}
Mathematica [A] time = 0.0492611, 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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Maple [A] time = 0.023, size = 31, normalized size = 0.9 \begin{align*}{\frac{\cos \left ( b{x}^{2}+a \right ) }{2\,{b}^{2}}}+{\frac{{x}^{2}\sin \left ( b{x}^{2}+a \right ) }{2\,b}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.33306, size = 36, normalized size = 1.06 \begin{align*} \frac{b x^{2} \sin \left (b x^{2} + a\right ) + \cos \left (b x^{2} + a\right )}{2 \, b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.56881, size = 66, normalized size = 1.94 \begin{align*} \frac{b x^{2} \sin \left (b x^{2} + a\right ) + \cos \left (b x^{2} + a\right )}{2 \, b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.08675, size = 36, normalized size = 1.06 \begin{align*} \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{\left (a \right )}}{4} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.12968, size = 36, normalized size = 1.06 \begin{align*} \frac{b x^{2} \sin \left (b x^{2} + a\right ) + \cos \left (b x^{2} + a\right )}{2 \, b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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