Optimal. Leaf size=33 \[ x \log \left (3 x^2+2\right )-2 x+2 \sqrt {\frac {2}{3}} \tan ^{-1}\left (\sqrt {\frac {3}{2}} x\right ) \]
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Rubi [A] time = 0.01, antiderivative size = 33, 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 = {2448, 321, 203} \[ x \log \left (3 x^2+2\right )-2 x+2 \sqrt {\frac {2}{3}} \tan ^{-1}\left (\sqrt {\frac {3}{2}} x\right ) \]
Antiderivative was successfully verified.
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Rule 203
Rule 321
Rule 2448
Rubi steps
\begin {align*} \int \log \left (2+3 x^2\right ) \, dx &=x \log \left (2+3 x^2\right )-6 \int \frac {x^2}{2+3 x^2} \, dx\\ &=-2 x+x \log \left (2+3 x^2\right )+4 \int \frac {1}{2+3 x^2} \, dx\\ &=-2 x+2 \sqrt {\frac {2}{3}} \tan ^{-1}\left (\sqrt {\frac {3}{2}} x\right )+x \log \left (2+3 x^2\right )\\ \end {align*}
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Mathematica [A] time = 0.01, size = 33, normalized size = 1.00 \[ x \log \left (3 x^2+2\right )-2 x+2 \sqrt {\frac {2}{3}} \tan ^{-1}\left (\sqrt {\frac {3}{2}} x\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 32, normalized size = 0.97 \[ \frac {2}{3} \, \sqrt {3} \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {3} \sqrt {2} x\right ) + x \log \left (3 \, x^{2} + 2\right ) - 2 \, x \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.12, size = 26, normalized size = 0.79 \[ x \log \left (3 \, x^{2} + 2\right ) + \frac {2}{3} \, \sqrt {6} \arctan \left (\frac {1}{2} \, \sqrt {6} x\right ) - 2 \, x \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 27, normalized size = 0.82 \[ x \ln \left (3 x^{2}+2\right )-2 x +\frac {2 \sqrt {6}\, \arctan \left (\frac {\sqrt {6}\, x}{2}\right )}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.27, size = 26, normalized size = 0.79 \[ x \log \left (3 \, x^{2} + 2\right ) + \frac {2}{3} \, \sqrt {6} \arctan \left (\frac {1}{2} \, \sqrt {6} x\right ) - 2 \, x \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.00, size = 26, normalized size = 0.79 \[ \frac {2\,\sqrt {6}\,\mathrm {atan}\left (\frac {\sqrt {6}\,x}{2}\right )}{3}-2\,x+x\,\ln \left (3\,x^2+2\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.14, size = 31, normalized size = 0.94 \[ x \log {\left (3 x^{2} + 2 \right )} - 2 x + \frac {2 \sqrt {6} \operatorname {atan}{\left (\frac {\sqrt {6} x}{2} \right )}}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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