Optimal. Leaf size=20 \[ \log \left (-10+\frac {\left (3+e^x\right ) \log (x)}{x (x+\log (3))}\right ) \]
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Rubi [F] time = 3.23, 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 {3 x+3 \log (3)+e^x (x+\log (3))+\left (-6 x-3 \log (3)+e^x \left (-2 x+x^2+(-1+x) \log (3)\right )\right ) \log (x)}{-10 x^4-20 x^3 \log (3)-10 x^2 \log ^2(3)+\left (3 x^2+3 x \log (3)+e^x \left (x^2+x \log (3)\right )\right ) \log (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 {-3 x-3 \log (3)-e^x (x+\log (3))-\left (-6 x-3 \log (3)+e^x \left (-2 x+x^2+(-1+x) \log (3)\right )\right ) \log (x)}{x (x+\log (3)) \left (10 x^2+10 x \log (3)-3 \log (x)-e^x \log (x)\right )} \, dx\\ &=\int \left (\frac {x+\log (3)+x^2 \log (x)-2 x \left (1-\frac {\log (3)}{2}\right ) \log (x)-\log (3) \log (x)}{x (x+\log (3)) \log (x)}+\frac {-10 x-10 \log (3)-10 x^2 \log (x)+20 x \left (1-\frac {\log (3)}{2}\right ) \log (x)+10 \log (3) \log (x)+3 \log ^2(x)}{\log (x) \left (10 x^2+10 x \log (3)-3 \log (x)-e^x \log (x)\right )}\right ) \, dx\\ &=\int \frac {x+\log (3)+x^2 \log (x)-2 x \left (1-\frac {\log (3)}{2}\right ) \log (x)-\log (3) \log (x)}{x (x+\log (3)) \log (x)} \, dx+\int \frac {-10 x-10 \log (3)-10 x^2 \log (x)+20 x \left (1-\frac {\log (3)}{2}\right ) \log (x)+10 \log (3) \log (x)+3 \log ^2(x)}{\log (x) \left (10 x^2+10 x \log (3)-3 \log (x)-e^x \log (x)\right )} \, dx\\ &=\int \frac {x+\log (3)+\left (x^2+x (-2+\log (3))-\log (3)\right ) \log (x)}{x (x+\log (3)) \log (x)} \, dx+\int \left (-\frac {10 x^2}{10 x^2+10 x \log (3)-3 \log (x)-e^x \log (x)}-\frac {10 x (-2+\log (3))}{10 x^2+10 x \log (3)-3 \log (x)-e^x \log (x)}+\frac {10 \log (3)}{10 x^2+10 x \log (3)-3 \log (x)-e^x \log (x)}-\frac {10 x}{\log (x) \left (10 x^2+10 x \log (3)-3 \log (x)-e^x \log (x)\right )}-\frac {10 \log (3)}{\log (x) \left (10 x^2+10 x \log (3)-3 \log (x)-e^x \log (x)\right )}-\frac {3 \log (x)}{-10 x^2-10 x \log (3)+3 \log (x)+e^x \log (x)}\right ) \, dx\\ &=-\left (3 \int \frac {\log (x)}{-10 x^2-10 x \log (3)+3 \log (x)+e^x \log (x)} \, dx\right )-10 \int \frac {x^2}{10 x^2+10 x \log (3)-3 \log (x)-e^x \log (x)} \, dx-10 \int \frac {x}{\log (x) \left (10 x^2+10 x \log (3)-3 \log (x)-e^x \log (x)\right )} \, dx+(10 (2-\log (3))) \int \frac {x}{10 x^2+10 x \log (3)-3 \log (x)-e^x \log (x)} \, dx+(10 \log (3)) \int \frac {1}{10 x^2+10 x \log (3)-3 \log (x)-e^x \log (x)} \, dx-(10 \log (3)) \int \frac {1}{\log (x) \left (10 x^2+10 x \log (3)-3 \log (x)-e^x \log (x)\right )} \, dx+\int \left (\frac {x^2-x (2-\log (3))-\log (3)}{x (x+\log (3))}+\frac {1}{x \log (x)}\right ) \, dx\\ &=-\left (3 \int \frac {\log (x)}{-10 x^2-10 x \log (3)+3 \log (x)+e^x \log (x)} \, dx\right )-10 \int \frac {x^2}{10 x^2+10 x \log (3)-3 \log (x)-e^x \log (x)} \, dx-10 \int \frac {x}{\log (x) \left (10 x^2+10 x \log (3)-3 \log (x)-e^x \log (x)\right )} \, dx+(10 (2-\log (3))) \int \frac {x}{10 x^2+10 x \log (3)-3 \log (x)-e^x \log (x)} \, dx+(10 \log (3)) \int \frac {1}{10 x^2+10 x \log (3)-3 \log (x)-e^x \log (x)} \, dx-(10 \log (3)) \int \frac {1}{\log (x) \left (10 x^2+10 x \log (3)-3 \log (x)-e^x \log (x)\right )} \, dx+\int \frac {x^2-x (2-\log (3))-\log (3)}{x (x+\log (3))} \, dx+\int \frac {1}{x \log (x)} \, dx\\ &=-\left (3 \int \frac {\log (x)}{-10 x^2-10 x \log (3)+3 \log (x)+e^x \log (x)} \, dx\right )-10 \int \frac {x^2}{10 x^2+10 x \log (3)-3 \log (x)-e^x \log (x)} \, dx-10 \int \frac {x}{\log (x) \left (10 x^2+10 x \log (3)-3 \log (x)-e^x \log (x)\right )} \, dx+(10 (2-\log (3))) \int \frac {x}{10 x^2+10 x \log (3)-3 \log (x)-e^x \log (x)} \, dx+(10 \log (3)) \int \frac {1}{10 x^2+10 x \log (3)-3 \log (x)-e^x \log (x)} \, dx-(10 \log (3)) \int \frac {1}{\log (x) \left (10 x^2+10 x \log (3)-3 \log (x)-e^x \log (x)\right )} \, dx+\int \left (1-\frac {1}{x}+\frac {1}{-x-\log (3)}\right ) \, dx+\operatorname {Subst}\left (\int \frac {1}{x} \, dx,x,\log (x)\right )\\ &=x-\log (x)-\log (x+\log (3))+\log (\log (x))-3 \int \frac {\log (x)}{-10 x^2-10 x \log (3)+3 \log (x)+e^x \log (x)} \, dx-10 \int \frac {x^2}{10 x^2+10 x \log (3)-3 \log (x)-e^x \log (x)} \, dx-10 \int \frac {x}{\log (x) \left (10 x^2+10 x \log (3)-3 \log (x)-e^x \log (x)\right )} \, dx+(10 (2-\log (3))) \int \frac {x}{10 x^2+10 x \log (3)-3 \log (x)-e^x \log (x)} \, dx+(10 \log (3)) \int \frac {1}{10 x^2+10 x \log (3)-3 \log (x)-e^x \log (x)} \, dx-(10 \log (3)) \int \frac {1}{\log (x) \left (10 x^2+10 x \log (3)-3 \log (x)-e^x \log (x)\right )} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [F] time = 1.65, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {3 x+3 \log (3)+e^x (x+\log (3))+\left (-6 x-3 \log (3)+e^x \left (-2 x+x^2+(-1+x) \log (3)\right )\right ) \log (x)}{-10 x^4-20 x^3 \log (3)-10 x^2 \log ^2(3)+\left (3 x^2+3 x \log (3)+e^x \left (x^2+x \log (3)\right )\right ) \log (x)} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [B] time = 1.10, size = 45, normalized size = 2.25 \begin {gather*} -\log \left (x^{2} + x \log \relax (3)\right ) + \log \left (-\frac {10 \, x^{2} + 10 \, x \log \relax (3) - {\left (e^{x} + 3\right )} \log \relax (x)}{e^{x} + 3}\right ) + \log \left (e^{x} + 3\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.23, size = 33, normalized size = 1.65 \begin {gather*} \log \left (-10 \, x^{2} - 10 \, x \log \relax (3) + e^{x} \log \relax (x) + 3 \, \log \relax (x)\right ) - \log \left (x + \log \relax (3)\right ) - \log \relax (x) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.32, size = 35, normalized size = 1.75
method | result | size |
norman | \(-\ln \relax (x )-\ln \left (\ln \relax (3)+x \right )+\ln \left (10 x \ln \relax (3)+10 x^{2}-{\mathrm e}^{x} \ln \relax (x )-3 \ln \relax (x )\right )\) | \(35\) |
risch | \(-\ln \left (x \ln \relax (3)+x^{2}\right )+\ln \left (3+{\mathrm e}^{x}\right )+\ln \left (\ln \relax (x )-\frac {10 x \left (\ln \relax (3)+x \right )}{3+{\mathrm e}^{x}}\right )\) | \(35\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.51, size = 43, normalized size = 2.15 \begin {gather*} -\log \left (x + \log \relax (3)\right ) - \log \relax (x) + \log \left (-\frac {10 \, x^{2} + 10 \, x \log \relax (3) - e^{x} \log \relax (x) - 3 \, \log \relax (x)}{\log \relax (x)}\right ) + \log \left (\log \relax (x)\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.05 \begin {gather*} -\int \frac {3\,x+3\,\ln \relax (3)-\ln \relax (x)\,\left (6\,x+3\,\ln \relax (3)-{\mathrm {e}}^x\,\left (\ln \relax (3)\,\left (x-1\right )-2\,x+x^2\right )\right )+{\mathrm {e}}^x\,\left (x+\ln \relax (3)\right )}{10\,x^2\,{\ln \relax (3)}^2+20\,x^3\,\ln \relax (3)-\ln \relax (x)\,\left (3\,x\,\ln \relax (3)+{\mathrm {e}}^x\,\left (x^2+\ln \relax (3)\,x\right )+3\,x^2\right )+10\,x^4} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.57, size = 39, normalized size = 1.95 \begin {gather*} - \log {\left (x^{2} + x \log {\relax (3 )} \right )} + \log {\left (\frac {- 10 x^{2} - 10 x \log {\relax (3 )} + 3 \log {\relax (x )}}{\log {\relax (x )}} + e^{x} \right )} + \log {\left (\log {\relax (x )} \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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