Optimal. Leaf size=29 \[ \log \left (x \left (256+\log \left (\frac {e^{2 x}-\log (x)}{1-e^{e^2}+x}\right )\right )\right ) \]
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Rubi [F]
time = 4.57, 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 {-1+e^{e^2} \left (1+e^{2 x} (-256-2 x)\right )-x+e^{2 x} \left (256+257 x+2 x^2\right )+\left (-256+256 e^{e^2}-255 x\right ) \log (x)+\left (-e^{e^2+2 x}+e^{2 x} (1+x)+\left (-1+e^{e^2}-x\right ) \log (x)\right ) \log \left (\frac {-e^{2 x}+\log (x)}{-1+e^{e^2}-x}\right )}{-256 e^{e^2+2 x} x+e^{2 x} \left (256 x+256 x^2\right )+\left (-256 x+256 e^{e^2} x-256 x^2\right ) \log (x)+\left (-e^{e^2+2 x} x+e^{2 x} \left (x+x^2\right )+\left (-x+e^{e^2} x-x^2\right ) \log (x)\right ) \log \left (\frac {-e^{2 x}+\log (x)}{-1+e^{e^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 {-1+e^{e^2} \left (1+e^{2 x} (-256-2 x)\right )-x+e^{2 x} \left (256+257 x+2 x^2\right )+\left (-256+256 e^{e^2}-255 x\right ) \log (x)+\left (-e^{e^2+2 x}+e^{2 x} (1+x)+\left (-1+e^{e^2}-x\right ) \log (x)\right ) \log \left (\frac {-e^{2 x}+\log (x)}{-1+e^{e^2}-x}\right )}{x \left (1-e^{e^2}+x\right ) \left (e^{2 x}-\log (x)\right ) \left (256+\log \left (\frac {e^{2 x}-\log (x)}{1-e^{e^2}+x}\right )\right )} \, dx\\ &=\int \left (\frac {-1+2 x \log (x)}{x \left (e^{2 x}-\log (x)\right ) \left (256+\log \left (\frac {e^{2 x}-\log (x)}{1-e^{e^2}+x}\right )\right )}+\frac {256 \left (1-e^{e^2}\right )+257 \left (1-\frac {2 e^{e^2}}{257}\right ) x+2 x^2+\left (1-e^{e^2}\right ) \log \left (\frac {e^{2 x}-\log (x)}{1-e^{e^2}+x}\right )+x \log \left (\frac {e^{2 x}-\log (x)}{1-e^{e^2}+x}\right )}{x \left (1-e^{e^2}+x\right ) \left (256+\log \left (\frac {e^{2 x}-\log (x)}{1-e^{e^2}+x}\right )\right )}\right ) \, dx\\ &=\int \frac {-1+2 x \log (x)}{x \left (e^{2 x}-\log (x)\right ) \left (256+\log \left (\frac {e^{2 x}-\log (x)}{1-e^{e^2}+x}\right )\right )} \, dx+\int \frac {256 \left (1-e^{e^2}\right )+257 \left (1-\frac {2 e^{e^2}}{257}\right ) x+2 x^2+\left (1-e^{e^2}\right ) \log \left (\frac {e^{2 x}-\log (x)}{1-e^{e^2}+x}\right )+x \log \left (\frac {e^{2 x}-\log (x)}{1-e^{e^2}+x}\right )}{x \left (1-e^{e^2}+x\right ) \left (256+\log \left (\frac {e^{2 x}-\log (x)}{1-e^{e^2}+x}\right )\right )} \, dx\\ &=\int \frac {256+257 x+2 x^2-2 e^{e^2} (128+x)+\left (1-e^{e^2}+x\right ) \log \left (\frac {e^{2 x}-\log (x)}{1-e^{e^2}+x}\right )}{x \left (1-e^{e^2}+x\right ) \left (256+\log \left (\frac {e^{2 x}-\log (x)}{1-e^{e^2}+x}\right )\right )} \, dx+\int \left (-\frac {1}{x \left (e^{2 x}-\log (x)\right ) \left (256+\log \left (\frac {e^{2 x}-\log (x)}{1-e^{e^2}+x}\right )\right )}+\frac {2 \log (x)}{\left (e^{2 x}-\log (x)\right ) \left (256+\log \left (\frac {e^{2 x}-\log (x)}{1-e^{e^2}+x}\right )\right )}\right ) \, dx\\ &=2 \int \frac {\log (x)}{\left (e^{2 x}-\log (x)\right ) \left (256+\log \left (\frac {e^{2 x}-\log (x)}{1-e^{e^2}+x}\right )\right )} \, dx-\int \frac {1}{x \left (e^{2 x}-\log (x)\right ) \left (256+\log \left (\frac {e^{2 x}-\log (x)}{1-e^{e^2}+x}\right )\right )} \, dx+\int \left (\frac {1}{x}+\frac {-1+2 e^{e^2}-2 x}{\left (-1+e^{e^2}-x\right ) \left (256+\log \left (\frac {e^{2 x}-\log (x)}{1-e^{e^2}+x}\right )\right )}\right ) \, dx\\ &=\log (x)+2 \int \frac {\log (x)}{\left (e^{2 x}-\log (x)\right ) \left (256+\log \left (\frac {e^{2 x}-\log (x)}{1-e^{e^2}+x}\right )\right )} \, dx+\int \frac {-1+2 e^{e^2}-2 x}{\left (-1+e^{e^2}-x\right ) \left (256+\log \left (\frac {e^{2 x}-\log (x)}{1-e^{e^2}+x}\right )\right )} \, dx-\int \frac {1}{x \left (e^{2 x}-\log (x)\right ) \left (256+\log \left (\frac {e^{2 x}-\log (x)}{1-e^{e^2}+x}\right )\right )} \, dx\\ &=\log (x)+2 \int \frac {\log (x)}{\left (e^{2 x}-\log (x)\right ) \left (256+\log \left (\frac {e^{2 x}-\log (x)}{1-e^{e^2}+x}\right )\right )} \, dx-\int \frac {1}{x \left (e^{2 x}-\log (x)\right ) \left (256+\log \left (\frac {e^{2 x}-\log (x)}{1-e^{e^2}+x}\right )\right )} \, dx+\int \left (\frac {2}{256+\log \left (\frac {e^{2 x}-\log (x)}{1-e^{e^2}+x}\right )}+\frac {1}{\left (-1+e^{e^2}-x\right ) \left (256+\log \left (\frac {e^{2 x}-\log (x)}{1-e^{e^2}+x}\right )\right )}\right ) \, dx\\ &=\log (x)+2 \int \frac {1}{256+\log \left (\frac {e^{2 x}-\log (x)}{1-e^{e^2}+x}\right )} \, dx+2 \int \frac {\log (x)}{\left (e^{2 x}-\log (x)\right ) \left (256+\log \left (\frac {e^{2 x}-\log (x)}{1-e^{e^2}+x}\right )\right )} \, dx+\int \frac {1}{\left (-1+e^{e^2}-x\right ) \left (256+\log \left (\frac {e^{2 x}-\log (x)}{1-e^{e^2}+x}\right )\right )} \, dx-\int \frac {1}{x \left (e^{2 x}-\log (x)\right ) \left (256+\log \left (\frac {e^{2 x}-\log (x)}{1-e^{e^2}+x}\right )\right )} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.18, size = 30, normalized size = 1.03 \begin {gather*} \log (x)+\log \left (256+\log \left (\frac {e^{2 x}-\log (x)}{1-e^{e^2}+x}\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
3.
time = 0.41, size = 226, normalized size = 7.79
method | result | size |
risch | \(\ln \left (x \right )+\ln \left (\ln \left ({\mathrm e}^{{\mathrm e}^{2}}-x -1\right )+\frac {i \left (\pi \,\mathrm {csgn}\left (i \left (\ln \left (x \right )-{\mathrm e}^{2 x}\right )\right ) \mathrm {csgn}\left (\frac {i}{1-{\mathrm e}^{{\mathrm e}^{2}}+x}\right ) \mathrm {csgn}\left (\frac {i \left (\ln \left (x \right )-{\mathrm e}^{2 x}\right )}{1-{\mathrm e}^{{\mathrm e}^{2}}+x}\right )+\pi \,\mathrm {csgn}\left (i \left (\ln \left (x \right )-{\mathrm e}^{2 x}\right )\right ) \mathrm {csgn}\left (\frac {i \left (\ln \left (x \right )-{\mathrm e}^{2 x}\right )}{1-{\mathrm e}^{{\mathrm e}^{2}}+x}\right )^{2}+\pi \,\mathrm {csgn}\left (\frac {i}{1-{\mathrm e}^{{\mathrm e}^{2}}+x}\right ) \mathrm {csgn}\left (\frac {i \left (\ln \left (x \right )-{\mathrm e}^{2 x}\right )}{1-{\mathrm e}^{{\mathrm e}^{2}}+x}\right )^{2}-\pi \mathrm {csgn}\left (\frac {i \left (\ln \left (x \right )-{\mathrm e}^{2 x}\right )}{1-{\mathrm e}^{{\mathrm e}^{2}}+x}\right )^{3}+2 \pi \mathrm {csgn}\left (\frac {i \left (\ln \left (x \right )-{\mathrm e}^{2 x}\right )}{1-{\mathrm e}^{{\mathrm e}^{2}}+x}\right )^{2}-2 \pi +2 i \ln \left ({\mathrm e}^{2 x}-\ln \left (x \right )\right )+512 i\right )}{2}\right )\) | \(226\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.40, size = 27, normalized size = 0.93 \begin {gather*} \log \left (x\right ) + \log \left (-\log \left (-x + e^{\left (e^{2}\right )} - 1\right ) + \log \left (-e^{\left (2 \, x\right )} + \log \left (x\right )\right ) + 256\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.39, size = 42, normalized size = 1.45 \begin {gather*} \log \left (x\right ) + \log \left (\log \left (-\frac {e^{\left (e^{2}\right )} \log \left (x\right ) - e^{\left (2 \, x + e^{2}\right )}}{{\left (x + 1\right )} e^{\left (e^{2}\right )} - e^{\left (2 \, e^{2}\right )}}\right ) + 256\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 2.35, size = 24, normalized size = 0.83 \begin {gather*} \log {\left (x \right )} + \log {\left (\log {\left (\frac {- e^{2 x} + \log {\left (x \right )}}{- x - 1 + e^{e^{2}}} \right )} + 256 \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.76, size = 27, normalized size = 0.93 \begin {gather*} \log \left (x\right ) + \log \left (\log \left (x - e^{\left (e^{2}\right )} + 1\right ) - \log \left (e^{\left (2 \, x\right )} - \log \left (x\right )\right ) - 256\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 4.70, size = 27, normalized size = 0.93 \begin {gather*} \ln \left (\ln \left (\frac {{\mathrm {e}}^{2\,x}-\ln \left (x\right )}{x-{\mathrm {e}}^{{\mathrm {e}}^2}+1}\right )+256\right )+\ln \left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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