Optimal. Leaf size=51 \[ \frac{\cos ^2\left (a+b x^2\right )}{8 b^2}+\frac{x^2 \sin \left (a+b x^2\right ) \cos \left (a+b x^2\right )}{4 b}+\frac{x^4}{8} \]
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Rubi [A] time = 0.0519296, antiderivative size = 51, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214, Rules used = {3380, 3310, 30} \[ \frac{\cos ^2\left (a+b x^2\right )}{8 b^2}+\frac{x^2 \sin \left (a+b x^2\right ) \cos \left (a+b x^2\right )}{4 b}+\frac{x^4}{8} \]
Antiderivative was successfully verified.
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Rule 3380
Rule 3310
Rule 30
Rubi steps
\begin{align*} \int x^3 \cos ^2\left (a+b x^2\right ) \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int x \cos ^2(a+b x) \, dx,x,x^2\right )\\ &=\frac{\cos ^2\left (a+b x^2\right )}{8 b^2}+\frac{x^2 \cos \left (a+b x^2\right ) \sin \left (a+b x^2\right )}{4 b}+\frac{1}{4} \operatorname{Subst}\left (\int x \, dx,x,x^2\right )\\ &=\frac{x^4}{8}+\frac{\cos ^2\left (a+b x^2\right )}{8 b^2}+\frac{x^2 \cos \left (a+b x^2\right ) \sin \left (a+b x^2\right )}{4 b}\\ \end{align*}
Mathematica [A] time = 0.129541, size = 40, normalized size = 0.78 \[ \frac{2 b x^2 \left (\sin \left (2 \left (a+b x^2\right )\right )+b x^2\right )+\cos \left (2 \left (a+b x^2\right )\right )}{16 b^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.03, size = 42, normalized size = 0.8 \begin{align*}{\frac{{x}^{4}}{8}}+{\frac{{x}^{2}\sin \left ( 2\,b{x}^{2}+2\,a \right ) }{8\,b}}+{\frac{\cos \left ( 2\,b{x}^{2}+2\,a \right ) }{16\,{b}^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.1601, size = 57, normalized size = 1.12 \begin{align*} \frac{2 \, b^{2} x^{4} + 2 \, b x^{2} \sin \left (2 \, b x^{2} + 2 \, a\right ) + \cos \left (2 \, b x^{2} + 2 \, a\right )}{16 \, b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.64402, size = 105, normalized size = 2.06 \begin{align*} \frac{b^{2} x^{4} + 2 \, b x^{2} \cos \left (b x^{2} + a\right ) \sin \left (b x^{2} + a\right ) + \cos \left (b x^{2} + a\right )^{2}}{8 \, b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 2.18639, size = 78, normalized size = 1.53 \begin{align*} \begin{cases} \frac{x^{4} \sin ^{2}{\left (a + b x^{2} \right )}}{8} + \frac{x^{4} \cos ^{2}{\left (a + b x^{2} \right )}}{8} + \frac{x^{2} \sin{\left (a + b x^{2} \right )} \cos{\left (a + b x^{2} \right )}}{4 b} - \frac{\sin ^{2}{\left (a + b x^{2} \right )}}{8 b^{2}} & \text{for}\: b \neq 0 \\\frac{x^{4} \cos ^{2}{\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 = 1.14041, size = 74, normalized size = 1.45 \begin{align*} \frac{2 \, b x^{2} \sin \left (2 \, b x^{2} + 2 \, a\right ) + 2 \,{\left (b x^{2} + a\right )}^{2} - 4 \,{\left (b x^{2} + a\right )} a + \cos \left (2 \, b x^{2} + 2 \, a\right )}{16 \, b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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