Optimal. Leaf size=23 \[ -\frac {x^2}{2}+\frac {1}{2} \left (a+x^2\right ) \log \left (a+x^2\right ) \]
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Rubi [A]
time = 0.01, antiderivative size = 23, normalized size of antiderivative = 1.00, 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 = {2504, 2436,
2332} \begin {gather*} \frac {1}{2} \left (a+x^2\right ) \log \left (a+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+x^2\right ) \, dx &=\frac {1}{2} \text {Subst}\left (\int \log (a+x) \, dx,x,x^2\right )\\ &=\frac {1}{2} \text {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*}
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Mathematica [A]
time = 0.00, size = 22, normalized size = 0.96 \begin {gather*} \frac {1}{2} \left (-x^2+\left (a+x^2\right ) \log \left (a+x^2\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.01, size = 23, normalized size = 1.00
method | result | size |
derivativedivides | \(\frac {\left (x^{2}+a \right ) \ln \left (x^{2}+a \right )}{2}-\frac {x^{2}}{2}-\frac {a}{2}\) | \(23\) |
default | \(\frac {\left (x^{2}+a \right ) \ln \left (x^{2}+a \right )}{2}-\frac {x^{2}}{2}-\frac {a}{2}\) | \(23\) |
norman | \(-\frac {x^{2}}{2}+\frac {\ln \left (x^{2}+a \right ) a}{2}+\frac {\ln \left (x^{2}+a \right ) x^{2}}{2}\) | \(27\) |
risch | \(-\frac {x^{2}}{2}+\frac {\ln \left (x^{2}+a \right ) a}{2}+\frac {\ln \left (x^{2}+a \right ) x^{2}}{2}\) | \(27\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 1.23, size = 22, normalized size = 0.96 \begin {gather*} -\frac {1}{2} \, x^{2} + \frac {1}{2} \, {\left (x^{2} + a\right )} \log \left (x^{2} + a\right ) - \frac {1}{2} \, a \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.56, size = 19, normalized size = 0.83 \begin {gather*} -\frac {1}{2} \, x^{2} + \frac {1}{2} \, {\left (x^{2} + a\right )} \log \left (x^{2} + a\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.06, size = 26, normalized size = 1.13 \begin {gather*} \frac {a \log {\left (a + x^{2} \right )}}{2} + \frac {x^{2} \log {\left (a + 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 = 0.51, size = 22, normalized size = 0.96 \begin {gather*} -\frac {1}{2} \, x^{2} + \frac {1}{2} \, {\left (x^{2} + a\right )} \log \left (x^{2} + a\right ) - \frac {1}{2} \, a \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.14, size = 41, normalized size = 1.78 \begin {gather*} \frac {a\,\ln \left (x+\sqrt {-a}\right )}{2}+\frac {x^2\,\ln \left (x^2+a\right )}{2}+\frac {a\,\ln \left (x-\sqrt {-a}\right )}{2}-\frac {x^2}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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