Optimal. Leaf size=28 \[ \left (-5+\frac {4}{5 (4+x)}\right ) \log \left (\frac {4 x^2}{2+e^x-x}\right ) \]
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Rubi [F] time = 1.74, 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 {-1536-400 x+96 x^2+25 x^3+e^x \left (-768-8 x+146 x^2+25 x^3\right )+\left (-8 x-4 e^x x+4 x^2\right ) \log \left (\frac {4 x^2}{2+e^x-x}\right )}{160 x-30 x^3-5 x^4+e^x \left (80 x+40 x^2+5 x^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 {-\frac {\left (-4+e^x (-2+x)+x\right ) \left (384+196 x+25 x^2\right )}{x \left (-2-e^x+x\right )}-4 \log \left (\frac {4 x^2}{2+e^x-x}\right )}{5 (4+x)^2} \, dx\\ &=\frac {1}{5} \int \frac {-\frac {\left (-4+e^x (-2+x)+x\right ) \left (384+196 x+25 x^2\right )}{x \left (-2-e^x+x\right )}-4 \log \left (\frac {4 x^2}{2+e^x-x}\right )}{(4+x)^2} \, dx\\ &=\frac {1}{5} \int \left (\frac {-288+21 x+25 x^2}{\left (2+e^x-x\right ) (4+x)}+\frac {-768-8 x+146 x^2+25 x^3-4 x \log \left (\frac {4 x^2}{2+e^x-x}\right )}{x (4+x)^2}\right ) \, dx\\ &=\frac {1}{5} \int \frac {-288+21 x+25 x^2}{\left (2+e^x-x\right ) (4+x)} \, dx+\frac {1}{5} \int \frac {-768-8 x+146 x^2+25 x^3-4 x \log \left (\frac {4 x^2}{2+e^x-x}\right )}{x (4+x)^2} \, dx\\ &=\frac {1}{5} \int \left (-\frac {79}{2+e^x-x}+\frac {25 x}{2+e^x-x}+\frac {28}{\left (2+e^x-x\right ) (4+x)}\right ) \, dx+\frac {1}{5} \int \left (-\frac {8}{(4+x)^2}-\frac {768}{x (4+x)^2}+\frac {146 x}{(4+x)^2}+\frac {25 x^2}{(4+x)^2}-\frac {4 \log \left (\frac {4 x^2}{2+e^x-x}\right )}{(4+x)^2}\right ) \, dx\\ &=\frac {8}{5 (4+x)}-\frac {4}{5} \int \frac {\log \left (\frac {4 x^2}{2+e^x-x}\right )}{(4+x)^2} \, dx+5 \int \frac {x}{2+e^x-x} \, dx+5 \int \frac {x^2}{(4+x)^2} \, dx+\frac {28}{5} \int \frac {1}{\left (2+e^x-x\right ) (4+x)} \, dx-\frac {79}{5} \int \frac {1}{2+e^x-x} \, dx+\frac {146}{5} \int \frac {x}{(4+x)^2} \, dx-\frac {768}{5} \int \frac {1}{x (4+x)^2} \, dx\\ &=\frac {8}{5 (4+x)}+\frac {4 \log \left (\frac {4 x^2}{2+e^x-x}\right )}{5 (4+x)}-\frac {4}{5} \int \frac {4-e^x (-2+x)-x}{\left (2+e^x-x\right ) x (4+x)} \, dx+5 \int \frac {x}{2+e^x-x} \, dx+5 \int \left (1+\frac {16}{(4+x)^2}-\frac {8}{4+x}\right ) \, dx+\frac {28}{5} \int \frac {1}{\left (2+e^x-x\right ) (4+x)} \, dx-\frac {79}{5} \int \frac {1}{2+e^x-x} \, dx+\frac {146}{5} \int \left (-\frac {4}{(4+x)^2}+\frac {1}{4+x}\right ) \, dx-\frac {768}{5} \int \left (\frac {1}{16 x}-\frac {1}{4 (4+x)^2}-\frac {1}{16 (4+x)}\right ) \, dx\\ &=5 x-\frac {48 \log (x)}{5}+\frac {4 \log \left (\frac {4 x^2}{2+e^x-x}\right )}{5 (4+x)}-\frac {6}{5} \log (4+x)-\frac {4}{5} \int \left (\frac {2-x}{x (4+x)}+\frac {-3+x}{(4+x) \left (-2-e^x+x\right )}\right ) \, dx+5 \int \frac {x}{2+e^x-x} \, dx+\frac {28}{5} \int \frac {1}{\left (2+e^x-x\right ) (4+x)} \, dx-\frac {79}{5} \int \frac {1}{2+e^x-x} \, dx\\ &=5 x-\frac {48 \log (x)}{5}+\frac {4 \log \left (\frac {4 x^2}{2+e^x-x}\right )}{5 (4+x)}-\frac {6}{5} \log (4+x)-\frac {4}{5} \int \frac {2-x}{x (4+x)} \, dx-\frac {4}{5} \int \frac {-3+x}{(4+x) \left (-2-e^x+x\right )} \, dx+5 \int \frac {x}{2+e^x-x} \, dx+\frac {28}{5} \int \frac {1}{\left (2+e^x-x\right ) (4+x)} \, dx-\frac {79}{5} \int \frac {1}{2+e^x-x} \, dx\\ &=5 x-\frac {48 \log (x)}{5}+\frac {4 \log \left (\frac {4 x^2}{2+e^x-x}\right )}{5 (4+x)}-\frac {6}{5} \log (4+x)-\frac {4}{5} \int \left (\frac {1}{2 x}-\frac {3}{2 (4+x)}\right ) \, dx-\frac {4}{5} \int \left (-\frac {1}{2+e^x-x}+\frac {7}{\left (2+e^x-x\right ) (4+x)}\right ) \, dx+5 \int \frac {x}{2+e^x-x} \, dx+\frac {28}{5} \int \frac {1}{\left (2+e^x-x\right ) (4+x)} \, dx-\frac {79}{5} \int \frac {1}{2+e^x-x} \, dx\\ &=5 x-10 \log (x)+\frac {4 \log \left (\frac {4 x^2}{2+e^x-x}\right )}{5 (4+x)}+\frac {4}{5} \int \frac {1}{2+e^x-x} \, dx+5 \int \frac {x}{2+e^x-x} \, dx-\frac {79}{5} \int \frac {1}{2+e^x-x} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.53, size = 43, normalized size = 1.54 \begin {gather*} \frac {1}{5} \left (25 \log \left (2+e^x-x\right )-50 \log (x)+\frac {4 \log \left (\frac {4 x^2}{2+e^x-x}\right )}{4+x}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.69, size = 27, normalized size = 0.96 \begin {gather*} -\frac {{\left (25 \, x + 96\right )} \log \left (-\frac {4 \, x^{2}}{x - e^{x} - 2}\right )}{5 \, {\left (x + 4\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.17, size = 55, normalized size = 1.96 \begin {gather*} \frac {25 \, x \log \left (x - e^{x} - 2\right ) - 50 \, x \log \relax (x) + 100 \, \log \left (x - e^{x} - 2\right ) - 200 \, \log \relax (x) + 4 \, \log \left (-\frac {4 \, x^{2}}{x - e^{x} - 2}\right )}{5 \, {\left (x + 4\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.27, size = 38, normalized size = 1.36
method | result | size |
norman | \(\frac {4 \ln \left (\frac {4 x^{2}}{{\mathrm e}^{x}+2-x}\right )}{5 \left (4+x \right )}-10 \ln \relax (x )+5 \ln \left (-{\mathrm e}^{x}+x -2\right )\) | \(38\) |
risch | \(-\frac {4 \ln \left (-{\mathrm e}^{x}+x -2\right )}{5 \left (4+x \right )}+\frac {-2 i \pi \mathrm {csgn}\left (i x^{2}\right )^{3}-2 i \pi \,\mathrm {csgn}\left (i x^{2}\right ) \mathrm {csgn}\left (i x \right )^{2}+2 i \pi \mathrm {csgn}\left (\frac {i x^{2}}{{\mathrm e}^{x}+2-x}\right )^{2} \mathrm {csgn}\left (i x^{2}\right )-2 i \pi \,\mathrm {csgn}\left (\frac {i}{{\mathrm e}^{x}+2-x}\right ) \mathrm {csgn}\left (\frac {i x^{2}}{{\mathrm e}^{x}+2-x}\right )^{2}-2 i \pi \,\mathrm {csgn}\left (\frac {i}{{\mathrm e}^{x}+2-x}\right ) \mathrm {csgn}\left (\frac {i x^{2}}{{\mathrm e}^{x}+2-x}\right ) \mathrm {csgn}\left (i x^{2}\right )-4 i \pi \mathrm {csgn}\left (\frac {i x^{2}}{{\mathrm e}^{x}+2-x}\right )^{2}+4 i \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{2}-2 i \pi \mathrm {csgn}\left (\frac {i x^{2}}{{\mathrm e}^{x}+2-x}\right )^{3}+4 i \pi -50 x \ln \relax (x )+25 \ln \left ({\mathrm e}^{x}+2-x \right ) x -192 \ln \relax (x )+8 \ln \relax (2)+100 \ln \left ({\mathrm e}^{x}+2-x \right )}{20+5 x}\) | \(260\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.49, size = 36, normalized size = 1.29 \begin {gather*} -\frac {2 \, {\left (25 \, x + 96\right )} \log \relax (x) - {\left (25 \, x + 96\right )} \log \left (-x + e^{x} + 2\right ) - 8 \, \log \relax (2)}{5 \, {\left (x + 4\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.18, size = 39, normalized size = 1.39 \begin {gather*} 5\,\ln \left ({\mathrm {e}}^x-x+2\right )-10\,\ln \relax (x)+\frac {4\,\ln \left (\frac {4\,x^2}{{\mathrm {e}}^x-x+2}\right )}{5\,\left (x+4\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.50, size = 34, normalized size = 1.21 \begin {gather*} - 10 \log {\relax (x )} + 5 \log {\left (- x + e^{x} + 2 \right )} + \frac {4 \log {\left (\frac {4 x^{2}}{- x + e^{x} + 2} \right )}}{5 x + 20} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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