Optimal. Leaf size=33 \[ \frac {\log \left (6-\log (x)+\frac {e^{-2 x} \left (1-\frac {x+\log (x)}{x}\right )^2}{x^2}\right )}{x} \]
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Rubi [F] time = 8.93, 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^{2 x} x^4+2 \log (x)+(-4-2 x) \log ^2(x)+\left (-6 e^{2 x} x^4+e^{2 x} x^4 \log (x)-\log ^2(x)\right ) \log \left (\frac {e^{-2 x} \left (6 e^{2 x} x^4-e^{2 x} x^4 \log (x)+\log ^2(x)\right )}{x^4}\right )}{6 e^{2 x} x^6-e^{2 x} x^6 \log (x)+x^2 \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 {\log (x) \left (12-25 \log (x)-12 x \log (x)+4 \log ^2(x)+2 x \log ^2(x)\right )}{x^2 (-6+\log (x)) \left (-6 e^{2 x} x^4+e^{2 x} x^4 \log (x)-\log ^2(x)\right )}+\frac {1+6 \log \left (6-\log (x)+\frac {e^{-2 x} \log ^2(x)}{x^4}\right )-\log (x) \log \left (6-\log (x)+\frac {e^{-2 x} \log ^2(x)}{x^4}\right )}{x^2 (-6+\log (x))}\right ) \, dx\\ &=\int \frac {\log (x) \left (12-25 \log (x)-12 x \log (x)+4 \log ^2(x)+2 x \log ^2(x)\right )}{x^2 (-6+\log (x)) \left (-6 e^{2 x} x^4+e^{2 x} x^4 \log (x)-\log ^2(x)\right )} \, dx+\int \frac {1+6 \log \left (6-\log (x)+\frac {e^{-2 x} \log ^2(x)}{x^4}\right )-\log (x) \log \left (6-\log (x)+\frac {e^{-2 x} \log ^2(x)}{x^4}\right )}{x^2 (-6+\log (x))} \, dx\\ &=\int \left (\frac {12 \log (x)}{x^2 (-6+\log (x)) \left (-6 e^{2 x} x^4+e^{2 x} x^4 \log (x)-\log ^2(x)\right )}-\frac {25 \log ^2(x)}{x^2 (-6+\log (x)) \left (-6 e^{2 x} x^4+e^{2 x} x^4 \log (x)-\log ^2(x)\right )}-\frac {12 \log ^2(x)}{x (-6+\log (x)) \left (-6 e^{2 x} x^4+e^{2 x} x^4 \log (x)-\log ^2(x)\right )}+\frac {4 \log ^3(x)}{x^2 (-6+\log (x)) \left (-6 e^{2 x} x^4+e^{2 x} x^4 \log (x)-\log ^2(x)\right )}+\frac {2 \log ^3(x)}{x (-6+\log (x)) \left (-6 e^{2 x} x^4+e^{2 x} x^4 \log (x)-\log ^2(x)\right )}\right ) \, dx+\int \frac {\frac {1}{-6+\log (x)}-\log \left (6-\log (x)+\frac {e^{-2 x} \log ^2(x)}{x^4}\right )}{x^2} \, dx\\ &=2 \int \frac {\log ^3(x)}{x (-6+\log (x)) \left (-6 e^{2 x} x^4+e^{2 x} x^4 \log (x)-\log ^2(x)\right )} \, dx+4 \int \frac {\log ^3(x)}{x^2 (-6+\log (x)) \left (-6 e^{2 x} x^4+e^{2 x} x^4 \log (x)-\log ^2(x)\right )} \, dx+12 \int \frac {\log (x)}{x^2 (-6+\log (x)) \left (-6 e^{2 x} x^4+e^{2 x} x^4 \log (x)-\log ^2(x)\right )} \, dx-12 \int \frac {\log ^2(x)}{x (-6+\log (x)) \left (-6 e^{2 x} x^4+e^{2 x} x^4 \log (x)-\log ^2(x)\right )} \, dx-25 \int \frac {\log ^2(x)}{x^2 (-6+\log (x)) \left (-6 e^{2 x} x^4+e^{2 x} x^4 \log (x)-\log ^2(x)\right )} \, dx+\int \left (\frac {1}{x^2 (-6+\log (x))}-\frac {\log \left (6-\log (x)+\frac {e^{-2 x} \log ^2(x)}{x^4}\right )}{x^2}\right ) \, dx\\ &=2 \int \frac {\log ^3(x)}{x (-6+\log (x)) \left (-6 e^{2 x} x^4+e^{2 x} x^4 \log (x)-\log ^2(x)\right )} \, dx+4 \int \frac {\log ^3(x)}{x^2 (-6+\log (x)) \left (-6 e^{2 x} x^4+e^{2 x} x^4 \log (x)-\log ^2(x)\right )} \, dx+12 \int \frac {\log (x)}{x^2 (-6+\log (x)) \left (-6 e^{2 x} x^4+e^{2 x} x^4 \log (x)-\log ^2(x)\right )} \, dx-12 \int \frac {\log ^2(x)}{x (-6+\log (x)) \left (-6 e^{2 x} x^4+e^{2 x} x^4 \log (x)-\log ^2(x)\right )} \, dx-25 \int \frac {\log ^2(x)}{x^2 (-6+\log (x)) \left (-6 e^{2 x} x^4+e^{2 x} x^4 \log (x)-\log ^2(x)\right )} \, dx+\int \frac {1}{x^2 (-6+\log (x))} \, dx-\int \frac {\log \left (6-\log (x)+\frac {e^{-2 x} \log ^2(x)}{x^4}\right )}{x^2} \, dx\\ &=2 \int \frac {\log ^3(x)}{x (-6+\log (x)) \left (-6 e^{2 x} x^4+e^{2 x} x^4 \log (x)-\log ^2(x)\right )} \, dx+4 \int \frac {\log ^3(x)}{x^2 (-6+\log (x)) \left (-6 e^{2 x} x^4+e^{2 x} x^4 \log (x)-\log ^2(x)\right )} \, dx+12 \int \frac {\log (x)}{x^2 (-6+\log (x)) \left (-6 e^{2 x} x^4+e^{2 x} x^4 \log (x)-\log ^2(x)\right )} \, dx-12 \int \frac {\log ^2(x)}{x (-6+\log (x)) \left (-6 e^{2 x} x^4+e^{2 x} x^4 \log (x)-\log ^2(x)\right )} \, dx-25 \int \frac {\log ^2(x)}{x^2 (-6+\log (x)) \left (-6 e^{2 x} x^4+e^{2 x} x^4 \log (x)-\log ^2(x)\right )} \, dx-\int \frac {\log \left (6-\log (x)+\frac {e^{-2 x} \log ^2(x)}{x^4}\right )}{x^2} \, dx+\operatorname {Subst}\left (\int \frac {e^{-x}}{-6+x} \, dx,x,\log (x)\right )\\ &=\frac {\text {Ei}(6-\log (x))}{e^6}+2 \int \frac {\log ^3(x)}{x (-6+\log (x)) \left (-6 e^{2 x} x^4+e^{2 x} x^4 \log (x)-\log ^2(x)\right )} \, dx+4 \int \frac {\log ^3(x)}{x^2 (-6+\log (x)) \left (-6 e^{2 x} x^4+e^{2 x} x^4 \log (x)-\log ^2(x)\right )} \, dx+12 \int \frac {\log (x)}{x^2 (-6+\log (x)) \left (-6 e^{2 x} x^4+e^{2 x} x^4 \log (x)-\log ^2(x)\right )} \, dx-12 \int \frac {\log ^2(x)}{x (-6+\log (x)) \left (-6 e^{2 x} x^4+e^{2 x} x^4 \log (x)-\log ^2(x)\right )} \, dx-25 \int \frac {\log ^2(x)}{x^2 (-6+\log (x)) \left (-6 e^{2 x} x^4+e^{2 x} x^4 \log (x)-\log ^2(x)\right )} \, dx-\int \frac {\log \left (6-\log (x)+\frac {e^{-2 x} \log ^2(x)}{x^4}\right )}{x^2} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.18, size = 26, normalized size = 0.79 \begin {gather*} 2+\frac {\log \left (6-\log (x)+\frac {e^{-2 x} \log ^2(x)}{x^4}\right )}{x} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.75, size = 40, normalized size = 1.21 \begin {gather*} \frac {\log \left (-\frac {{\left (x^{4} e^{\left (2 \, x\right )} \log \relax (x) - 6 \, x^{4} e^{\left (2 \, x\right )} - \log \relax (x)^{2}\right )} e^{\left (-2 \, x\right )}}{x^{4}}\right )}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.37, size = 42, normalized size = 1.27 \begin {gather*} \frac {\log \left (-{\left (x^{4} e^{\left (2 \, x\right )} \log \relax (x) - 6 \, x^{4} e^{\left (2 \, x\right )} - \log \relax (x)^{2}\right )} e^{\left (-2 \, x\right )}\right ) - 4 \, \log \relax (x)}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.71, size = 722, normalized size = 21.88
method | result | size |
risch | \(-\frac {2 \ln \left ({\mathrm e}^{x}\right )}{x}-\frac {8 \ln \relax (x )+2 i \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{2}-i \pi \mathrm {csgn}\left (\frac {i \left (-{\mathrm e}^{2 x} \left (\ln \relax (x )-6\right ) x^{4}+\ln \relax (x )^{2}\right ) {\mathrm e}^{-2 x}}{x^{4}}\right )^{2} \mathrm {csgn}\left (\frac {i}{x^{4}}\right )+i \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{4}\right )^{2}+i \pi \,\mathrm {csgn}\left (i x^{3}\right ) \mathrm {csgn}\left (i x^{4}\right )^{2}+i \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{3}\right )^{2}+i \pi \,\mathrm {csgn}\left (i x^{2}\right ) \mathrm {csgn}\left (i x^{3}\right )^{2}+i \pi \,\mathrm {csgn}\left (i {\mathrm e}^{-2 x} \left (-{\mathrm e}^{2 x} \left (\ln \relax (x )-6\right ) x^{4}+\ln \relax (x )^{2}\right )\right ) \mathrm {csgn}\left (\frac {i \left (-{\mathrm e}^{2 x} \left (\ln \relax (x )-6\right ) x^{4}+\ln \relax (x )^{2}\right ) {\mathrm e}^{-2 x}}{x^{4}}\right )^{2}+i \pi \,\mathrm {csgn}\left (i \left (-{\mathrm e}^{2 x} \left (\ln \relax (x )-6\right ) x^{4}+\ln \relax (x )^{2}\right )\right ) \mathrm {csgn}\left (i {\mathrm e}^{-2 x} \left (-{\mathrm e}^{2 x} \left (\ln \relax (x )-6\right ) x^{4}+\ln \relax (x )^{2}\right )\right )^{2}-i \pi \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right )+2 i \pi \,\mathrm {csgn}\left (i {\mathrm e}^{x}\right ) \mathrm {csgn}\left (i {\mathrm e}^{2 x}\right )^{2}-i \pi \mathrm {csgn}\left (i {\mathrm e}^{x}\right )^{2} \mathrm {csgn}\left (i {\mathrm e}^{2 x}\right )-i \pi \,\mathrm {csgn}\left (i {\mathrm e}^{-2 x}\right ) \mathrm {csgn}\left (i {\mathrm e}^{-2 x} \left (-{\mathrm e}^{2 x} \left (\ln \relax (x )-6\right ) x^{4}+\ln \relax (x )^{2}\right )\right )^{2}-2 i \pi +i \pi \,\mathrm {csgn}\left (i {\mathrm e}^{-2 x}\right ) \mathrm {csgn}\left (i \left (-{\mathrm e}^{2 x} \left (\ln \relax (x )-6\right ) x^{4}+\ln \relax (x )^{2}\right )\right ) \mathrm {csgn}\left (i {\mathrm e}^{-2 x} \left (-{\mathrm e}^{2 x} \left (\ln \relax (x )-6\right ) x^{4}+\ln \relax (x )^{2}\right )\right )-2 \ln \left ({\mathrm e}^{2 x} \left (\ln \relax (x )-6\right ) x^{4}-\ln \relax (x )^{2}\right )+i \pi \mathrm {csgn}\left (\frac {i \left (-{\mathrm e}^{2 x} \left (\ln \relax (x )-6\right ) x^{4}+\ln \relax (x )^{2}\right ) {\mathrm e}^{-2 x}}{x^{4}}\right )^{3}-i \pi \mathrm {csgn}\left (i {\mathrm e}^{-2 x} \left (-{\mathrm e}^{2 x} \left (\ln \relax (x )-6\right ) x^{4}+\ln \relax (x )^{2}\right )\right )^{3}-i \pi \mathrm {csgn}\left (i x^{4}\right )^{3}+2 i \pi \mathrm {csgn}\left (\frac {i \left (-{\mathrm e}^{2 x} \left (\ln \relax (x )-6\right ) x^{4}+\ln \relax (x )^{2}\right ) {\mathrm e}^{-2 x}}{x^{4}}\right )^{2}-i \pi \mathrm {csgn}\left (i x^{3}\right )^{3}-i \pi \mathrm {csgn}\left (i {\mathrm e}^{2 x}\right )^{3}-i \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right ) \mathrm {csgn}\left (i x^{3}\right )-i \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{3}\right ) \mathrm {csgn}\left (i x^{4}\right )+i \pi \,\mathrm {csgn}\left (i {\mathrm e}^{-2 x} \left (-{\mathrm e}^{2 x} \left (\ln \relax (x )-6\right ) x^{4}+\ln \relax (x )^{2}\right )\right ) \mathrm {csgn}\left (\frac {i \left (-{\mathrm e}^{2 x} \left (\ln \relax (x )-6\right ) x^{4}+\ln \relax (x )^{2}\right ) {\mathrm e}^{-2 x}}{x^{4}}\right ) \mathrm {csgn}\left (\frac {i}{x^{4}}\right )-i \pi \mathrm {csgn}\left (i x^{2}\right )^{3}}{2 x}\) | \(722\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.41, size = 35, normalized size = 1.06 \begin {gather*} \frac {\log \left (-x^{4} e^{\left (2 \, x\right )} \log \relax (x) + 6 \, x^{4} e^{\left (2 \, x\right )} + \log \relax (x)^{2}\right ) - 4 \, \log \relax (x)}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 7.49, size = 32, normalized size = 0.97 \begin {gather*} \frac {\ln \left (\frac {1}{x^4}\right )+\ln \left (6\,x^4-x^4\,\ln \relax (x)+{\mathrm {e}}^{-2\,x}\,{\ln \relax (x)}^2\right )}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.16, size = 37, normalized size = 1.12 \begin {gather*} \frac {\log {\left (\frac {\left (- x^{4} e^{2 x} \log {\relax (x )} + 6 x^{4} e^{2 x} + \log {\relax (x )}^{2}\right ) e^{- 2 x}}{x^{4}} \right )}}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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