Optimal. Leaf size=15 \[ \frac{\sin \left (a+b x^2\right )}{2 b} \]
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Rubi [A] time = 0.0143389, antiderivative size = 15, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {3380, 2637} \[ \frac{\sin \left (a+b x^2\right )}{2 b} \]
Antiderivative was successfully verified.
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Rule 3380
Rule 2637
Rubi steps
\begin{align*} \int x \cos \left (a+b x^2\right ) \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \cos (a+b x) \, dx,x,x^2\right )\\ &=\frac{\sin \left (a+b x^2\right )}{2 b}\\ \end{align*}
Mathematica [A] time = 0.0027367, size = 15, normalized size = 1. \[ \frac{\sin \left (a+b x^2\right )}{2 b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.023, size = 14, normalized size = 0.9 \begin{align*}{\frac{\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 = 1.35392, size = 18, normalized size = 1.2 \begin{align*} \frac{\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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Fricas [A] time = 1.57076, size = 30, normalized size = 2. \begin{align*} \frac{\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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Sympy [A] time = 0.208326, size = 19, normalized size = 1.27 \begin{align*} \begin{cases} \frac{\sin{\left (a + b x^{2} \right )}}{2 b} & \text{for}\: b \neq 0 \\\frac{x^{2} \cos{\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 = 1.15867, size = 18, normalized size = 1.2 \begin{align*} \frac{\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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