Optimal. Leaf size=79 \[ \frac{\cos ^3\left (a+b x^2\right )}{18 b^2}+\frac{\cos \left (a+b x^2\right )}{3 b^2}+\frac{x^2 \sin \left (a+b x^2\right )}{3 b}+\frac{x^2 \sin \left (a+b x^2\right ) \cos ^2\left (a+b x^2\right )}{6 b} \]
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Rubi [A] time = 0.0738608, antiderivative size = 79, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286, Rules used = {3380, 3310, 3296, 2638} \[ \frac{\cos ^3\left (a+b x^2\right )}{18 b^2}+\frac{\cos \left (a+b x^2\right )}{3 b^2}+\frac{x^2 \sin \left (a+b x^2\right )}{3 b}+\frac{x^2 \sin \left (a+b x^2\right ) \cos ^2\left (a+b x^2\right )}{6 b} \]
Antiderivative was successfully verified.
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Rule 3380
Rule 3310
Rule 3296
Rule 2638
Rubi steps
\begin{align*} \int x^3 \cos ^3\left (a+b x^2\right ) \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int x \cos ^3(a+b x) \, dx,x,x^2\right )\\ &=\frac{\cos ^3\left (a+b x^2\right )}{18 b^2}+\frac{x^2 \cos ^2\left (a+b x^2\right ) \sin \left (a+b x^2\right )}{6 b}+\frac{1}{3} \operatorname{Subst}\left (\int x \cos (a+b x) \, dx,x,x^2\right )\\ &=\frac{\cos ^3\left (a+b x^2\right )}{18 b^2}+\frac{x^2 \sin \left (a+b x^2\right )}{3 b}+\frac{x^2 \cos ^2\left (a+b x^2\right ) \sin \left (a+b x^2\right )}{6 b}-\frac{\operatorname{Subst}\left (\int \sin (a+b x) \, dx,x,x^2\right )}{3 b}\\ &=\frac{\cos \left (a+b x^2\right )}{3 b^2}+\frac{\cos ^3\left (a+b x^2\right )}{18 b^2}+\frac{x^2 \sin \left (a+b x^2\right )}{3 b}+\frac{x^2 \cos ^2\left (a+b x^2\right ) \sin \left (a+b x^2\right )}{6 b}\\ \end{align*}
Mathematica [A] time = 0.163611, size = 55, normalized size = 0.7 \[ \frac{3 b x^2 \left (9 \sin \left (a+b x^2\right )+\sin \left (3 \left (a+b x^2\right )\right )\right )+27 \cos \left (a+b x^2\right )+\cos \left (3 \left (a+b x^2\right )\right )}{72 b^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.033, size = 66, normalized size = 0.8 \begin{align*}{\frac{3\,{x}^{2}\sin \left ( b{x}^{2}+a \right ) }{8\,b}}+{\frac{3\,\cos \left ( b{x}^{2}+a \right ) }{8\,{b}^{2}}}+{\frac{{x}^{2}\sin \left ( 3\,b{x}^{2}+3\,a \right ) }{24\,b}}+{\frac{\cos \left ( 3\,b{x}^{2}+3\,a \right ) }{72\,{b}^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.08759, size = 78, normalized size = 0.99 \begin{align*} \frac{3 \, b x^{2} \sin \left (3 \, b x^{2} + 3 \, a\right ) + 27 \, b x^{2} \sin \left (b x^{2} + a\right ) + \cos \left (3 \, b x^{2} + 3 \, a\right ) + 27 \, \cos \left (b x^{2} + a\right )}{72 \, b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.61153, size = 138, normalized size = 1.75 \begin{align*} \frac{\cos \left (b x^{2} + a\right )^{3} + 3 \,{\left (b x^{2} \cos \left (b x^{2} + a\right )^{2} + 2 \, b x^{2}\right )} \sin \left (b x^{2} + a\right ) + 6 \, \cos \left (b x^{2} + a\right )}{18 \, b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 4.01279, size = 92, normalized size = 1.16 \begin{align*} \begin{cases} \frac{x^{2} \sin ^{3}{\left (a + b x^{2} \right )}}{3 b} + \frac{x^{2} \sin{\left (a + b x^{2} \right )} \cos ^{2}{\left (a + b x^{2} \right )}}{2 b} + \frac{\sin ^{2}{\left (a + b x^{2} \right )} \cos{\left (a + b x^{2} \right )}}{3 b^{2}} + \frac{7 \cos ^{3}{\left (a + b x^{2} \right )}}{18 b^{2}} & \text{for}\: b \neq 0 \\\frac{x^{4} \cos ^{3}{\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 = 1.1368, size = 78, normalized size = 0.99 \begin{align*} \frac{3 \, b x^{2} \sin \left (3 \, b x^{2} + 3 \, a\right ) + 27 \, b x^{2} \sin \left (b x^{2} + a\right ) + \cos \left (3 \, b x^{2} + 3 \, a\right ) + 27 \, \cos \left (b x^{2} + a\right )}{72 \, b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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