Optimal. Leaf size=26 \[ 1+x-\left (x+(-x+\log (x)) \left (5+\log \left (-e^x+x\right )\right )\right )^2 \]
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Rubi [F]
time = 6.80, 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 {-41 x^2+40 x^3+e^x \left (41 x-32 x^2-8 x^3\right )+\left (50 x-58 x^2+e^x \left (-50+40 x+18 x^2\right )\right ) \log (x)+\left (10 x-10 e^x x\right ) \log ^2(x)+\left (-18 x^2+18 x^3+e^x \left (18 x-16 x^2-2 x^3\right )+\left (20 x-22 x^2+e^x \left (-20+18 x+4 x^2\right )\right ) \log (x)+\left (2 x-2 e^x x\right ) \log ^2(x)\right ) \log \left (-e^x+x\right )+\left (-2 x^2+2 x^3+e^x \left (2 x-2 x^2\right )+\left (2 x-2 x^2+e^x (-2+2 x)\right ) \log (x)\right ) \log ^2\left (-e^x+x\right )}{e^x x-x^2} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \left (-\frac {2 (-1+x) (x-\log (x)) \left (4 x-5 \log (x)+x \log \left (-e^x+x\right )-\log (x) \log \left (-e^x+x\right )\right )}{e^x-x}+\frac {41 x-32 x^2-8 x^3-50 \log (x)+40 x \log (x)+18 x^2 \log (x)-10 x \log ^2(x)+18 x \log \left (-e^x+x\right )-16 x^2 \log \left (-e^x+x\right )-2 x^3 \log \left (-e^x+x\right )-20 \log (x) \log \left (-e^x+x\right )+18 x \log (x) \log \left (-e^x+x\right )+4 x^2 \log (x) \log \left (-e^x+x\right )-2 x \log ^2(x) \log \left (-e^x+x\right )+2 x \log ^2\left (-e^x+x\right )-2 x^2 \log ^2\left (-e^x+x\right )-2 \log (x) \log ^2\left (-e^x+x\right )+2 x \log (x) \log ^2\left (-e^x+x\right )}{x}\right ) \, dx\\ &=-\left (2 \int \frac {(-1+x) (x-\log (x)) \left (4 x-5 \log (x)+x \log \left (-e^x+x\right )-\log (x) \log \left (-e^x+x\right )\right )}{e^x-x} \, dx\right )+\int \frac {41 x-32 x^2-8 x^3-50 \log (x)+40 x \log (x)+18 x^2 \log (x)-10 x \log ^2(x)+18 x \log \left (-e^x+x\right )-16 x^2 \log \left (-e^x+x\right )-2 x^3 \log \left (-e^x+x\right )-20 \log (x) \log \left (-e^x+x\right )+18 x \log (x) \log \left (-e^x+x\right )+4 x^2 \log (x) \log \left (-e^x+x\right )-2 x \log ^2(x) \log \left (-e^x+x\right )+2 x \log ^2\left (-e^x+x\right )-2 x^2 \log ^2\left (-e^x+x\right )-2 \log (x) \log ^2\left (-e^x+x\right )+2 x \log (x) \log ^2\left (-e^x+x\right )}{x} \, dx\\ &=-\left (2 \int \left (-\frac {(x-\log (x)) \left (4 x-5 \log (x)+x \log \left (-e^x+x\right )-\log (x) \log \left (-e^x+x\right )\right )}{e^x-x}+\frac {x (x-\log (x)) \left (4 x-5 \log (x)+x \log \left (-e^x+x\right )-\log (x) \log \left (-e^x+x\right )\right )}{e^x-x}\right ) \, dx\right )+\int \left (\frac {41 x-32 x^2-8 x^3-50 \log (x)+40 x \log (x)+18 x^2 \log (x)-10 x \log ^2(x)}{x}-\frac {2 \left (-9 x+8 x^2+x^3+10 \log (x)-9 x \log (x)-2 x^2 \log (x)+x \log ^2(x)\right ) \log \left (-e^x+x\right )}{x}-\frac {2 (-1+x) (x-\log (x)) \log ^2\left (-e^x+x\right )}{x}\right ) \, dx\\ &=-\left (2 \int \frac {\left (-9 x+8 x^2+x^3+10 \log (x)-9 x \log (x)-2 x^2 \log (x)+x \log ^2(x)\right ) \log \left (-e^x+x\right )}{x} \, dx\right )-2 \int \frac {(-1+x) (x-\log (x)) \log ^2\left (-e^x+x\right )}{x} \, dx+2 \int \frac {(x-\log (x)) \left (4 x-5 \log (x)+x \log \left (-e^x+x\right )-\log (x) \log \left (-e^x+x\right )\right )}{e^x-x} \, dx-2 \int \frac {x (x-\log (x)) \left (4 x-5 \log (x)+x \log \left (-e^x+x\right )-\log (x) \log \left (-e^x+x\right )\right )}{e^x-x} \, dx+\int \frac {41 x-32 x^2-8 x^3-50 \log (x)+40 x \log (x)+18 x^2 \log (x)-10 x \log ^2(x)}{x} \, dx\\ &=-\left (2 \int \left (-9 \log \left (-e^x+x\right )+8 x \log \left (-e^x+x\right )+x^2 \log \left (-e^x+x\right )-9 \log (x) \log \left (-e^x+x\right )+\frac {10 \log (x) \log \left (-e^x+x\right )}{x}-2 x \log (x) \log \left (-e^x+x\right )+\log ^2(x) \log \left (-e^x+x\right )\right ) \, dx\right )+2 \int \left (\frac {4 x^2}{e^x-x}-\frac {9 x \log (x)}{e^x-x}+\frac {5 \log ^2(x)}{e^x-x}+\frac {x^2 \log \left (-e^x+x\right )}{e^x-x}-\frac {2 x \log (x) \log \left (-e^x+x\right )}{e^x-x}+\frac {\log ^2(x) \log \left (-e^x+x\right )}{e^x-x}\right ) \, dx-2 \int \left (\frac {4 x^3}{e^x-x}-\frac {9 x^2 \log (x)}{e^x-x}+\frac {5 x \log ^2(x)}{e^x-x}+\frac {x^3 \log \left (-e^x+x\right )}{e^x-x}-\frac {2 x^2 \log (x) \log \left (-e^x+x\right )}{e^x-x}+\frac {x \log ^2(x) \log \left (-e^x+x\right )}{e^x-x}\right ) \, dx-2 \int \left ((x-\log (x)) \log ^2\left (-e^x+x\right )-\frac {(x-\log (x)) \log ^2\left (-e^x+x\right )}{x}\right ) \, dx+\int \left (41-32 x-8 x^2+\frac {2 \left (-25+20 x+9 x^2\right ) \log (x)}{x}-10 \log ^2(x)\right ) \, dx\\ &=41 x-16 x^2-\frac {8 x^3}{3}+2 \int \frac {\left (-25+20 x+9 x^2\right ) \log (x)}{x} \, dx-2 \int x^2 \log \left (-e^x+x\right ) \, dx+2 \int \frac {x^2 \log \left (-e^x+x\right )}{e^x-x} \, dx-2 \int \frac {x^3 \log \left (-e^x+x\right )}{e^x-x} \, dx-2 \int \log ^2(x) \log \left (-e^x+x\right ) \, dx+2 \int \frac {\log ^2(x) \log \left (-e^x+x\right )}{e^x-x} \, dx-2 \int \frac {x \log ^2(x) \log \left (-e^x+x\right )}{e^x-x} \, dx-2 \int (x-\log (x)) \log ^2\left (-e^x+x\right ) \, dx+2 \int \frac {(x-\log (x)) \log ^2\left (-e^x+x\right )}{x} \, dx+4 \int x \log (x) \log \left (-e^x+x\right ) \, dx-4 \int \frac {x \log (x) \log \left (-e^x+x\right )}{e^x-x} \, dx+4 \int \frac {x^2 \log (x) \log \left (-e^x+x\right )}{e^x-x} \, dx+8 \int \frac {x^2}{e^x-x} \, dx-8 \int \frac {x^3}{e^x-x} \, dx-10 \int \log ^2(x) \, dx+10 \int \frac {\log ^2(x)}{e^x-x} \, dx-10 \int \frac {x \log ^2(x)}{e^x-x} \, dx-16 \int x \log \left (-e^x+x\right ) \, dx-18 \int \frac {x \log (x)}{e^x-x} \, dx+18 \int \frac {x^2 \log (x)}{e^x-x} \, dx+18 \int \log \left (-e^x+x\right ) \, dx+18 \int \log (x) \log \left (-e^x+x\right ) \, dx-20 \int \frac {\log (x) \log \left (-e^x+x\right )}{x} \, dx\\ &=\text {Rest of rules removed due to large latex content} \end {aligned} \end {gather*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(65\) vs. \(2(26)=52\).
time = 0.15, size = 65, normalized size = 2.50 \begin {gather*} x-16 x^2+40 x \log (x)-25 \log ^2(x)-2 \left (4 x^2-9 x \log (x)+5 \log ^2(x)\right ) \log \left (-e^x+x\right )-(x-\log (x))^2 \log ^2\left (-e^x+x\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(70\) vs.
\(2(25)=50\).
time = 0.06, size = 71, normalized size = 2.73
method | result | size |
risch | \(\left (-x^{2}+2 x \ln \left (x \right )-\ln \left (x \right )^{2}\right ) \ln \left (x -{\mathrm e}^{x}\right )^{2}+\left (-8 x^{2}+18 x \ln \left (x \right )-10 \ln \left (x \right )^{2}\right ) \ln \left (x -{\mathrm e}^{x}\right )-16 x^{2}+40 x \ln \left (x \right )-25 \ln \left (x \right )^{2}+x\) | \(71\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 68 vs.
\(2 (27) = 54\).
time = 0.30, size = 68, normalized size = 2.62 \begin {gather*} -{\left (x^{2} - 2 \, x \log \left (x\right ) + \log \left (x\right )^{2}\right )} \log \left (x - e^{x}\right )^{2} - 16 \, x^{2} - 2 \, {\left (4 \, x^{2} - 9 \, x \log \left (x\right ) + 5 \, \log \left (x\right )^{2}\right )} \log \left (x - e^{x}\right ) + 40 \, x \log \left (x\right ) - 25 \, \log \left (x\right )^{2} + x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 68 vs.
\(2 (27) = 54\).
time = 0.38, size = 68, normalized size = 2.62 \begin {gather*} -{\left (x^{2} - 2 \, x \log \left (x\right ) + \log \left (x\right )^{2}\right )} \log \left (x - e^{x}\right )^{2} - 16 \, x^{2} - 2 \, {\left (4 \, x^{2} - 9 \, x \log \left (x\right ) + 5 \, \log \left (x\right )^{2}\right )} \log \left (x - e^{x}\right ) + 40 \, x \log \left (x\right ) - 25 \, \log \left (x\right )^{2} + x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 68 vs.
\(2 (19) = 38\).
time = 0.43, size = 68, normalized size = 2.62 \begin {gather*} - 16 x^{2} + 40 x \log {\left (x \right )} + x + \left (- 8 x^{2} + 18 x \log {\left (x \right )} - 10 \log {\left (x \right )}^{2}\right ) \log {\left (x - e^{x} \right )} + \left (- x^{2} + 2 x \log {\left (x \right )} - \log {\left (x \right )}^{2}\right ) \log {\left (x - e^{x} \right )}^{2} - 25 \log {\left (x \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. 98 vs.
\(2 (27) = 54\).
time = 0.44, size = 98, normalized size = 3.77 \begin {gather*} -x^{2} \log \left (x - e^{x}\right )^{2} + 2 \, x \log \left (x - e^{x}\right )^{2} \log \left (x\right ) - \log \left (x - e^{x}\right )^{2} \log \left (x\right )^{2} - 8 \, x^{2} \log \left (x - e^{x}\right ) + 18 \, x \log \left (x - e^{x}\right ) \log \left (x\right ) - 10 \, \log \left (x - e^{x}\right ) \log \left (x\right )^{2} - 16 \, x^{2} + 40 \, x \log \left (x\right ) - 25 \, \log \left (x\right )^{2} + x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.20, size = 98, normalized size = 3.77 \begin {gather*} x-25\,{\ln \left (x\right )}^2-\ln \left (x-{\mathrm {e}}^x\right )\,\left (18\,x-\frac {18\,x^2-8\,x^3}{x}+10\,{\ln \left (x\right )}^2-18\,x\,\ln \left (x\right )\right )-{\ln \left (x-{\mathrm {e}}^x\right )}^2\,\left (2\,x-\frac {2\,x^2-x^3}{x}+{\ln \left (x\right )}^2-2\,x\,\ln \left (x\right )\right )+40\,x\,\ln \left (x\right )-16\,x^2 \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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