Optimal. Leaf size=19 \[ 2 x^2 \log \left (x+\log \left (\frac {x}{\log (2-x)}\right )\right ) \]
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Rubi [F]
time = 1.83, 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 {-2 x^2+\left (-4 x-2 x^2+2 x^3\right ) \log (2-x)+\left (\left (-8 x^2+4 x^3\right ) \log (2-x)+\left (-8 x+4 x^2\right ) \log (2-x) \log \left (\frac {x}{\log (2-x)}\right )\right ) \log \left (x+\log \left (\frac {x}{\log (2-x)}\right )\right )}{\left (-2 x+x^2\right ) \log (2-x)+(-2+x) \log (2-x) \log \left (\frac {x}{\log (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 {2 x \left (x-(-2+x) \log (2-x) \left (1+x+2 \left (x+\log \left (\frac {x}{\log (2-x)}\right )\right ) \log \left (x+\log \left (\frac {x}{\log (2-x)}\right )\right )\right )\right )}{(2-x) \log (2-x) \left (x+\log \left (\frac {x}{\log (2-x)}\right )\right )} \, dx\\ &=2 \int \frac {x \left (x-(-2+x) \log (2-x) \left (1+x+2 \left (x+\log \left (\frac {x}{\log (2-x)}\right )\right ) \log \left (x+\log \left (\frac {x}{\log (2-x)}\right )\right )\right )\right )}{(2-x) \log (2-x) \left (x+\log \left (\frac {x}{\log (2-x)}\right )\right )} \, dx\\ &=2 \int \left (\frac {x \left (-x-2 \log (2-x)-x \log (2-x)+x^2 \log (2-x)\right )}{(-2+x) \log (2-x) \left (x+\log \left (\frac {x}{\log (2-x)}\right )\right )}+2 x \log \left (x+\log \left (\frac {x}{\log (2-x)}\right )\right )\right ) \, dx\\ &=2 \int \frac {x \left (-x-2 \log (2-x)-x \log (2-x)+x^2 \log (2-x)\right )}{(-2+x) \log (2-x) \left (x+\log \left (\frac {x}{\log (2-x)}\right )\right )} \, dx+4 \int x \log \left (x+\log \left (\frac {x}{\log (2-x)}\right )\right ) \, dx\\ &=2 \int \frac {x \left (x-\left (-2-x+x^2\right ) \log (2-x)\right )}{(2-x) \log (2-x) \left (x+\log \left (\frac {x}{\log (2-x)}\right )\right )} \, dx+4 \int x \log \left (x+\log \left (\frac {x}{\log (2-x)}\right )\right ) \, dx\\ &=2 \int \left (\frac {-x-2 \log (2-x)-x \log (2-x)+x^2 \log (2-x)}{\log (2-x) \left (x+\log \left (\frac {x}{\log (2-x)}\right )\right )}+\frac {2 \left (-x-2 \log (2-x)-x \log (2-x)+x^2 \log (2-x)\right )}{(-2+x) \log (2-x) \left (x+\log \left (\frac {x}{\log (2-x)}\right )\right )}\right ) \, dx+4 \int x \log \left (x+\log \left (\frac {x}{\log (2-x)}\right )\right ) \, dx\\ &=2 \int \frac {-x-2 \log (2-x)-x \log (2-x)+x^2 \log (2-x)}{\log (2-x) \left (x+\log \left (\frac {x}{\log (2-x)}\right )\right )} \, dx+4 \int \frac {-x-2 \log (2-x)-x \log (2-x)+x^2 \log (2-x)}{(-2+x) \log (2-x) \left (x+\log \left (\frac {x}{\log (2-x)}\right )\right )} \, dx+4 \int x \log \left (x+\log \left (\frac {x}{\log (2-x)}\right )\right ) \, dx\\ &=2 \int \frac {-x+\left (-2-x+x^2\right ) \log (2-x)}{\log (2-x) \left (x+\log \left (\frac {x}{\log (2-x)}\right )\right )} \, dx+4 \int \frac {x-\left (-2-x+x^2\right ) \log (2-x)}{(2-x) \log (2-x) \left (x+\log \left (\frac {x}{\log (2-x)}\right )\right )} \, dx+4 \int x \log \left (x+\log \left (\frac {x}{\log (2-x)}\right )\right ) \, dx\\ &=2 \int \left (-\frac {2}{x+\log \left (\frac {x}{\log (2-x)}\right )}-\frac {x}{x+\log \left (\frac {x}{\log (2-x)}\right )}+\frac {x^2}{x+\log \left (\frac {x}{\log (2-x)}\right )}-\frac {x}{\log (2-x) \left (x+\log \left (\frac {x}{\log (2-x)}\right )\right )}\right ) \, dx+4 \int \left (-\frac {2}{(-2+x) \left (x+\log \left (\frac {x}{\log (2-x)}\right )\right )}-\frac {x}{(-2+x) \left (x+\log \left (\frac {x}{\log (2-x)}\right )\right )}+\frac {x^2}{(-2+x) \left (x+\log \left (\frac {x}{\log (2-x)}\right )\right )}-\frac {x}{(-2+x) \log (2-x) \left (x+\log \left (\frac {x}{\log (2-x)}\right )\right )}\right ) \, dx+4 \int x \log \left (x+\log \left (\frac {x}{\log (2-x)}\right )\right ) \, dx\\ &=-\left (2 \int \frac {x}{x+\log \left (\frac {x}{\log (2-x)}\right )} \, dx\right )+2 \int \frac {x^2}{x+\log \left (\frac {x}{\log (2-x)}\right )} \, dx-2 \int \frac {x}{\log (2-x) \left (x+\log \left (\frac {x}{\log (2-x)}\right )\right )} \, dx-4 \int \frac {1}{x+\log \left (\frac {x}{\log (2-x)}\right )} \, dx-4 \int \frac {x}{(-2+x) \left (x+\log \left (\frac {x}{\log (2-x)}\right )\right )} \, dx+4 \int \frac {x^2}{(-2+x) \left (x+\log \left (\frac {x}{\log (2-x)}\right )\right )} \, dx-4 \int \frac {x}{(-2+x) \log (2-x) \left (x+\log \left (\frac {x}{\log (2-x)}\right )\right )} \, dx+4 \int x \log \left (x+\log \left (\frac {x}{\log (2-x)}\right )\right ) \, dx-8 \int \frac {1}{(-2+x) \left (x+\log \left (\frac {x}{\log (2-x)}\right )\right )} \, dx\\ &=-\left (2 \int \frac {x}{x+\log \left (\frac {x}{\log (2-x)}\right )} \, dx\right )+2 \int \frac {x^2}{x+\log \left (\frac {x}{\log (2-x)}\right )} \, dx-2 \int \frac {x}{\log (2-x) \left (x+\log \left (\frac {x}{\log (2-x)}\right )\right )} \, dx-4 \int \frac {1}{x+\log \left (\frac {x}{\log (2-x)}\right )} \, dx-4 \int \left (\frac {1}{x+\log \left (\frac {x}{\log (2-x)}\right )}+\frac {2}{(-2+x) \left (x+\log \left (\frac {x}{\log (2-x)}\right )\right )}\right ) \, dx+4 \int \left (\frac {2}{x+\log \left (\frac {x}{\log (2-x)}\right )}+\frac {4}{(-2+x) \left (x+\log \left (\frac {x}{\log (2-x)}\right )\right )}+\frac {x}{x+\log \left (\frac {x}{\log (2-x)}\right )}\right ) \, dx-4 \int \left (\frac {1}{\log (2-x) \left (x+\log \left (\frac {x}{\log (2-x)}\right )\right )}+\frac {2}{(-2+x) \log (2-x) \left (x+\log \left (\frac {x}{\log (2-x)}\right )\right )}\right ) \, dx+4 \int x \log \left (x+\log \left (\frac {x}{\log (2-x)}\right )\right ) \, dx-8 \int \frac {1}{(-2+x) \left (x+\log \left (\frac {x}{\log (2-x)}\right )\right )} \, dx\\ &=-\left (2 \int \frac {x}{x+\log \left (\frac {x}{\log (2-x)}\right )} \, dx\right )+2 \int \frac {x^2}{x+\log \left (\frac {x}{\log (2-x)}\right )} \, dx-2 \int \frac {x}{\log (2-x) \left (x+\log \left (\frac {x}{\log (2-x)}\right )\right )} \, dx-2 \left (4 \int \frac {1}{x+\log \left (\frac {x}{\log (2-x)}\right )} \, dx\right )+4 \int \frac {x}{x+\log \left (\frac {x}{\log (2-x)}\right )} \, dx-4 \int \frac {1}{\log (2-x) \left (x+\log \left (\frac {x}{\log (2-x)}\right )\right )} \, dx+4 \int x \log \left (x+\log \left (\frac {x}{\log (2-x)}\right )\right ) \, dx+8 \int \frac {1}{x+\log \left (\frac {x}{\log (2-x)}\right )} \, dx-2 \left (8 \int \frac {1}{(-2+x) \left (x+\log \left (\frac {x}{\log (2-x)}\right )\right )} \, dx\right )-8 \int \frac {1}{(-2+x) \log (2-x) \left (x+\log \left (\frac {x}{\log (2-x)}\right )\right )} \, dx+16 \int \frac {1}{(-2+x) \left (x+\log \left (\frac {x}{\log (2-x)}\right )\right )} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.11, size = 19, normalized size = 1.00 \begin {gather*} 2 x^2 \log \left (x+\log \left (\frac {x}{\log (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 = 0.46, size = 86, normalized size = 4.53
method | result | size |
risch | \(2 x^{2} \ln \left (\ln \left (x \right )-\ln \left (\ln \left (2-x \right )\right )-\frac {i \pi \,\mathrm {csgn}\left (\frac {i x}{\ln \left (2-x \right )}\right ) \left (-\mathrm {csgn}\left (\frac {i x}{\ln \left (2-x \right )}\right )+\mathrm {csgn}\left (i x \right )\right ) \left (-\mathrm {csgn}\left (\frac {i x}{\ln \left (2-x \right )}\right )+\mathrm {csgn}\left (\frac {i}{\ln \left (2-x \right )}\right )\right )}{2}+x \right )\) | \(86\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.35, size = 19, normalized size = 1.00 \begin {gather*} 2 \, x^{2} \log \left (x + \log \left (x\right ) - \log \left (\log \left (-x + 2\right )\right )\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.30, size = 19, normalized size = 1.00 \begin {gather*} 2 \, x^{2} \log \left (x + \log \left (\frac {x}{\log \left (-x + 2\right )}\right )\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 1.95, size = 15, normalized size = 0.79 \begin {gather*} 2 x^{2} \log {\left (x + \log {\left (\frac {x}{\log {\left (2 - x \right )}} \right )} \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.55, size = 19, normalized size = 1.00 \begin {gather*} 2 \, x^{2} \log \left (x + \log \left (x\right ) - \log \left (\log \left (-x + 2\right )\right )\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.14, size = 19, normalized size = 1.00 \begin {gather*} 2\,x^2\,\ln \left (x+\ln \left (\frac {x}{\ln \left (2-x\right )}\right )\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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