Optimal. Leaf size=31 \[ \frac{\sin \left (a+b x^2\right ) \cos \left (a+b x^2\right )}{4 b}+\frac{x^2}{4} \]
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Rubi [A] time = 0.028877, antiderivative size = 31, 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, 2635, 8} \[ \frac{\sin \left (a+b x^2\right ) \cos \left (a+b x^2\right )}{4 b}+\frac{x^2}{4} \]
Antiderivative was successfully verified.
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Rule 3380
Rule 2635
Rule 8
Rubi steps
\begin{align*} \int x \cos ^2\left (a+b x^2\right ) \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \cos ^2(a+b x) \, dx,x,x^2\right )\\ &=\frac{\cos \left (a+b x^2\right ) \sin \left (a+b x^2\right )}{4 b}+\frac{1}{4} \operatorname{Subst}\left (\int 1 \, dx,x,x^2\right )\\ &=\frac{x^2}{4}+\frac{\cos \left (a+b x^2\right ) \sin \left (a+b x^2\right )}{4 b}\\ \end{align*}
Mathematica [A] time = 0.0451212, size = 27, normalized size = 0.87 \[ \frac{2 \left (a+b x^2\right )+\sin \left (2 \left (a+b x^2\right )\right )}{8 b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.026, size = 34, normalized size = 1.1 \begin{align*}{\frac{1}{2\,b} \left ({\frac{\cos \left ( b{x}^{2}+a \right ) \sin \left ( b{x}^{2}+a \right ) }{2}}+{\frac{b{x}^{2}}{2}}+{\frac{a}{2}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.17817, size = 31, normalized size = 1. \begin{align*} \frac{2 \, b x^{2} + \sin \left (2 \, b x^{2} + 2 \, a\right )}{8 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.61826, size = 63, normalized size = 2.03 \begin{align*} \frac{b x^{2} + \cos \left (b x^{2} + a\right ) \sin \left (b x^{2} + a\right )}{4 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.517633, size = 60, normalized size = 1.94 \begin{align*} \begin{cases} \frac{x^{2} \sin ^{2}{\left (a + b x^{2} \right )}}{4} + \frac{x^{2} \cos ^{2}{\left (a + b x^{2} \right )}}{4} + \frac{\sin{\left (a + b x^{2} \right )} \cos{\left (a + b x^{2} \right )}}{4 b} & \text{for}\: b \neq 0 \\\frac{x^{2} \cos ^{2}{\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 = 1.11567, size = 35, normalized size = 1.13 \begin{align*} \frac{2 \, b x^{2} + 2 \, a + \sin \left (2 \, b x^{2} + 2 \, a\right )}{8 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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