Optimal. Leaf size=27 \[ \log \left (6-4 e^{-\frac {1}{3} x \left (5-e^4+3 x-\log (x)\right )} x\right ) \]
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Rubi [F] time = 3.27, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {-6+8 x-2 e^4 x+12 x^2-2 x \log (x)}{9 e^{\frac {1}{3} \left (5 x-e^4 x+3 x^2-x \log (x)\right )}-6 x} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-6+\left (8-2 e^4\right ) x+12 x^2-2 x \log (x)}{9 e^{\frac {1}{3} \left (5 x-e^4 x+3 x^2-x \log (x)\right )}-6 x} \, dx\\ &=\int \frac {2 \left (-3+4 \left (1-\frac {e^4}{4}\right ) x+6 x^2-x \log (x)\right )}{9 e^{\frac {1}{3} \left (5 x-e^4 x+3 x^2-x \log (x)\right )}-6 x} \, dx\\ &=2 \int \frac {-3+4 \left (1-\frac {e^4}{4}\right ) x+6 x^2-x \log (x)}{9 e^{\frac {1}{3} \left (5 x-e^4 x+3 x^2-x \log (x)\right )}-6 x} \, dx\\ &=2 \int \left (\frac {e^{\frac {5 x}{3}+x^2} \left (-3+4 \left (1-\frac {e^4}{4}\right ) x+6 x^2-x \log (x)\right )}{2 x \left (3 e^{\frac {5 x}{3}+x^2}-2 e^{\frac {e^4 x}{3}} x^{1+\frac {x}{3}}\right )}+\frac {3-4 \left (1-\frac {e^4}{4}\right ) x-6 x^2+x \log (x)}{6 x}\right ) \, dx\\ &=\frac {1}{3} \int \frac {3-4 \left (1-\frac {e^4}{4}\right ) x-6 x^2+x \log (x)}{x} \, dx+\int \frac {e^{\frac {5 x}{3}+x^2} \left (-3+4 \left (1-\frac {e^4}{4}\right ) x+6 x^2-x \log (x)\right )}{x \left (3 e^{\frac {5 x}{3}+x^2}-2 e^{\frac {e^4 x}{3}} x^{1+\frac {x}{3}}\right )} \, dx\\ &=\frac {1}{3} \int \left (\frac {3-\left (4-e^4\right ) x-6 x^2}{x}+\log (x)\right ) \, dx+\int \left (-\frac {e^{\frac {5 x}{3}+x^2} \left (-4+e^4\right )}{3 e^{\frac {5 x}{3}+x^2}-2 e^{\frac {e^4 x}{3}} x^{1+\frac {x}{3}}}+\frac {3 e^{\frac {5 x}{3}+x^2}}{x \left (-3 e^{\frac {5 x}{3}+x^2}+2 e^{\frac {e^4 x}{3}} x^{1+\frac {x}{3}}\right )}-\frac {6 e^{\frac {5 x}{3}+x^2} x}{-3 e^{\frac {5 x}{3}+x^2}+2 e^{\frac {e^4 x}{3}} x^{1+\frac {x}{3}}}-\frac {e^{\frac {5 x}{3}+x^2} \log (x)}{3 e^{\frac {5 x}{3}+x^2}-2 e^{\frac {e^4 x}{3}} x^{1+\frac {x}{3}}}\right ) \, dx\\ &=\frac {1}{3} \int \frac {3-\left (4-e^4\right ) x-6 x^2}{x} \, dx+\frac {1}{3} \int \log (x) \, dx+3 \int \frac {e^{\frac {5 x}{3}+x^2}}{x \left (-3 e^{\frac {5 x}{3}+x^2}+2 e^{\frac {e^4 x}{3}} x^{1+\frac {x}{3}}\right )} \, dx-6 \int \frac {e^{\frac {5 x}{3}+x^2} x}{-3 e^{\frac {5 x}{3}+x^2}+2 e^{\frac {e^4 x}{3}} x^{1+\frac {x}{3}}} \, dx+\left (4-e^4\right ) \int \frac {e^{\frac {5 x}{3}+x^2}}{3 e^{\frac {5 x}{3}+x^2}-2 e^{\frac {e^4 x}{3}} x^{1+\frac {x}{3}}} \, dx-\int \frac {e^{\frac {5 x}{3}+x^2} \log (x)}{3 e^{\frac {5 x}{3}+x^2}-2 e^{\frac {e^4 x}{3}} x^{1+\frac {x}{3}}} \, dx\\ &=-\frac {x}{3}+\frac {1}{3} x \log (x)+\frac {1}{3} \int \left (-4+e^4+\frac {3}{x}-6 x\right ) \, dx+3 \int \frac {e^{\frac {5 x}{3}+x^2}}{x \left (-3 e^{\frac {5 x}{3}+x^2}+2 e^{\frac {e^4 x}{3}} x^{1+\frac {x}{3}}\right )} \, dx-6 \int \frac {e^{\frac {5 x}{3}+x^2} x}{-3 e^{\frac {5 x}{3}+x^2}+2 e^{\frac {e^4 x}{3}} x^{1+\frac {x}{3}}} \, dx+\left (4-e^4\right ) \int \frac {e^{\frac {5 x}{3}+x^2}}{3 e^{\frac {5 x}{3}+x^2}-2 e^{\frac {e^4 x}{3}} x^{1+\frac {x}{3}}} \, dx-\log (x) \int \frac {e^{\frac {5 x}{3}+x^2}}{3 e^{\frac {5 x}{3}+x^2}-2 e^{\frac {e^4 x}{3}} x^{1+\frac {x}{3}}} \, dx+\int \frac {\int \frac {e^{\frac {5 x}{3}+x^2}}{3 e^{\frac {5 x}{3}+x^2}-2 e^{\frac {e^4 x}{3}} x^{1+\frac {x}{3}}} \, dx}{x} \, dx\\ &=-\frac {x}{3}-\frac {1}{3} \left (4-e^4\right ) x-x^2+\log (x)+\frac {1}{3} x \log (x)+3 \int \frac {e^{\frac {5 x}{3}+x^2}}{x \left (-3 e^{\frac {5 x}{3}+x^2}+2 e^{\frac {e^4 x}{3}} x^{1+\frac {x}{3}}\right )} \, dx-6 \int \frac {e^{\frac {5 x}{3}+x^2} x}{-3 e^{\frac {5 x}{3}+x^2}+2 e^{\frac {e^4 x}{3}} x^{1+\frac {x}{3}}} \, dx+\left (4-e^4\right ) \int \frac {e^{\frac {5 x}{3}+x^2}}{3 e^{\frac {5 x}{3}+x^2}-2 e^{\frac {e^4 x}{3}} x^{1+\frac {x}{3}}} \, dx-\log (x) \int \frac {e^{\frac {5 x}{3}+x^2}}{3 e^{\frac {5 x}{3}+x^2}-2 e^{\frac {e^4 x}{3}} x^{1+\frac {x}{3}}} \, dx+\int \frac {\int \frac {e^{\frac {5 x}{3}+x^2}}{3 e^{\frac {5 x}{3}+x^2}-2 e^{\frac {e^4 x}{3}} x^{1+\frac {x}{3}}} \, dx}{x} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [B] time = 1.31, size = 57, normalized size = 2.11 \begin {gather*} -\frac {1}{3} x \left (6-e^4+3 x-\log (x)\right )+\frac {1}{3} \left (x+3 \log \left (-2 x+3 e^{-\frac {1}{3} \left (-5+e^4-3 x\right ) x} x^{-x/3}\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 44, normalized size = 1.63 \begin {gather*} -x^{2} + \frac {1}{3} \, x e^{4} + \frac {1}{3} \, x \log \relax (x) - \frac {5}{3} \, x + \log \left (-2 \, x + 3 \, e^{\left (x^{2} - \frac {1}{3} \, x e^{4} - \frac {1}{3} \, x \log \relax (x) + \frac {5}{3} \, x\right )}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.18, size = 44, normalized size = 1.63 \begin {gather*} -x^{2} + \frac {1}{3} \, x e^{4} + \frac {1}{3} \, x \log \relax (x) - \frac {5}{3} \, x + \log \left (2 \, x - 3 \, e^{\left (x^{2} - \frac {1}{3} \, x e^{4} - \frac {1}{3} \, x \log \relax (x) + \frac {5}{3} \, x\right )}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.08, size = 42, normalized size = 1.56
method | result | size |
risch | \(\frac {x \ln \relax (x )}{3}+\frac {x \,{\mathrm e}^{4}}{3}-x^{2}-\frac {5 x}{3}+\ln \left (x^{-\frac {x}{3}} {\mathrm e}^{-\frac {x \left ({\mathrm e}^{4}-3 x -5\right )}{3}}-\frac {2 x}{3}\right )\) | \(42\) |
norman | \(\left (-\frac {5}{3}+\frac {{\mathrm e}^{4}}{3}\right ) x -x^{2}+\frac {x \ln \relax (x )}{3}+\ln \left (-9 \,{\mathrm e}^{-\frac {x \ln \relax (x )}{3}-\frac {x \,{\mathrm e}^{4}}{3}+x^{2}+\frac {5 x}{3}}+6 x \right )\) | \(45\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.41, size = 55, normalized size = 2.04 \begin {gather*} -x^{2} + \frac {1}{3} \, {\left (x + 3\right )} \log \relax (x) - \frac {5}{3} \, x + \log \left (\frac {2 \, x e^{\left (\frac {1}{3} \, x e^{4} + \frac {1}{3} \, x \log \relax (x)\right )} - 3 \, e^{\left (x^{2} + \frac {5}{3} \, x\right )}}{2 \, x x^{\frac {1}{3} \, x}}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 6.81, size = 46, normalized size = 1.70 \begin {gather*} \ln \left (x-\frac {3\,{\mathrm {e}}^{x^2}\,{\left ({\mathrm {e}}^x\right )}^{5/3}}{2\,x^{x/3}\,{\left ({\mathrm {e}}^{x\,{\mathrm {e}}^4}\right )}^{1/3}}\right )+\frac {x\,\ln \relax (x)}{3}-x^2+x\,\left (\frac {{\mathrm {e}}^4}{3}-\frac {5}{3}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.36, size = 53, normalized size = 1.96 \begin {gather*} - \frac {x^{2}}{3} + \frac {x \log {\relax (x )}}{9} + x \left (- \frac {5}{9} + \frac {e^{4}}{9}\right ) + \frac {\log {\left (- \frac {2 x}{3} + e^{x^{2} - \frac {x \log {\relax (x )}}{3} - \frac {x e^{4}}{3} + \frac {5 x}{3}} \right )}}{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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