Optimal. Leaf size=27 \[ -\frac {x^2}{2}+\frac {1}{2} \left (a^2+x^2\right ) \log \left (a^2+x^2\right ) \]
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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 = {2504, 2436,
2332} \begin {gather*} \frac {1}{2} \left (a^2+x^2\right ) \log \left (a^2+x^2\right )-\frac {x^2}{2} \end {gather*}
Antiderivative was successfully verified.
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Rule 2332
Rule 2436
Rule 2504
Rubi steps
\begin {align*} \int x \log \left (a^2+x^2\right ) \, dx &=\frac {1}{2} \text {Subst}\left (\int \log \left (a^2+x\right ) \, dx,x,x^2\right )\\ &=\frac {1}{2} \text {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 \begin {gather*} \frac {1}{2} \left (-x^2+\left (a^2+x^2\right ) \log \left (a^2+x^2\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.01, size = 29, normalized size = 1.07
method | result | size |
derivativedivides | \(\frac {\left (a^{2}+x^{2}\right ) \ln \left (a^{2}+x^{2}\right )}{2}-\frac {a^{2}}{2}-\frac {x^{2}}{2}\) | \(29\) |
default | \(\frac {\left (a^{2}+x^{2}\right ) \ln \left (a^{2}+x^{2}\right )}{2}-\frac {a^{2}}{2}-\frac {x^{2}}{2}\) | \(29\) |
norman | \(-\frac {x^{2}}{2}+\frac {x^{2} \ln \left (a^{2}+x^{2}\right )}{2}+\frac {\ln \left (a^{2}+x^{2}\right ) a^{2}}{2}\) | \(33\) |
risch | \(-\frac {x^{2}}{2}+\frac {x^{2} \ln \left (a^{2}+x^{2}\right )}{2}+\frac {\ln \left (a^{2}+x^{2}\right ) a^{2}}{2}\) | \(33\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 4.06, size = 28, normalized size = 1.04 \begin {gather*} -\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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.60, size = 23, normalized size = 0.85 \begin {gather*} -\frac {1}{2} \, x^{2} + \frac {1}{2} \, {\left (a^{2} + x^{2}\right )} \log \left (a^{2} + x^{2}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.06, size = 31, normalized size = 1.15 \begin {gather*} \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.17, size = 28, normalized size = 1.04 \begin {gather*} -\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 {gather*}
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 \begin {gather*} \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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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