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 = 0.02, antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, 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 2295
Rule 2389
Rule 2454
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*}
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Mathematica [A] time = 0.00, 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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fricas [A] time = 0.42, size = 23, normalized size = 0.85 \[ -\frac {1}{2} \, x^{2} + \frac {1}{2} \, {\left (a^{2} + x^{2}\right )} \log \left (a^{2} + x^{2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.07, size = 28, normalized size = 1.04 \[ -\frac {1}{2} \, a^{2} - \frac {1}{2} \, x^{2} + \frac {1}{2} \, {\left (a^{2} + x^{2}\right )} \log \left (a^{2} + x^{2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 29, normalized size = 1.07 \[ -\frac {a^{2}}{2}-\frac {x^{2}}{2}+\frac {\left (a^{2}+x^{2}\right ) \ln \left (a^{2}+x^{2}\right )}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.42, size = 28, normalized size = 1.04 \[ -\frac {1}{2} \, a^{2} - \frac {1}{2} \, x^{2} + \frac {1}{2} \, {\left (a^{2} + x^{2}\right )} \log \left (a^{2} + x^{2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.04, size = 51, normalized size = 1.89 \[ \frac {a^2\,\ln \left (x-\sqrt {-a^2}\right )}{2}+\frac {x^2\,\ln \left (a^2+x^2\right )}{2}-\frac {x^2}{2}+\frac {a^2\,\ln \left (x+\sqrt {-a^2}\right )}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.16, size = 31, normalized size = 1.15 \[ \frac {a^{2} \log {\left (a^{2} + x^{2} \right )}}{2} + \frac {x^{2} \log {\left (a^{2} + x^{2} \right )}}{2} - \frac {x^{2}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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