Optimal. Leaf size=33 \[ \frac{\cos ^3\left (a+b x^2\right )}{6 b}-\frac{\cos \left (a+b x^2\right )}{2 b} \]
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Rubi [A] time = 0.0307496, antiderivative size = 33, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {3379, 2633} \[ \frac{\cos ^3\left (a+b x^2\right )}{6 b}-\frac{\cos \left (a+b x^2\right )}{2 b} \]
Antiderivative was successfully verified.
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Rule 3379
Rule 2633
Rubi steps
\begin{align*} \int x \sin ^3\left (a+b x^2\right ) \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \sin ^3(a+b x) \, dx,x,x^2\right )\\ &=-\frac{\operatorname{Subst}\left (\int \left (1-x^2\right ) \, dx,x,\cos \left (a+b x^2\right )\right )}{2 b}\\ &=-\frac{\cos \left (a+b x^2\right )}{2 b}+\frac{\cos ^3\left (a+b x^2\right )}{6 b}\\ \end{align*}
Mathematica [A] time = 0.0281061, size = 33, normalized size = 1. \[ \frac{\cos \left (3 \left (a+b x^2\right )\right )}{24 b}-\frac{3 \cos \left (a+b x^2\right )}{8 b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 26, normalized size = 0.8 \begin{align*} -{\frac{ \left ( 2+ \left ( \sin \left ( b{x}^{2}+a \right ) \right ) ^{2} \right ) \cos \left ( b{x}^{2}+a \right ) }{6\,b}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.963564, size = 36, normalized size = 1.09 \begin{align*} \frac{\cos \left (3 \, b x^{2} + 3 \, a\right ) - 9 \, \cos \left (b x^{2} + a\right )}{24 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.16631, size = 61, normalized size = 1.85 \begin{align*} \frac{\cos \left (b x^{2} + a\right )^{3} - 3 \, \cos \left (b x^{2} + a\right )}{6 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.03725, size = 46, normalized size = 1.39 \begin{align*} \begin{cases} - \frac{\sin ^{2}{\left (a + b x^{2} \right )} \cos{\left (a + b x^{2} \right )}}{2 b} - \frac{\cos ^{3}{\left (a + b x^{2} \right )}}{3 b} & \text{for}\: b \neq 0 \\\frac{x^{2} \sin ^{3}{\left (a \right )}}{2} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.10891, size = 35, normalized size = 1.06 \begin{align*} \frac{\cos \left (b x^{2} + a\right )^{3} - 3 \, \cos \left (b x^{2} + a\right )}{6 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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