Optimal. Leaf size=25 \[ e^{x/3}-\frac {(-3+x-x \log (2))^2}{x^2}+\log (x) \]
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Rubi [A]
time = 0.03, antiderivative size = 26, normalized size of antiderivative = 1.04, number of steps
used = 7, number of rules used = 4, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.121, Rules used = {6, 12, 14, 2225}
\begin {gather*} -\frac {9}{x^2}+e^{x/3}+\log (x)+\frac {6 (1-\log (2))}{x} \end {gather*}
Antiderivative was successfully verified.
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Rule 6
Rule 12
Rule 14
Rule 2225
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {54+3 x^2+e^{x/3} x^3+x (-18+18 \log (2))}{3 x^3} \, dx\\ &=\frac {1}{3} \int \frac {54+3 x^2+e^{x/3} x^3+x (-18+18 \log (2))}{x^3} \, dx\\ &=\frac {1}{3} \int \left (e^{x/3}+\frac {3 \left (18+x^2-6 x (1-\log (2))\right )}{x^3}\right ) \, dx\\ &=\frac {1}{3} \int e^{x/3} \, dx+\int \frac {18+x^2-6 x (1-\log (2))}{x^3} \, dx\\ &=e^{x/3}+\int \left (\frac {18}{x^3}+\frac {1}{x}+\frac {6 (-1+\log (2))}{x^2}\right ) \, dx\\ &=e^{x/3}-\frac {9}{x^2}+\frac {6 (1-\log (2))}{x}+\log (x)\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.03, size = 24, normalized size = 0.96 \begin {gather*} e^{x/3}-\frac {9}{x^2}-\frac {6 (-1+\log (2))}{x}+\log (x) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.29, size = 27, normalized size = 1.08
method | result | size |
risch | \(\frac {\left (-18 \ln \left (2\right )+18\right ) x -27}{3 x^{2}}+\ln \left (x \right )+{\mathrm e}^{\frac {x}{3}}\) | \(23\) |
norman | \(\frac {-9+x^{2} {\mathrm e}^{\frac {x}{3}}+\left (-6 \ln \left (2\right )+6\right ) x}{x^{2}}+\ln \left (x \right )\) | \(26\) |
derivativedivides | \(\ln \left (\frac {x}{3}\right )-\frac {9}{x^{2}}+\frac {6}{x}-\frac {6 \ln \left (2\right )}{x}+{\mathrm e}^{\frac {x}{3}}\) | \(27\) |
default | \(\ln \left (\frac {x}{3}\right )-\frac {9}{x^{2}}+\frac {6}{x}-\frac {6 \ln \left (2\right )}{x}+{\mathrm e}^{\frac {x}{3}}\) | \(27\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.26, size = 24, normalized size = 0.96 \begin {gather*} -\frac {6 \, \log \left (2\right )}{x} + \frac {6}{x} - \frac {9}{x^{2}} + e^{\left (\frac {1}{3} \, x\right )} + \log \left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.55, size = 28, normalized size = 1.12 \begin {gather*} \frac {x^{2} e^{\left (\frac {1}{3} \, x\right )} + x^{2} \log \left (x\right ) - 6 \, x \log \left (2\right ) + 6 \, x - 9}{x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.10, size = 20, normalized size = 0.80 \begin {gather*} e^{\frac {x}{3}} + \log {\left (x \right )} + \frac {x \left (6 - 6 \log {\left (2 \right )}\right ) - 9}{x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.41, size = 30, normalized size = 1.20 \begin {gather*} \frac {x^{2} e^{\left (\frac {1}{3} \, x\right )} + x^{2} \log \left (\frac {1}{3} \, x\right ) - 6 \, x \log \left (2\right ) + 6 \, x - 9}{x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.87, size = 20, normalized size = 0.80 \begin {gather*} {\mathrm {e}}^{x/3}+\ln \left (x\right )-\frac {x\,\left (\ln \left (64\right )-6\right )+9}{x^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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