Optimal. Leaf size=21 \[ x+\log ^2\left (4 \left (-4+\frac {2}{\log \left (e^x-x\right )}\right )\right ) \]
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Rubi [F]
time = 3.53, 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 {\left (-e^x+x\right ) \log \left (e^x-x\right )+\left (2 e^x-2 x\right ) \log ^2\left (e^x-x\right )+\left (-2+2 e^x\right ) \log \left (\frac {8-16 \log \left (e^x-x\right )}{\log \left (e^x-x\right )}\right )}{\left (-e^x+x\right ) \log \left (e^x-x\right )+\left (2 e^x-2 x\right ) \log ^2\left (e^x-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 {-\left (\left (-e^x+x\right ) \log \left (e^x-x\right )\right )-\left (2 e^x-2 x\right ) \log ^2\left (e^x-x\right )-\left (-2+2 e^x\right ) \log \left (\frac {8-16 \log \left (e^x-x\right )}{\log \left (e^x-x\right )}\right )}{\left (e^x-x\right ) \left (1-2 \log \left (e^x-x\right )\right ) \log \left (e^x-x\right )} \, dx\\ &=\int \left (\frac {2 (-1+x) \log \left (8 \left (-2+\frac {1}{\log \left (e^x-x\right )}\right )\right )}{\left (e^x-x\right ) \log \left (e^x-x\right ) \left (-1+2 \log \left (e^x-x\right )\right )}+\frac {-\log \left (e^x-x\right )+2 \log ^2\left (e^x-x\right )+2 \log \left (8 \left (-2+\frac {1}{\log \left (e^x-x\right )}\right )\right )}{\log \left (e^x-x\right ) \left (-1+2 \log \left (e^x-x\right )\right )}\right ) \, dx\\ &=2 \int \frac {(-1+x) \log \left (8 \left (-2+\frac {1}{\log \left (e^x-x\right )}\right )\right )}{\left (e^x-x\right ) \log \left (e^x-x\right ) \left (-1+2 \log \left (e^x-x\right )\right )} \, dx+\int \frac {-\log \left (e^x-x\right )+2 \log ^2\left (e^x-x\right )+2 \log \left (8 \left (-2+\frac {1}{\log \left (e^x-x\right )}\right )\right )}{\log \left (e^x-x\right ) \left (-1+2 \log \left (e^x-x\right )\right )} \, dx\\ &=2 \int \left (-\frac {\log \left (8 \left (-2+\frac {1}{\log \left (e^x-x\right )}\right )\right )}{\left (e^x-x\right ) \log \left (e^x-x\right ) \left (-1+2 \log \left (e^x-x\right )\right )}+\frac {x \log \left (8 \left (-2+\frac {1}{\log \left (e^x-x\right )}\right )\right )}{\left (e^x-x\right ) \log \left (e^x-x\right ) \left (-1+2 \log \left (e^x-x\right )\right )}\right ) \, dx+\int \left (1+\frac {2 \log \left (8 \left (-2+\frac {1}{\log \left (e^x-x\right )}\right )\right )}{\log \left (e^x-x\right ) \left (-1+2 \log \left (e^x-x\right )\right )}\right ) \, dx\\ &=x+2 \int \frac {\log \left (8 \left (-2+\frac {1}{\log \left (e^x-x\right )}\right )\right )}{\log \left (e^x-x\right ) \left (-1+2 \log \left (e^x-x\right )\right )} \, dx-2 \int \frac {\log \left (8 \left (-2+\frac {1}{\log \left (e^x-x\right )}\right )\right )}{\left (e^x-x\right ) \log \left (e^x-x\right ) \left (-1+2 \log \left (e^x-x\right )\right )} \, dx+2 \int \frac {x \log \left (8 \left (-2+\frac {1}{\log \left (e^x-x\right )}\right )\right )}{\left (e^x-x\right ) \log \left (e^x-x\right ) \left (-1+2 \log \left (e^x-x\right )\right )} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.05, size = 19, normalized size = 0.90 \begin {gather*} x+\log ^2\left (8 \left (-2+\frac {1}{\log \left (e^x-x\right )}\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 = 1.14, size = 567, normalized size = 27.00
method | result | size |
risch | \(\ln \left (\ln \left ({\mathrm e}^{x}-x \right )-\frac {1}{2}\right )^{2}-2 \ln \left (\ln \left ({\mathrm e}^{x}-x \right )\right ) \ln \left (\ln \left ({\mathrm e}^{x}-x \right )-\frac {1}{2}\right )+\ln \left (\ln \left ({\mathrm e}^{x}-x \right )\right )^{2}+2 i \pi \ln \left (\ln \left ({\mathrm e}^{x}-x \right )\right ) \mathrm {csgn}\left (\frac {i \left (\ln \left ({\mathrm e}^{x}-x \right )-\frac {1}{2}\right )}{\ln \left ({\mathrm e}^{x}-x \right )}\right )^{2}-i \pi \ln \left (\ln \left ({\mathrm e}^{x}-x \right )\right ) \mathrm {csgn}\left (\frac {i}{\ln \left ({\mathrm e}^{x}-x \right )}\right ) \mathrm {csgn}\left (\frac {i \left (\ln \left ({\mathrm e}^{x}-x \right )-\frac {1}{2}\right )}{\ln \left ({\mathrm e}^{x}-x \right )}\right )^{2}+i \pi \ln \left (\ln \left ({\mathrm e}^{x}-x \right )-\frac {1}{2}\right ) \mathrm {csgn}\left (i \left (\ln \left ({\mathrm e}^{x}-x \right )-\frac {1}{2}\right )\right ) \mathrm {csgn}\left (\frac {i \left (\ln \left ({\mathrm e}^{x}-x \right )-\frac {1}{2}\right )}{\ln \left ({\mathrm e}^{x}-x \right )}\right )^{2}+i \pi \ln \left (\ln \left ({\mathrm e}^{x}-x \right )-\frac {1}{2}\right ) \mathrm {csgn}\left (\frac {i \left (\ln \left ({\mathrm e}^{x}-x \right )-\frac {1}{2}\right )}{\ln \left ({\mathrm e}^{x}-x \right )}\right )^{3}-i \pi \ln \left (\ln \left ({\mathrm e}^{x}-x \right )-\frac {1}{2}\right ) \mathrm {csgn}\left (i \left (\ln \left ({\mathrm e}^{x}-x \right )-\frac {1}{2}\right )\right ) \mathrm {csgn}\left (\frac {i}{\ln \left ({\mathrm e}^{x}-x \right )}\right ) \mathrm {csgn}\left (\frac {i \left (\ln \left ({\mathrm e}^{x}-x \right )-\frac {1}{2}\right )}{\ln \left ({\mathrm e}^{x}-x \right )}\right )-i \pi \ln \left (\ln \left ({\mathrm e}^{x}-x \right )\right ) \mathrm {csgn}\left (\frac {i \left (\ln \left ({\mathrm e}^{x}-x \right )-\frac {1}{2}\right )}{\ln \left ({\mathrm e}^{x}-x \right )}\right )^{3}+i \pi \ln \left (\ln \left ({\mathrm e}^{x}-x \right )-\frac {1}{2}\right ) \mathrm {csgn}\left (\frac {i}{\ln \left ({\mathrm e}^{x}-x \right )}\right ) \mathrm {csgn}\left (\frac {i \left (\ln \left ({\mathrm e}^{x}-x \right )-\frac {1}{2}\right )}{\ln \left ({\mathrm e}^{x}-x \right )}\right )^{2}-i \pi \ln \left (\ln \left ({\mathrm e}^{x}-x \right )\right ) \mathrm {csgn}\left (i \left (\ln \left ({\mathrm e}^{x}-x \right )-\frac {1}{2}\right )\right ) \mathrm {csgn}\left (\frac {i \left (\ln \left ({\mathrm e}^{x}-x \right )-\frac {1}{2}\right )}{\ln \left ({\mathrm e}^{x}-x \right )}\right )^{2}-2 i \pi \ln \left (\ln \left ({\mathrm e}^{x}-x \right )\right )+2 i \pi \ln \left (\ln \left ({\mathrm e}^{x}-x \right )-\frac {1}{2}\right )-2 i \pi \ln \left (\ln \left ({\mathrm e}^{x}-x \right )-\frac {1}{2}\right ) \mathrm {csgn}\left (\frac {i \left (\ln \left ({\mathrm e}^{x}-x \right )-\frac {1}{2}\right )}{\ln \left ({\mathrm e}^{x}-x \right )}\right )^{2}+i \pi \ln \left (\ln \left ({\mathrm e}^{x}-x \right )\right ) \mathrm {csgn}\left (i \left (\ln \left ({\mathrm e}^{x}-x \right )-\frac {1}{2}\right )\right ) \mathrm {csgn}\left (\frac {i}{\ln \left ({\mathrm e}^{x}-x \right )}\right ) \mathrm {csgn}\left (\frac {i \left (\ln \left ({\mathrm e}^{x}-x \right )-\frac {1}{2}\right )}{\ln \left ({\mathrm e}^{x}-x \right )}\right )-8 \ln \left (2\right ) \ln \left (\ln \left ({\mathrm e}^{x}-x \right )\right )+8 \ln \left (2\right ) \ln \left (\ln \left ({\mathrm e}^{x}-x \right )-\frac {1}{2}\right )+x\) | \(567\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [C] Result contains complex when optimal does not.
time = 0.56, size = 74, normalized size = 3.52 \begin {gather*} -2 \, {\left (-i \, \pi - 3 \, \log \left (2\right ) + \log \left (\log \left (-x + e^{x}\right )\right )\right )} \log \left (2 \, \log \left (-x + e^{x}\right ) - 1\right ) + \log \left (2 \, \log \left (-x + e^{x}\right ) - 1\right )^{2} - 2 \, {\left (i \, \pi + 3 \, \log \left (2\right )\right )} \log \left (\log \left (-x + e^{x}\right )\right ) + \log \left (\log \left (-x + e^{x}\right )\right )^{2} + x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 27, normalized size = 1.29 \begin {gather*} \log \left (-\frac {8 \, {\left (2 \, \log \left (-x + e^{x}\right ) - 1\right )}}{\log \left (-x + e^{x}\right )}\right )^{2} + x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.88, size = 20, normalized size = 0.95 \begin {gather*} x + \log {\left (\frac {8 - 16 \log {\left (- x + e^{x} \right )}}{\log {\left (- x + e^{x} \right )}} \right )}^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 76 vs.
\(2 (18) = 36\).
time = 0.51, size = 76, normalized size = 3.62 \begin {gather*} -\log \left (2 \, \log \left (-x + e^{x}\right ) - 1\right )^{2} + 2 \, \log \left (2 \, \log \left (-x + e^{x}\right ) - 1\right ) \log \left (-16 \, \log \left (-x + e^{x}\right ) + 8\right ) - 2 \, \log \left (-16 \, \log \left (-x + e^{x}\right ) + 8\right ) \log \left (\log \left (-x + e^{x}\right )\right ) + \log \left (\log \left (-x + e^{x}\right )\right )^{2} + x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 2.39, size = 18, normalized size = 0.86 \begin {gather*} {\ln \left (\frac {8}{\ln \left ({\mathrm {e}}^x-x\right )}-16\right )}^2+x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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