Optimal. Leaf size=32 \[ -e^{\frac {1}{4} (2-\log (4))} x+\frac {x^2}{\log (4)}+2 \log \left (9+x^2\right ) \]
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Rubi [A]
time = 0.11, antiderivative size = 29, normalized size of antiderivative = 0.91, number of steps
used = 4, number of rules used = 3, integrand size = 60, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {12, 1824, 266}
\begin {gather*} \frac {x^2}{\log (4)}+2 \log \left (x^2+9\right )-\sqrt {\frac {e}{2}} x \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 266
Rule 1824
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\frac {\sqrt {\frac {e}{2}} \int \frac {\left (-9-x^2\right ) \log (4)+e^{\frac {1}{4} (-2+\log (4))} \left (18 x+2 x^3+4 x \log (4)\right )}{9+x^2} \, dx}{\log (4)}\\ &=\frac {\sqrt {\frac {e}{2}} \int \left (2 \sqrt {\frac {2}{e}} x-\log (4)+\frac {4 \sqrt {\frac {2}{e}} x \log (4)}{9+x^2}\right ) \, dx}{\log (4)}\\ &=-\sqrt {\frac {e}{2}} x+\frac {x^2}{\log (4)}+4 \int \frac {x}{9+x^2} \, dx\\ &=-\sqrt {\frac {e}{2}} x+\frac {x^2}{\log (4)}+2 \log \left (9+x^2\right )\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.04, size = 33, normalized size = 1.03 \begin {gather*} \frac {18+2 x^2-\sqrt {2 e} x \log (4)+\log (256) \log \left (9+x^2\right )}{\log (16)} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.73, size = 50, normalized size = 1.56
method | result | size |
risch | \(\frac {x^{2}}{2 \ln \left (2\right )}-\frac {\sqrt {2}\, {\mathrm e}^{\frac {1}{2}} x}{2}+2 \ln \left (x^{2}+9\right )\) | \(27\) |
norman | \(\frac {x^{2}}{2 \ln \left (2\right )}-\frac {\sqrt {2}\, {\mathrm e}^{\frac {1}{2}} x}{2}+2 \ln \left (x^{2}+9\right )\) | \(29\) |
default | \(\frac {{\mathrm e}^{\frac {1}{2}-\frac {\ln \left (2\right )}{2}} \left ({\mathrm e}^{\frac {\ln \left (2\right )}{2}-\frac {1}{2}} x^{2}-2 x \ln \left (2\right )+4 \ln \left (2\right ) {\mathrm e}^{\frac {\ln \left (2\right )}{2}-\frac {1}{2}} \ln \left (x^{2}+9\right )\right )}{2 \ln \left (2\right )}\) | \(50\) |
meijerg | \(-3 \,{\mathrm e}^{\frac {1}{2}-\frac {\ln \left (2\right )}{2}} \arctan \left (\frac {x}{3}\right )+\frac {\frac {x^{2}}{2}-\frac {9 \ln \left (1+\frac {x^{2}}{9}\right )}{2}}{\ln \left (2\right )}+\frac {9 \left (\frac {4 \ln \left (2\right )}{9}+1\right ) \ln \left (1+\frac {x^{2}}{9}\right )}{2 \ln \left (2\right )}-\frac {3 \,{\mathrm e}^{\frac {1}{2}-\frac {\ln \left (2\right )}{2}} \left (\frac {2 x}{3}-2 \arctan \left (\frac {x}{3}\right )\right )}{2}\) | \(76\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 51, normalized size = 1.59 \begin {gather*} \frac {{\left (4 \, e^{\left (\frac {1}{2} \, \log \left (2\right ) - \frac {1}{2}\right )} \log \left (2\right ) \log \left (x^{2} + 9\right ) - {\left (2 \, x e^{\frac {1}{2}} \log \left (2\right ) - \sqrt {2} x^{2}\right )} e^{\left (-\frac {1}{2}\right )}\right )} e^{\left (-\frac {1}{2} \, \log \left (2\right ) + \frac {1}{2}\right )}}{2 \, \log \left (2\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 47, normalized size = 1.47 \begin {gather*} \frac {{\left (x^{2} e^{\left (\frac {1}{2} \, \log \left (2\right ) - \frac {1}{2}\right )} + 4 \, e^{\left (\frac {1}{2} \, \log \left (2\right ) - \frac {1}{2}\right )} \log \left (2\right ) \log \left (x^{2} + 9\right ) - 2 \, x \log \left (2\right )\right )} e^{\left (-\frac {1}{2} \, \log \left (2\right ) + \frac {1}{2}\right )}}{2 \, \log \left (2\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.09, size = 29, normalized size = 0.91 \begin {gather*} \frac {x^{2}}{2 \log {\left (2 \right )}} - \frac {\sqrt {2} x e^{\frac {1}{2}}}{2} + 2 \log {\left (x^{2} + 9 \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.41, size = 47, normalized size = 1.47 \begin {gather*} \frac {{\left (x^{2} e^{\left (\frac {1}{2} \, \log \left (2\right ) - \frac {1}{2}\right )} + 4 \, e^{\left (\frac {1}{2} \, \log \left (2\right ) - \frac {1}{2}\right )} \log \left (2\right ) \log \left (x^{2} + 9\right ) - 2 \, x \log \left (2\right )\right )} e^{\left (-\frac {1}{2} \, \log \left (2\right ) + \frac {1}{2}\right )}}{2 \, \log \left (2\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.15, size = 26, normalized size = 0.81 \begin {gather*} 2\,\ln \left (x^2+9\right )+\frac {x^2}{2\,\ln \left (2\right )}-\frac {\sqrt {2}\,x\,\sqrt {\mathrm {e}}}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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