Optimal. Leaf size=23 \[ x+\log \left (-\frac {1}{25} e^{-x} x^4+\log (x (x+\log (3)))\right ) \]
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Rubi [F] time = 11.45, 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 {-4 x^5-4 x^4 \log (3)+e^x (50 x+25 \log (3))+e^x \left (25 x^2+25 x \log (3)\right ) \log \left (x^2+x \log (3)\right )}{-x^6-x^5 \log (3)+e^x \left (25 x^2+25 x \log (3)\right ) \log \left (x^2+x \log (3)\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 {4 x^5+4 x^4 \log (3)-e^x (50 x+25 \log (3))-e^x \left (25 x^2+25 x \log (3)\right ) \log \left (x^2+x \log (3)\right )}{x (x+\log (3)) \left (x^4-25 e^x \log (x (x+\log (3)))\right )} \, dx\\ &=\int \left (\frac {x^3 \left (-2 x-\log (3)-x^2 \log (x (x+\log (3)))+4 x \left (1-\frac {\log (3)}{4}\right ) \log (x (x+\log (3)))+4 \log (3) \log (x (x+\log (3)))\right )}{(x+\log (3)) \log (x (x+\log (3))) \left (x^4-25 e^x \log (x (x+\log (3)))\right )}+\frac {2 x+\log (3)+x^2 \log (x (x+\log (3)))+x \log (3) \log (x (x+\log (3)))}{x (x+\log (3)) \log (x (x+\log (3)))}\right ) \, dx\\ &=\int \frac {x^3 \left (-2 x-\log (3)-x^2 \log (x (x+\log (3)))+4 x \left (1-\frac {\log (3)}{4}\right ) \log (x (x+\log (3)))+4 \log (3) \log (x (x+\log (3)))\right )}{(x+\log (3)) \log (x (x+\log (3))) \left (x^4-25 e^x \log (x (x+\log (3)))\right )} \, dx+\int \frac {2 x+\log (3)+x^2 \log (x (x+\log (3)))+x \log (3) \log (x (x+\log (3)))}{x (x+\log (3)) \log (x (x+\log (3)))} \, dx\\ &=\int \frac {x^3 (-2 x-\log (3)-(-4+x) (x+\log (3)) \log (x (x+\log (3))))}{(x+\log (3)) \log (x (x+\log (3))) \left (x^4-25 e^x \log (x (x+\log (3)))\right )} \, dx+\int \frac {2 x+\log (3)+x (x+\log (3)) \log (x (x+\log (3)))}{x (x+\log (3)) \log (x (x+\log (3)))} \, dx\\ &=\int \left (1+\frac {2 x+\log (3)}{x (x+\log (3)) \log (x (x+\log (3)))}\right ) \, dx+\int \left (\frac {x \log (3) \left (2 x+\log (3)+x^2 \log (x (x+\log (3)))-4 x \left (1-\frac {\log (3)}{4}\right ) \log (x (x+\log (3)))-4 \log (3) \log (x (x+\log (3)))\right )}{\log (x (x+\log (3))) \left (x^4-25 e^x \log (x (x+\log (3)))\right )}+\frac {\log ^3(3) \left (2 x+\log (3)+x^2 \log (x (x+\log (3)))-4 x \left (1-\frac {\log (3)}{4}\right ) \log (x (x+\log (3)))-4 \log (3) \log (x (x+\log (3)))\right )}{(x+\log (3)) \log (x (x+\log (3))) \left (x^4-25 e^x \log (x (x+\log (3)))\right )}+\frac {x^2 \left (-2 x-\log (3)-x^2 \log (x (x+\log (3)))+4 x \left (1-\frac {\log (3)}{4}\right ) \log (x (x+\log (3)))+4 \log (3) \log (x (x+\log (3)))\right )}{\log (x (x+\log (3))) \left (x^4-25 e^x \log (x (x+\log (3)))\right )}+\frac {\log ^2(3) \left (-2 x-\log (3)-x^2 \log (x (x+\log (3)))+4 x \left (1-\frac {\log (3)}{4}\right ) \log (x (x+\log (3)))+4 \log (3) \log (x (x+\log (3)))\right )}{\log (x (x+\log (3))) \left (x^4-25 e^x \log (x (x+\log (3)))\right )}\right ) \, dx\\ &=x+\log (3) \int \frac {x \left (2 x+\log (3)+x^2 \log (x (x+\log (3)))-4 x \left (1-\frac {\log (3)}{4}\right ) \log (x (x+\log (3)))-4 \log (3) \log (x (x+\log (3)))\right )}{\log (x (x+\log (3))) \left (x^4-25 e^x \log (x (x+\log (3)))\right )} \, dx+\log ^2(3) \int \frac {-2 x-\log (3)-x^2 \log (x (x+\log (3)))+4 x \left (1-\frac {\log (3)}{4}\right ) \log (x (x+\log (3)))+4 \log (3) \log (x (x+\log (3)))}{\log (x (x+\log (3))) \left (x^4-25 e^x \log (x (x+\log (3)))\right )} \, dx+\log ^3(3) \int \frac {2 x+\log (3)+x^2 \log (x (x+\log (3)))-4 x \left (1-\frac {\log (3)}{4}\right ) \log (x (x+\log (3)))-4 \log (3) \log (x (x+\log (3)))}{(x+\log (3)) \log (x (x+\log (3))) \left (x^4-25 e^x \log (x (x+\log (3)))\right )} \, dx+\int \frac {2 x+\log (3)}{x (x+\log (3)) \log (x (x+\log (3)))} \, dx+\int \frac {x^2 \left (-2 x-\log (3)-x^2 \log (x (x+\log (3)))+4 x \left (1-\frac {\log (3)}{4}\right ) \log (x (x+\log (3)))+4 \log (3) \log (x (x+\log (3)))\right )}{\log (x (x+\log (3))) \left (x^4-25 e^x \log (x (x+\log (3)))\right )} \, dx\\ &=x+\log (3) \int \frac {x (2 x+\log (3)+(-4+x) (x+\log (3)) \log (x (x+\log (3))))}{\log (x (x+\log (3))) \left (x^4-25 e^x \log (x (x+\log (3)))\right )} \, dx+\log ^2(3) \int \frac {-2 x-\log (3)-(-4+x) (x+\log (3)) \log (x (x+\log (3)))}{\log (x (x+\log (3))) \left (x^4-25 e^x \log (x (x+\log (3)))\right )} \, dx+\log ^3(3) \int \frac {2 x+\log (3)+(-4+x) (x+\log (3)) \log (x (x+\log (3)))}{(x+\log (3)) \log (x (x+\log (3))) \left (x^4-25 e^x \log (x (x+\log (3)))\right )} \, dx+\int \frac {2 x+\log (3)}{x (x+\log (3)) \log (x (x+\log (3)))} \, dx+\int \frac {x^2 (-2 x-\log (3)-(-4+x) (x+\log (3)) \log (x (x+\log (3))))}{\log (x (x+\log (3))) \left (x^4-25 e^x \log (x (x+\log (3)))\right )} \, dx\\ &=x+\log (3) \int \left (\frac {x^3}{x^4-25 e^x \log (x (x+\log (3)))}-\frac {4 x^2 \left (1-\frac {\log (3)}{4}\right )}{x^4-25 e^x \log (x (x+\log (3)))}-\frac {4 x \log (3)}{x^4-25 e^x \log (x (x+\log (3)))}+\frac {2 x^2}{\log (x (x+\log (3))) \left (x^4-25 e^x \log (x (x+\log (3)))\right )}+\frac {x \log (3)}{\log (x (x+\log (3))) \left (x^4-25 e^x \log (x (x+\log (3)))\right )}\right ) \, dx+\log ^2(3) \int \left (-\frac {x^2}{x^4-25 e^x \log (x (x+\log (3)))}+\frac {4 x \left (1-\frac {\log (3)}{4}\right )}{x^4-25 e^x \log (x (x+\log (3)))}+\frac {\log (81)}{x^4-25 e^x \log (x (x+\log (3)))}-\frac {2 x}{\log (x (x+\log (3))) \left (x^4-25 e^x \log (x (x+\log (3)))\right )}+\frac {\log (3)}{\log (x (x+\log (3))) \left (-x^4+25 e^x \log (x (x+\log (3)))\right )}\right ) \, dx+\log ^3(3) \int \left (\frac {x^2}{(x+\log (3)) \left (x^4-25 e^x \log (x (x+\log (3)))\right )}-\frac {4 x \left (1-\frac {\log (3)}{4}\right )}{(x+\log (3)) \left (x^4-25 e^x \log (x (x+\log (3)))\right )}-\frac {4 \log (3)}{(x+\log (3)) \left (x^4-25 e^x \log (x (x+\log (3)))\right )}+\frac {2 x}{(x+\log (3)) \log (x (x+\log (3))) \left (x^4-25 e^x \log (x (x+\log (3)))\right )}+\frac {\log (3)}{(x+\log (3)) \log (x (x+\log (3))) \left (x^4-25 e^x \log (x (x+\log (3)))\right )}\right ) \, dx+\int \frac {2 x+\log (3)}{x (x+\log (3)) \log (x (x+\log (3)))} \, dx+\int \left (-\frac {x^4}{x^4-25 e^x \log (x (x+\log (3)))}+\frac {4 x^3 \left (1-\frac {\log (3)}{4}\right )}{x^4-25 e^x \log (x (x+\log (3)))}+\frac {x^2 \log (81)}{x^4-25 e^x \log (x (x+\log (3)))}-\frac {2 x^3}{\log (x (x+\log (3))) \left (x^4-25 e^x \log (x (x+\log (3)))\right )}-\frac {x^2 \log (3)}{\log (x (x+\log (3))) \left (x^4-25 e^x \log (x (x+\log (3)))\right )}\right ) \, dx\\ &=x-2 \int \frac {x^3}{\log (x (x+\log (3))) \left (x^4-25 e^x \log (x (x+\log (3)))\right )} \, dx+(4-\log (3)) \int \frac {x^3}{x^4-25 e^x \log (x (x+\log (3)))} \, dx+\log (3) \int \frac {x^3}{x^4-25 e^x \log (x (x+\log (3)))} \, dx-\log (3) \int \frac {x^2}{\log (x (x+\log (3))) \left (x^4-25 e^x \log (x (x+\log (3)))\right )} \, dx+(2 \log (3)) \int \frac {x^2}{\log (x (x+\log (3))) \left (x^4-25 e^x \log (x (x+\log (3)))\right )} \, dx-((4-\log (3)) \log (3)) \int \frac {x^2}{x^4-25 e^x \log (x (x+\log (3)))} \, dx-\log ^2(3) \int \frac {x^2}{x^4-25 e^x \log (x (x+\log (3)))} \, dx+\log ^2(3) \int \frac {x}{\log (x (x+\log (3))) \left (x^4-25 e^x \log (x (x+\log (3)))\right )} \, dx-\left (2 \log ^2(3)\right ) \int \frac {x}{\log (x (x+\log (3))) \left (x^4-25 e^x \log (x (x+\log (3)))\right )} \, dx-\left (4 \log ^2(3)\right ) \int \frac {x}{x^4-25 e^x \log (x (x+\log (3)))} \, dx+\left ((4-\log (3)) \log ^2(3)\right ) \int \frac {x}{x^4-25 e^x \log (x (x+\log (3)))} \, dx+\log ^3(3) \int \frac {x^2}{(x+\log (3)) \left (x^4-25 e^x \log (x (x+\log (3)))\right )} \, dx+\log ^3(3) \int \frac {1}{\log (x (x+\log (3))) \left (-x^4+25 e^x \log (x (x+\log (3)))\right )} \, dx+\left (2 \log ^3(3)\right ) \int \frac {x}{(x+\log (3)) \log (x (x+\log (3))) \left (x^4-25 e^x \log (x (x+\log (3)))\right )} \, dx-\left ((4-\log (3)) \log ^3(3)\right ) \int \frac {x}{(x+\log (3)) \left (x^4-25 e^x \log (x (x+\log (3)))\right )} \, dx+\log ^4(3) \int \frac {1}{(x+\log (3)) \log (x (x+\log (3))) \left (x^4-25 e^x \log (x (x+\log (3)))\right )} \, dx-\left (4 \log ^4(3)\right ) \int \frac {1}{(x+\log (3)) \left (x^4-25 e^x \log (x (x+\log (3)))\right )} \, dx+\log (81) \int \frac {x^2}{x^4-25 e^x \log (x (x+\log (3)))} \, dx+\left (\log ^2(3) \log (81)\right ) \int \frac {1}{x^4-25 e^x \log (x (x+\log (3)))} \, dx+\int \frac {2 x+\log (3)}{x (x+\log (3)) \log (x (x+\log (3)))} \, dx-\int \frac {x^4}{x^4-25 e^x \log (x (x+\log (3)))} \, dx\\ &=\text {Rest of rules removed due to large latex content} \end {aligned} \end {gather*}
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Mathematica [F] time = 2.32, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {-4 x^5-4 x^4 \log (3)+e^x (50 x+25 \log (3))+e^x \left (25 x^2+25 x \log (3)\right ) \log \left (x^2+x \log (3)\right )}{-x^6-x^5 \log (3)+e^x \left (25 x^2+25 x \log (3)\right ) \log \left (x^2+x \log (3)\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.55, size = 26, normalized size = 1.13 \begin {gather*} x + \log \left (-{\left (x^{4} - 25 \, e^{x} \log \left (x^{2} + x \log \relax (3)\right )\right )} e^{\left (-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 = 20, normalized size = 0.87 \begin {gather*} \log \left (-x^{4} + 25 \, e^{x} \log \left (x^{2} + x \log \relax (3)\right )\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.08, size = 119, normalized size = 5.17
method | result | size |
risch | \(x +\ln \left (\ln \left (\ln \relax (3)+x \right )+\frac {i \left (2 i x^{4}-25 \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i \left (\ln \relax (3)+x \right )\right ) \mathrm {csgn}\left (i x \left (\ln \relax (3)+x \right )\right ) {\mathrm e}^{x}+25 \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x \left (\ln \relax (3)+x \right )\right )^{2} {\mathrm e}^{x}+25 \pi \,\mathrm {csgn}\left (i \left (\ln \relax (3)+x \right )\right ) \mathrm {csgn}\left (i x \left (\ln \relax (3)+x \right )\right )^{2} {\mathrm e}^{x}-25 \pi \mathrm {csgn}\left (i x \left (\ln \relax (3)+x \right )\right )^{3} {\mathrm e}^{x}-50 i {\mathrm e}^{x} \ln \relax (x )\right ) {\mathrm e}^{-x}}{50}\right )\) | \(119\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.50, size = 28, normalized size = 1.22 \begin {gather*} x + \log \left (-\frac {1}{25} \, {\left (x^{4} - 25 \, e^{x} \log \left (x + \log \relax (3)\right ) - 25 \, e^{x} \log \relax (x)\right )} e^{\left (-x\right )}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.82, size = 22, normalized size = 0.96 \begin {gather*} x+\ln \left (\ln \left (x^2+\ln \relax (3)\,x\right )-\frac {x^4\,{\mathrm {e}}^{-x}}{25}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.85, size = 31, normalized size = 1.35 \begin {gather*} \log {\left (- \frac {x^{4}}{25 \log {\left (x^{2} + x \log {\relax (3 )} \right )}} + e^{x} \right )} + \log {\left (\log {\left (x^{2} + x \log {\relax (3 )} \right )} \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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