Optimal. Leaf size=25 \[ \log \left (\log (x)-\frac {3 \left (-e^x \log \left (5 e^x\right )+\log (x)\right )^2}{e}\right ) \]
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Rubi [F] time = 8.48, 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 {-e+6 e^{2 x} x \log ^2\left (5 e^x\right )+\left (6-6 e^x x\right ) \log (x)+\log \left (5 e^x\right ) \left (-6 e^x+6 e^{2 x} x-6 e^x x \log (x)\right )}{3 e^{2 x} x \log ^2\left (5 e^x\right )-e x \log (x)-6 e^x x \log \left (5 e^x\right ) \log (x)+3 x \log ^2(x)} \, dx \end {gather*}
Verification is not applicable to the result.
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\begin {gather*} \begin {aligned} \text {integral} &=\int \left (\frac {2 \left (1+\log \left (5 e^x\right )\right )}{\log \left (5 e^x\right )}+\frac {-e \log \left (5 e^x\right )-6 e^x \log ^2\left (5 e^x\right )+2 e x \log (x)+6 \log \left (5 e^x\right ) \log (x)+2 e x \log \left (5 e^x\right ) \log (x)+6 e^x x \log \left (5 e^x\right ) \log (x)+6 e^x x \log ^2\left (5 e^x\right ) \log (x)-6 x \log ^2(x)-6 x \log \left (5 e^x\right ) \log ^2(x)}{x \log \left (5 e^x\right ) \left (3 e^{2 x} \log ^2\left (5 e^x\right )-e \log (x)-6 e^x \log \left (5 e^x\right ) \log (x)+3 \log ^2(x)\right )}\right ) \, dx\\ &=2 \int \frac {1+\log \left (5 e^x\right )}{\log \left (5 e^x\right )} \, dx+\int \frac {-e \log \left (5 e^x\right )-6 e^x \log ^2\left (5 e^x\right )+2 e x \log (x)+6 \log \left (5 e^x\right ) \log (x)+2 e x \log \left (5 e^x\right ) \log (x)+6 e^x x \log \left (5 e^x\right ) \log (x)+6 e^x x \log ^2\left (5 e^x\right ) \log (x)-6 x \log ^2(x)-6 x \log \left (5 e^x\right ) \log ^2(x)}{x \log \left (5 e^x\right ) \left (3 e^{2 x} \log ^2\left (5 e^x\right )-e \log (x)-6 e^x \log \left (5 e^x\right ) \log (x)+3 \log ^2(x)\right )} \, dx\\ &=2 \operatorname {Subst}\left (\int \frac {1+\frac {1}{\log (5 x)}}{x} \, dx,x,e^x\right )+\int \frac {2 x (e-3 \log (x)) \log (x)+6 e^x \log ^2\left (5 e^x\right ) (-1+x \log (x))+\log \left (5 e^x\right ) \left (-e+2 \left (3+e x+3 e^x x\right ) \log (x)-6 x \log ^2(x)\right )}{x \log \left (5 e^x\right ) \left (3 e^{2 x} \log ^2\left (5 e^x\right )-6 e^x \log \left (5 e^x\right ) \log (x)+\log (x) (-e+3 \log (x))\right )} \, dx\\ &=2 \operatorname {Subst}\left (\int \left (\frac {1}{x}+\frac {1}{x \log (5 x)}\right ) \, dx,x,e^x\right )+\int \left (\frac {e}{x \left (-3 e^{2 x} \log ^2\left (5 e^x\right )+e \log (x)+6 e^x \log \left (5 e^x\right ) \log (x)-3 \log ^2(x)\right )}-\frac {2 e \log (x)}{-3 e^{2 x} \log ^2\left (5 e^x\right )+e \log (x)+6 e^x \log \left (5 e^x\right ) \log (x)-3 \log ^2(x)}-\frac {2 e \log (x)}{\log \left (5 e^x\right ) \left (-3 e^{2 x} \log ^2\left (5 e^x\right )+e \log (x)+6 e^x \log \left (5 e^x\right ) \log (x)-3 \log ^2(x)\right )}-\frac {6 e^x \log \left (5 e^x\right )}{x \left (3 e^{2 x} \log ^2\left (5 e^x\right )-e \log (x)-6 e^x \log \left (5 e^x\right ) \log (x)+3 \log ^2(x)\right )}+\frac {6 e^x \log (x)}{3 e^{2 x} \log ^2\left (5 e^x\right )-e \log (x)-6 e^x \log \left (5 e^x\right ) \log (x)+3 \log ^2(x)}+\frac {6 \log (x)}{x \left (3 e^{2 x} \log ^2\left (5 e^x\right )-e \log (x)-6 e^x \log \left (5 e^x\right ) \log (x)+3 \log ^2(x)\right )}+\frac {6 e^x \log \left (5 e^x\right ) \log (x)}{3 e^{2 x} \log ^2\left (5 e^x\right )-e \log (x)-6 e^x \log \left (5 e^x\right ) \log (x)+3 \log ^2(x)}-\frac {6 \log ^2(x)}{3 e^{2 x} \log ^2\left (5 e^x\right )-e \log (x)-6 e^x \log \left (5 e^x\right ) \log (x)+3 \log ^2(x)}-\frac {6 \log ^2(x)}{\log \left (5 e^x\right ) \left (3 e^{2 x} \log ^2\left (5 e^x\right )-e \log (x)-6 e^x \log \left (5 e^x\right ) \log (x)+3 \log ^2(x)\right )}\right ) \, dx\\ &=2 x+2 \operatorname {Subst}\left (\int \frac {1}{x \log (5 x)} \, dx,x,e^x\right )-6 \int \frac {e^x \log \left (5 e^x\right )}{x \left (3 e^{2 x} \log ^2\left (5 e^x\right )-e \log (x)-6 e^x \log \left (5 e^x\right ) \log (x)+3 \log ^2(x)\right )} \, dx+6 \int \frac {e^x \log (x)}{3 e^{2 x} \log ^2\left (5 e^x\right )-e \log (x)-6 e^x \log \left (5 e^x\right ) \log (x)+3 \log ^2(x)} \, dx+6 \int \frac {\log (x)}{x \left (3 e^{2 x} \log ^2\left (5 e^x\right )-e \log (x)-6 e^x \log \left (5 e^x\right ) \log (x)+3 \log ^2(x)\right )} \, dx+6 \int \frac {e^x \log \left (5 e^x\right ) \log (x)}{3 e^{2 x} \log ^2\left (5 e^x\right )-e \log (x)-6 e^x \log \left (5 e^x\right ) \log (x)+3 \log ^2(x)} \, dx-6 \int \frac {\log ^2(x)}{3 e^{2 x} \log ^2\left (5 e^x\right )-e \log (x)-6 e^x \log \left (5 e^x\right ) \log (x)+3 \log ^2(x)} \, dx-6 \int \frac {\log ^2(x)}{\log \left (5 e^x\right ) \left (3 e^{2 x} \log ^2\left (5 e^x\right )-e \log (x)-6 e^x \log \left (5 e^x\right ) \log (x)+3 \log ^2(x)\right )} \, dx+e \int \frac {1}{x \left (-3 e^{2 x} \log ^2\left (5 e^x\right )+e \log (x)+6 e^x \log \left (5 e^x\right ) \log (x)-3 \log ^2(x)\right )} \, dx-(2 e) \int \frac {\log (x)}{-3 e^{2 x} \log ^2\left (5 e^x\right )+e \log (x)+6 e^x \log \left (5 e^x\right ) \log (x)-3 \log ^2(x)} \, dx-(2 e) \int \frac {\log (x)}{\log \left (5 e^x\right ) \left (-3 e^{2 x} \log ^2\left (5 e^x\right )+e \log (x)+6 e^x \log \left (5 e^x\right ) \log (x)-3 \log ^2(x)\right )} \, dx\\ &=2 x+2 \operatorname {Subst}\left (\int \frac {1}{x} \, dx,x,\log \left (5 e^x\right )\right )-6 \int \frac {e^x \log \left (5 e^x\right )}{x \left (3 e^{2 x} \log ^2\left (5 e^x\right )-e \log (x)-6 e^x \log \left (5 e^x\right ) \log (x)+3 \log ^2(x)\right )} \, dx+6 \int \frac {e^x \log (x)}{3 e^{2 x} \log ^2\left (5 e^x\right )-e \log (x)-6 e^x \log \left (5 e^x\right ) \log (x)+3 \log ^2(x)} \, dx+6 \int \frac {\log (x)}{x \left (3 e^{2 x} \log ^2\left (5 e^x\right )-e \log (x)-6 e^x \log \left (5 e^x\right ) \log (x)+3 \log ^2(x)\right )} \, dx+6 \int \frac {e^x \log \left (5 e^x\right ) \log (x)}{3 e^{2 x} \log ^2\left (5 e^x\right )-e \log (x)-6 e^x \log \left (5 e^x\right ) \log (x)+3 \log ^2(x)} \, dx-6 \int \frac {\log ^2(x)}{3 e^{2 x} \log ^2\left (5 e^x\right )-e \log (x)-6 e^x \log \left (5 e^x\right ) \log (x)+3 \log ^2(x)} \, dx-6 \int \frac {\log ^2(x)}{\log \left (5 e^x\right ) \left (3 e^{2 x} \log ^2\left (5 e^x\right )-e \log (x)-6 e^x \log \left (5 e^x\right ) \log (x)+3 \log ^2(x)\right )} \, dx+e \int \frac {1}{x \left (-3 e^{2 x} \log ^2\left (5 e^x\right )+e \log (x)+6 e^x \log \left (5 e^x\right ) \log (x)-3 \log ^2(x)\right )} \, dx-(2 e) \int \frac {\log (x)}{-3 e^{2 x} \log ^2\left (5 e^x\right )+e \log (x)+6 e^x \log \left (5 e^x\right ) \log (x)-3 \log ^2(x)} \, dx-(2 e) \int \frac {\log (x)}{\log \left (5 e^x\right ) \left (-3 e^{2 x} \log ^2\left (5 e^x\right )+e \log (x)+6 e^x \log \left (5 e^x\right ) \log (x)-3 \log ^2(x)\right )} \, dx\\ &=2 x+2 \log \left (\log \left (5 e^x\right )\right )-6 \int \frac {e^x \log \left (5 e^x\right )}{x \left (3 e^{2 x} \log ^2\left (5 e^x\right )-e \log (x)-6 e^x \log \left (5 e^x\right ) \log (x)+3 \log ^2(x)\right )} \, dx+6 \int \frac {e^x \log (x)}{3 e^{2 x} \log ^2\left (5 e^x\right )-e \log (x)-6 e^x \log \left (5 e^x\right ) \log (x)+3 \log ^2(x)} \, dx+6 \int \frac {\log (x)}{x \left (3 e^{2 x} \log ^2\left (5 e^x\right )-e \log (x)-6 e^x \log \left (5 e^x\right ) \log (x)+3 \log ^2(x)\right )} \, dx+6 \int \frac {e^x \log \left (5 e^x\right ) \log (x)}{3 e^{2 x} \log ^2\left (5 e^x\right )-e \log (x)-6 e^x \log \left (5 e^x\right ) \log (x)+3 \log ^2(x)} \, dx-6 \int \frac {\log ^2(x)}{3 e^{2 x} \log ^2\left (5 e^x\right )-e \log (x)-6 e^x \log \left (5 e^x\right ) \log (x)+3 \log ^2(x)} \, dx-6 \int \frac {\log ^2(x)}{\log \left (5 e^x\right ) \left (3 e^{2 x} \log ^2\left (5 e^x\right )-e \log (x)-6 e^x \log \left (5 e^x\right ) \log (x)+3 \log ^2(x)\right )} \, dx+e \int \frac {1}{x \left (-3 e^{2 x} \log ^2\left (5 e^x\right )+e \log (x)+6 e^x \log \left (5 e^x\right ) \log (x)-3 \log ^2(x)\right )} \, dx-(2 e) \int \frac {\log (x)}{-3 e^{2 x} \log ^2\left (5 e^x\right )+e \log (x)+6 e^x \log \left (5 e^x\right ) \log (x)-3 \log ^2(x)} \, dx-(2 e) \int \frac {\log (x)}{\log \left (5 e^x\right ) \left (-3 e^{2 x} \log ^2\left (5 e^x\right )+e \log (x)+6 e^x \log \left (5 e^x\right ) \log (x)-3 \log ^2(x)\right )} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [F] time = 0.46, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {-e+6 e^{2 x} x \log ^2\left (5 e^x\right )+\left (6-6 e^x x\right ) \log (x)+\log \left (5 e^x\right ) \left (-6 e^x+6 e^{2 x} x-6 e^x x \log (x)\right )}{3 e^{2 x} x \log ^2\left (5 e^x\right )-e x \log (x)-6 e^x x \log \left (5 e^x\right ) \log (x)+3 x \log ^2(x)} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.76, size = 42, normalized size = 1.68 \begin {gather*} \log \left (3 \, {\left (x^{2} + 2 \, x \log \relax (5) + \log \relax (5)^{2}\right )} e^{\left (2 \, x\right )} - {\left (6 \, {\left (x + \log \relax (5)\right )} e^{x} + e\right )} \log \relax (x) + 3 \, \log \relax (x)^{2}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.47, size = 56, normalized size = 2.24 \begin {gather*} \log \left (-3 \, x^{2} e^{\left (2 \, x\right )} - 6 \, x e^{\left (2 \, x\right )} \log \relax (5) - 3 \, e^{\left (2 \, x\right )} \log \relax (5)^{2} + 6 \, x e^{x} \log \relax (x) + 6 \, e^{x} \log \relax (5) \log \relax (x) + e \log \relax (x) - 3 \, \log \relax (x)^{2}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.28, size = 69, normalized size = 2.76
method | result | size |
risch | \(2 x +\ln \left (\ln \left ({\mathrm e}^{x}\right )^{2}-{\mathrm e}^{-x} \left (-2 \,{\mathrm e}^{x} \ln \relax (5)+2 \ln \relax (x )\right ) \ln \left ({\mathrm e}^{x}\right )-\frac {\left (-12 \ln \relax (5)^{2} {\mathrm e}^{2 x}+24 \ln \relax (5) {\mathrm e}^{x} \ln \relax (x )+4 \,{\mathrm e} \ln \relax (x )-12 \ln \relax (x )^{2}\right ) {\mathrm e}^{-2 x}}{12}\right )\) | \(69\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.52, size = 67, normalized size = 2.68 \begin {gather*} 2 \, \log \left (x + \log \relax (5)\right ) + \log \left (-\frac {6 \, {\left (x + \log \relax (5)\right )} e^{x} \log \relax (x) - 3 \, {\left (x^{2} + 2 \, x \log \relax (5) + \log \relax (5)^{2}\right )} e^{\left (2 \, x\right )} + e \log \relax (x) - 3 \, \log \relax (x)^{2}}{3 \, {\left (x^{2} + 2 \, x \log \relax (5) + \log \relax (5)^{2}\right )}}\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 = -0.04 \begin {gather*} \int -\frac {-6\,x\,{\mathrm {e}}^{2\,x}\,{\ln \left (5\,{\mathrm {e}}^x\right )}^2+\left (6\,{\mathrm {e}}^x-6\,x\,{\mathrm {e}}^{2\,x}+6\,x\,{\mathrm {e}}^x\,\ln \relax (x)\right )\,\ln \left (5\,{\mathrm {e}}^x\right )+\mathrm {e}+\ln \relax (x)\,\left (6\,x\,{\mathrm {e}}^x-6\right )}{3\,x\,{\mathrm {e}}^{2\,x}\,{\ln \left (5\,{\mathrm {e}}^x\right )}^2-6\,x\,{\mathrm {e}}^x\,\ln \left (5\,{\mathrm {e}}^x\right )\,\ln \relax (x)+3\,x\,{\ln \relax (x)}^2-x\,\mathrm {e}\,\ln \relax (x)} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 3.67, size = 60, normalized size = 2.40 \begin {gather*} 2 \log {\left (x + \log {\relax (5 )} \right )} + \log {\left (\frac {3 \log {\relax (x )}^{2} - e \log {\relax (x )}}{3 x^{2} + 6 x \log {\relax (5 )} + 3 \log {\relax (5 )}^{2}} + e^{2 x} - \frac {2 e^{x} \log {\relax (x )}}{x + \log {\relax (5 )}} \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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