Optimal. Leaf size=23 \[ \frac{1}{2} \left (a+x^2\right ) \log \left (a+x^2\right )-\frac{x^2}{2} \]
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Rubi [A] time = 0.0185245, antiderivative size = 23, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.375, Rules used = {2454, 2389, 2295} \[ \frac{1}{2} \left (a+x^2\right ) \log \left (a+x^2\right )-\frac{x^2}{2} \]
Antiderivative was successfully verified.
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Rule 2454
Rule 2389
Rule 2295
Rubi steps
\begin{align*} \int x \log \left (a+x^2\right ) \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \log (a+x) \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \log (x) \, dx,x,a+x^2\right )\\ &=-\frac{x^2}{2}+\frac{1}{2} \left (a+x^2\right ) \log \left (a+x^2\right )\\ \end{align*}
Mathematica [A] time = 0.0032048, size = 22, normalized size = 0.96 \[ \frac{1}{2} \left (\left (a+x^2\right ) \log \left (a+x^2\right )-x^2\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.002, size = 23, normalized size = 1. \begin{align*}{\frac{ \left ({x}^{2}+a \right ) \ln \left ({x}^{2}+a \right ) }{2}}-{\frac{{x}^{2}}{2}}-{\frac{a}{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.965159, size = 30, normalized size = 1.3 \begin{align*} -\frac{1}{2} \, x^{2} + \frac{1}{2} \,{\left (x^{2} + a\right )} \log \left (x^{2} + a\right ) - \frac{1}{2} \, a \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.60257, size = 53, normalized size = 2.3 \begin{align*} -\frac{1}{2} \, x^{2} + \frac{1}{2} \,{\left (x^{2} + a\right )} \log \left (x^{2} + a\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.29671, size = 26, normalized size = 1.13 \begin{align*} \frac{a \log{\left (a + x^{2} \right )}}{2} + \frac{x^{2} \log{\left (a + x^{2} \right )}}{2} - \frac{x^{2}}{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.05921, size = 30, normalized size = 1.3 \begin{align*} -\frac{1}{2} \, x^{2} + \frac{1}{2} \,{\left (x^{2} + a\right )} \log \left (x^{2} + a\right ) - \frac{1}{2} \, a \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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