Optimal. Leaf size=32 \[ \log \left (\left (4+\frac {-1-e^{1-x}+2 x}{\log (x)}\right ) \log \left (-x+\log ^2(x)\right )\right ) \]
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Rubi [F] time = 14.84, 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 {\left (x+e^{1-x} x-2 x^2\right ) \log (x)+\left (-2-2 e^{1-x}\right ) \log ^2(x)+8 \log ^3(x)+\left (-x-e^{1-x} x+2 x^2+\left (-2 x^2-e^{1-x} x^2\right ) \log (x)+\left (1+e^{1-x}-2 x\right ) \log ^2(x)+\left (2 x+e^{1-x} x\right ) \log ^3(x)\right ) \log \left (-x+\log ^2(x)\right )}{\left (\left (x^2+e^{1-x} x^2-2 x^3\right ) \log (x)-4 x^2 \log ^2(x)+\left (-x-e^{1-x} x+2 x^2\right ) \log ^3(x)+4 x \log ^4(x)\right ) \log \left (-x+\log ^2(x)\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {e^x \left (\left (x+e^{1-x} x-2 x^2\right ) \log (x)+\left (-2-2 e^{1-x}\right ) \log ^2(x)+8 \log ^3(x)+\left (-x-e^{1-x} x+2 x^2+\left (-2 x^2-e^{1-x} x^2\right ) \log (x)+\left (1+e^{1-x}-2 x\right ) \log ^2(x)+\left (2 x+e^{1-x} x\right ) \log ^3(x)\right ) \log \left (-x+\log ^2(x)\right )\right )}{x \log (x) \left (e+e^x-2 e^x x-4 e^x \log (x)\right ) \left (x-\log ^2(x)\right ) \log \left (-x+\log ^2(x)\right )} \, dx\\ &=\int \left (\frac {e^x \left (4+x+2 x^2+4 x \log (x)\right )}{x \left (-e-e^x+2 e^x x+4 e^x \log (x)\right )}-\frac {-x \log (x)+2 \log ^2(x)+x \log \left (-x+\log ^2(x)\right )+x^2 \log (x) \log \left (-x+\log ^2(x)\right )-\log ^2(x) \log \left (-x+\log ^2(x)\right )-x \log ^3(x) \log \left (-x+\log ^2(x)\right )}{x \log (x) \left (x-\log ^2(x)\right ) \log \left (-x+\log ^2(x)\right )}\right ) \, dx\\ &=\int \frac {e^x \left (4+x+2 x^2+4 x \log (x)\right )}{x \left (-e-e^x+2 e^x x+4 e^x \log (x)\right )} \, dx-\int \frac {-x \log (x)+2 \log ^2(x)+x \log \left (-x+\log ^2(x)\right )+x^2 \log (x) \log \left (-x+\log ^2(x)\right )-\log ^2(x) \log \left (-x+\log ^2(x)\right )-x \log ^3(x) \log \left (-x+\log ^2(x)\right )}{x \log (x) \left (x-\log ^2(x)\right ) \log \left (-x+\log ^2(x)\right )} \, dx\\ &=\int \left (\frac {e^x}{-e-e^x+2 e^x x+4 e^x \log (x)}+\frac {4 e^x}{x \left (-e-e^x+2 e^x x+4 e^x \log (x)\right )}+\frac {2 e^x x}{-e-e^x+2 e^x x+4 e^x \log (x)}+\frac {4 e^x \log (x)}{-e-e^x+2 e^x x+4 e^x \log (x)}\right ) \, dx-\int \left (\frac {1+x \log (x)}{x \log (x)}+\frac {-x+2 \log (x)}{x \left (x-\log ^2(x)\right ) \log \left (-x+\log ^2(x)\right )}\right ) \, dx\\ &=2 \int \frac {e^x x}{-e-e^x+2 e^x x+4 e^x \log (x)} \, dx+4 \int \frac {e^x}{x \left (-e-e^x+2 e^x x+4 e^x \log (x)\right )} \, dx+4 \int \frac {e^x \log (x)}{-e-e^x+2 e^x x+4 e^x \log (x)} \, dx+\int \frac {e^x}{-e-e^x+2 e^x x+4 e^x \log (x)} \, dx-\int \frac {1+x \log (x)}{x \log (x)} \, dx-\int \frac {-x+2 \log (x)}{x \left (x-\log ^2(x)\right ) \log \left (-x+\log ^2(x)\right )} \, dx\\ &=\log \left (\log \left (-x+\log ^2(x)\right )\right )+2 \int \frac {e^x x}{-e-e^x+2 e^x x+4 e^x \log (x)} \, dx+4 \int \frac {e^x}{x \left (-e-e^x+2 e^x x+4 e^x \log (x)\right )} \, dx+4 \int \frac {e^x \log (x)}{-e-e^x+2 e^x x+4 e^x \log (x)} \, dx-\int \left (1+\frac {1}{x \log (x)}\right ) \, dx+\int \frac {e^x}{-e-e^x+2 e^x x+4 e^x \log (x)} \, dx\\ &=-x+\log \left (\log \left (-x+\log ^2(x)\right )\right )+2 \int \frac {e^x x}{-e-e^x+2 e^x x+4 e^x \log (x)} \, dx+4 \int \frac {e^x}{x \left (-e-e^x+2 e^x x+4 e^x \log (x)\right )} \, dx+4 \int \frac {e^x \log (x)}{-e-e^x+2 e^x x+4 e^x \log (x)} \, dx-\int \frac {1}{x \log (x)} \, dx+\int \frac {e^x}{-e-e^x+2 e^x x+4 e^x \log (x)} \, dx\\ &=-x+\log \left (\log \left (-x+\log ^2(x)\right )\right )+2 \int \frac {e^x x}{-e-e^x+2 e^x x+4 e^x \log (x)} \, dx+4 \int \frac {e^x}{x \left (-e-e^x+2 e^x x+4 e^x \log (x)\right )} \, dx+4 \int \frac {e^x \log (x)}{-e-e^x+2 e^x x+4 e^x \log (x)} \, dx+\int \frac {e^x}{-e-e^x+2 e^x x+4 e^x \log (x)} \, dx-\operatorname {Subst}\left (\int \frac {1}{x} \, dx,x,\log (x)\right )\\ &=-x-\log (\log (x))+\log \left (\log \left (-x+\log ^2(x)\right )\right )+2 \int \frac {e^x x}{-e-e^x+2 e^x x+4 e^x \log (x)} \, dx+4 \int \frac {e^x}{x \left (-e-e^x+2 e^x x+4 e^x \log (x)\right )} \, dx+4 \int \frac {e^x \log (x)}{-e-e^x+2 e^x x+4 e^x \log (x)} \, dx+\int \frac {e^x}{-e-e^x+2 e^x x+4 e^x \log (x)} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.30, size = 42, normalized size = 1.31 \begin {gather*} -x-\log (\log (x))+\log \left (-e-e^x+2 e^x x+4 e^x \log (x)\right )+\log \left (\log \left (-x+\log ^2(x)\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.20, size = 34, normalized size = 1.06 \begin {gather*} \log \left (2 \, x - e^{\left (-x + 1\right )} + 4 \, \log \relax (x) - 1\right ) + \log \left (\log \left (\log \relax (x)^{2} - x\right )\right ) - \log \left (\log \relax (x)\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.92, size = 34, normalized size = 1.06 \begin {gather*} \log \left (2 \, x - e^{\left (-x + 1\right )} + 4 \, \log \relax (x) - 1\right ) + \log \left (\log \left (\log \relax (x)^{2} - x\right )\right ) - \log \left (\log \relax (x)\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 33, normalized size = 1.03
method | result | size |
risch | \(\ln \left (\frac {x}{2}-\frac {{\mathrm e}^{1-x}}{4}+\ln \relax (x )-\frac {1}{4}\right )-\ln \left (\ln \relax (x )\right )+\ln \left (\ln \left (\ln \relax (x )^{2}-x \right )\right )\) | \(33\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.55, size = 57, normalized size = 1.78 \begin {gather*} -x + \log \left (\frac {1}{2} \, x + \log \relax (x) - \frac {1}{4}\right ) + \log \left (\frac {{\left (2 \, x + 4 \, \log \relax (x) - 1\right )} e^{x} - e}{2 \, x + 4 \, \log \relax (x) - 1}\right ) + \log \left (\log \left (\log \relax (x)^{2} - x\right )\right ) - \log \left (\log \relax (x)\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.01, size = 80, normalized size = 2.50 \begin {gather*} \ln \left (\frac {2\,x-{\mathrm {e}}^{1-x}+4\,\ln \relax (x)-1}{x}\right )+\ln \left (\frac {2\,x+x\,{\mathrm {e}}^{1-x}+4}{x}\right )+\ln \left (\ln \left ({\ln \relax (x)}^2-x\right )\right )-\ln \left (\frac {4\,\ln \relax (x)+2\,x\,\ln \relax (x)+x\,{\mathrm {e}}^{1-x}\,\ln \relax (x)}{x}\right )+\ln \relax (x) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.78, size = 31, normalized size = 0.97 \begin {gather*} \log {\left (- 2 x + e^{1 - x} - 4 \log {\relax (x )} + 1 \right )} - \log {\left (\log {\relax (x )} \right )} + \log {\left (\log {\left (- x + \log {\relax (x )}^{2} \right )} \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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