Optimal. Leaf size=27 \[ \frac{1}{2} \left (a^2+x^2\right ) \log \left (a^2+x^2\right )-\frac{x^2}{2} \]
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Rubi [A] time = 0.0204105, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {2454, 2389, 2295} \[ \frac{1}{2} \left (a^2+x^2\right ) \log \left (a^2+x^2\right )-\frac{x^2}{2} \]
Antiderivative was successfully verified.
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Rule 2454
Rule 2389
Rule 2295
Rubi steps
\begin{align*} \int x \log \left (a^2+x^2\right ) \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \log \left (a^2+x\right ) \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \log (x) \, dx,x,a^2+x^2\right )\\ &=-\frac{x^2}{2}+\frac{1}{2} \left (a^2+x^2\right ) \log \left (a^2+x^2\right )\\ \end{align*}
Mathematica [A] time = 0.0029061, size = 26, normalized size = 0.96 \[ \frac{1}{2} \left (\left (a^2+x^2\right ) \log \left (a^2+x^2\right )-x^2\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.001, size = 29, normalized size = 1.1 \begin{align*}{\frac{ \left ({a}^{2}+{x}^{2} \right ) \ln \left ({a}^{2}+{x}^{2} \right ) }{2}}-{\frac{{x}^{2}}{2}}-{\frac{{a}^{2}}{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.965512, size = 38, normalized size = 1.41 \begin{align*} -\frac{1}{2} \, a^{2} - \frac{1}{2} \, x^{2} + \frac{1}{2} \,{\left (a^{2} + x^{2}\right )} \log \left (a^{2} + x^{2}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.02125, size = 58, normalized size = 2.15 \begin{align*} -\frac{1}{2} \, x^{2} + \frac{1}{2} \,{\left (a^{2} + x^{2}\right )} \log \left (a^{2} + x^{2}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.307117, size = 31, normalized size = 1.15 \begin{align*} \frac{a^{2} \log{\left (a^{2} + x^{2} \right )}}{2} + \frac{x^{2} \log{\left (a^{2} + 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 = 1.08371, size = 38, normalized size = 1.41 \begin{align*} -\frac{1}{2} \, a^{2} - \frac{1}{2} \, x^{2} + \frac{1}{2} \,{\left (a^{2} + x^{2}\right )} \log \left (a^{2} + x^{2}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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