Optimal. Leaf size=25 \[ x+\log \left (4 \left (x+\left (3+\frac {x^2}{\log \left (-x+x^2\right )}\right )^2\right )\right ) \]
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Rubi [F] time = 15.01, 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^3-4 x^4+\left (6 x-12 x^2-4 x^3+3 x^4+x^5\right ) \log \left (-x+x^2\right )+\left (-12 x+6 x^2+6 x^3\right ) \log ^2\left (-x+x^2\right )+\left (-10+9 x+x^2\right ) \log ^3\left (-x+x^2\right )}{\left (-x^4+x^5\right ) \log \left (-x+x^2\right )+\left (-6 x^2+6 x^3\right ) \log ^2\left (-x+x^2\right )+\left (-9+8 x+x^2\right ) \log ^3\left (-x+x^2\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^3+4 x^4-\left (6 x-12 x^2-4 x^3+3 x^4+x^5\right ) \log \left (-x+x^2\right )-\left (-12 x+6 x^2+6 x^3\right ) \log ^2\left (-x+x^2\right )-\left (-10+9 x+x^2\right ) \log ^3\left (-x+x^2\right )}{(1-x) \log ((-1+x) x) \left (x^4+6 x^2 \log ((-1+x) x)+9 \log ^2((-1+x) x)+x \log ^2((-1+x) x)\right )} \, dx\\ &=\int \left (\frac {x \left (6-12 x-4 x^2+3 x^3+x^4\right )}{(-1+x) \left (x^4+6 x^2 \log ((-1+x) x)+9 \log ^2((-1+x) x)+x \log ^2((-1+x) x)\right )}-\frac {2 x^3 (-1+2 x)}{(-1+x) \log ((-1+x) x) \left (x^4+6 x^2 \log ((-1+x) x)+9 \log ^2((-1+x) x)+x \log ^2((-1+x) x)\right )}+\frac {6 x (2+x) \log ((-1+x) x)}{x^4+6 x^2 \log ((-1+x) x)+9 \log ^2((-1+x) x)+x \log ^2((-1+x) x)}+\frac {(10+x) \log ^2((-1+x) x)}{x^4+6 x^2 \log ((-1+x) x)+9 \log ^2((-1+x) x)+x \log ^2((-1+x) x)}\right ) \, dx\\ &=-\left (2 \int \frac {x^3 (-1+2 x)}{(-1+x) \log ((-1+x) x) \left (x^4+6 x^2 \log ((-1+x) x)+9 \log ^2((-1+x) x)+x \log ^2((-1+x) x)\right )} \, dx\right )+6 \int \frac {x (2+x) \log ((-1+x) x)}{x^4+6 x^2 \log ((-1+x) x)+9 \log ^2((-1+x) x)+x \log ^2((-1+x) x)} \, dx+\int \frac {x \left (6-12 x-4 x^2+3 x^3+x^4\right )}{(-1+x) \left (x^4+6 x^2 \log ((-1+x) x)+9 \log ^2((-1+x) x)+x \log ^2((-1+x) x)\right )} \, dx+\int \frac {(10+x) \log ^2((-1+x) x)}{x^4+6 x^2 \log ((-1+x) x)+9 \log ^2((-1+x) x)+x \log ^2((-1+x) x)} \, dx\\ &=-\left (2 \int \left (\frac {-1+2 x}{(-1+x) x \log ((-1+x) x)}-\frac {(-1+2 x) \left (6 x^2+9 \log ((-1+x) x)+x \log ((-1+x) x)\right )}{(-1+x) x \left (x^4+6 x^2 \log ((-1+x) x)+9 \log ^2((-1+x) x)+x \log ^2((-1+x) x)\right )}\right ) \, dx\right )+6 \int \left (\frac {2 x \log ((-1+x) x)}{x^4+6 x^2 \log ((-1+x) x)+9 \log ^2((-1+x) x)+x \log ^2((-1+x) x)}+\frac {x^2 \log ((-1+x) x)}{x^4+6 x^2 \log ((-1+x) x)+9 \log ^2((-1+x) x)+x \log ^2((-1+x) x)}\right ) \, dx+\int \left (-\frac {6}{x^4+6 x^2 \log ((-1+x) x)+9 \log ^2((-1+x) x)+x \log ^2((-1+x) x)}-\frac {6}{(-1+x) \left (x^4+6 x^2 \log ((-1+x) x)+9 \log ^2((-1+x) x)+x \log ^2((-1+x) x)\right )}-\frac {12 x}{x^4+6 x^2 \log ((-1+x) x)+9 \log ^2((-1+x) x)+x \log ^2((-1+x) x)}+\frac {4 x^3}{x^4+6 x^2 \log ((-1+x) x)+9 \log ^2((-1+x) x)+x \log ^2((-1+x) x)}+\frac {x^4}{x^4+6 x^2 \log ((-1+x) x)+9 \log ^2((-1+x) x)+x \log ^2((-1+x) x)}\right ) \, dx+\int \left (\frac {10+x}{9+x}-\frac {x^2 (10+x) \left (x^2+6 \log ((-1+x) x)\right )}{(9+x) \left (x^4+6 x^2 \log ((-1+x) x)+9 \log ^2((-1+x) x)+x \log ^2((-1+x) x)\right )}\right ) \, dx\\ &=-\left (2 \int \frac {-1+2 x}{(-1+x) x \log ((-1+x) x)} \, dx\right )+2 \int \frac {(-1+2 x) \left (6 x^2+9 \log ((-1+x) x)+x \log ((-1+x) x)\right )}{(-1+x) x \left (x^4+6 x^2 \log ((-1+x) x)+9 \log ^2((-1+x) x)+x \log ^2((-1+x) x)\right )} \, dx+4 \int \frac {x^3}{x^4+6 x^2 \log ((-1+x) x)+9 \log ^2((-1+x) x)+x \log ^2((-1+x) x)} \, dx-6 \int \frac {1}{x^4+6 x^2 \log ((-1+x) x)+9 \log ^2((-1+x) x)+x \log ^2((-1+x) x)} \, dx-6 \int \frac {1}{(-1+x) \left (x^4+6 x^2 \log ((-1+x) x)+9 \log ^2((-1+x) x)+x \log ^2((-1+x) x)\right )} \, dx+6 \int \frac {x^2 \log ((-1+x) x)}{x^4+6 x^2 \log ((-1+x) x)+9 \log ^2((-1+x) x)+x \log ^2((-1+x) x)} \, dx-12 \int \frac {x}{x^4+6 x^2 \log ((-1+x) x)+9 \log ^2((-1+x) x)+x \log ^2((-1+x) x)} \, dx+12 \int \frac {x \log ((-1+x) x)}{x^4+6 x^2 \log ((-1+x) x)+9 \log ^2((-1+x) x)+x \log ^2((-1+x) x)} \, dx+\int \frac {10+x}{9+x} \, dx+\int \frac {x^4}{x^4+6 x^2 \log ((-1+x) x)+9 \log ^2((-1+x) x)+x \log ^2((-1+x) x)} \, dx-\int \frac {x^2 (10+x) \left (x^2+6 \log ((-1+x) x)\right )}{(9+x) \left (x^4+6 x^2 \log ((-1+x) x)+9 \log ^2((-1+x) x)+x \log ^2((-1+x) x)\right )} \, dx\\ &=-\left (2 \int \frac {-1+2 x}{(-1+x) x \log ((-1+x) x)} \, dx\right )+2 \int \left (\frac {6 x^2+9 \log ((-1+x) x)+x \log ((-1+x) x)}{(-1+x) \left (x^4+6 x^2 \log ((-1+x) x)+9 \log ^2((-1+x) x)+x \log ^2((-1+x) x)\right )}+\frac {6 x^2+9 \log ((-1+x) x)+x \log ((-1+x) x)}{x \left (x^4+6 x^2 \log ((-1+x) x)+9 \log ^2((-1+x) x)+x \log ^2((-1+x) x)\right )}\right ) \, dx+4 \int \frac {x^3}{x^4+6 x^2 \log ((-1+x) x)+9 \log ^2((-1+x) x)+x \log ^2((-1+x) x)} \, dx-6 \int \frac {1}{x^4+6 x^2 \log ((-1+x) x)+9 \log ^2((-1+x) x)+x \log ^2((-1+x) x)} \, dx-6 \int \frac {1}{(-1+x) \left (x^4+6 x^2 \log ((-1+x) x)+9 \log ^2((-1+x) x)+x \log ^2((-1+x) x)\right )} \, dx+6 \int \frac {x^2 \log ((-1+x) x)}{x^4+6 x^2 \log ((-1+x) x)+9 \log ^2((-1+x) x)+x \log ^2((-1+x) x)} \, dx-12 \int \frac {x}{x^4+6 x^2 \log ((-1+x) x)+9 \log ^2((-1+x) x)+x \log ^2((-1+x) x)} \, dx+12 \int \frac {x \log ((-1+x) x)}{x^4+6 x^2 \log ((-1+x) x)+9 \log ^2((-1+x) x)+x \log ^2((-1+x) x)} \, dx+\int \left (1+\frac {1}{9+x}\right ) \, dx+\int \frac {x^4}{x^4+6 x^2 \log ((-1+x) x)+9 \log ^2((-1+x) x)+x \log ^2((-1+x) x)} \, dx-\int \left (-\frac {9 \left (x^2+6 \log ((-1+x) x)\right )}{x^4+6 x^2 \log ((-1+x) x)+9 \log ^2((-1+x) x)+x \log ^2((-1+x) x)}+\frac {81 \left (x^2+6 \log ((-1+x) x)\right )}{(9+x) \left (x^4+6 x^2 \log ((-1+x) x)+9 \log ^2((-1+x) x)+x \log ^2((-1+x) x)\right )}+\frac {x^3+6 x \log ((-1+x) x)}{x^4+6 x^2 \log ((-1+x) x)+9 \log ^2((-1+x) x)+x \log ^2((-1+x) x)}+\frac {x^4+6 x^2 \log ((-1+x) x)}{x^4+6 x^2 \log ((-1+x) x)+9 \log ^2((-1+x) x)+x \log ^2((-1+x) x)}\right ) \, dx\\ &=x+\log (9+x)-2 \int \frac {-1+2 x}{(-1+x) x \log ((-1+x) x)} \, dx+2 \int \frac {6 x^2+9 \log ((-1+x) x)+x \log ((-1+x) x)}{(-1+x) \left (x^4+6 x^2 \log ((-1+x) x)+9 \log ^2((-1+x) x)+x \log ^2((-1+x) x)\right )} \, dx+2 \int \frac {6 x^2+9 \log ((-1+x) x)+x \log ((-1+x) x)}{x \left (x^4+6 x^2 \log ((-1+x) x)+9 \log ^2((-1+x) x)+x \log ^2((-1+x) x)\right )} \, dx+4 \int \frac {x^3}{x^4+6 x^2 \log ((-1+x) x)+9 \log ^2((-1+x) x)+x \log ^2((-1+x) x)} \, dx-6 \int \frac {1}{x^4+6 x^2 \log ((-1+x) x)+9 \log ^2((-1+x) x)+x \log ^2((-1+x) x)} \, dx-6 \int \frac {1}{(-1+x) \left (x^4+6 x^2 \log ((-1+x) x)+9 \log ^2((-1+x) x)+x \log ^2((-1+x) x)\right )} \, dx+6 \int \frac {x^2 \log ((-1+x) x)}{x^4+6 x^2 \log ((-1+x) x)+9 \log ^2((-1+x) x)+x \log ^2((-1+x) x)} \, dx+9 \int \frac {x^2+6 \log ((-1+x) x)}{x^4+6 x^2 \log ((-1+x) x)+9 \log ^2((-1+x) x)+x \log ^2((-1+x) x)} \, dx-12 \int \frac {x}{x^4+6 x^2 \log ((-1+x) x)+9 \log ^2((-1+x) x)+x \log ^2((-1+x) x)} \, dx+12 \int \frac {x \log ((-1+x) x)}{x^4+6 x^2 \log ((-1+x) x)+9 \log ^2((-1+x) x)+x \log ^2((-1+x) x)} \, dx-81 \int \frac {x^2+6 \log ((-1+x) x)}{(9+x) \left (x^4+6 x^2 \log ((-1+x) x)+9 \log ^2((-1+x) x)+x \log ^2((-1+x) x)\right )} \, dx+\int \frac {x^4}{x^4+6 x^2 \log ((-1+x) x)+9 \log ^2((-1+x) x)+x \log ^2((-1+x) x)} \, dx-\int \frac {x^3+6 x \log ((-1+x) x)}{x^4+6 x^2 \log ((-1+x) x)+9 \log ^2((-1+x) x)+x \log ^2((-1+x) x)} \, dx-\int \frac {x^4+6 x^2 \log ((-1+x) x)}{x^4+6 x^2 \log ((-1+x) x)+9 \log ^2((-1+x) x)+x \log ^2((-1+x) x)} \, dx\\ &=\text {Rest of rules removed due to large latex content} \end {aligned} \end {gather*}
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Mathematica [A] time = 0.09, size = 47, normalized size = 1.88 \begin {gather*} x-2 \log (\log ((-1+x) x))+\log \left (x^4+6 x^2 \log ((-1+x) x)+9 \log ^2((-1+x) x)+x \log ^2((-1+x) x)\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.90, size = 55, normalized size = 2.20 \begin {gather*} x + \log \left (x + 9\right ) + \log \left (\frac {x^{4} + 6 \, x^{2} \log \left (x^{2} - x\right ) + {\left (x + 9\right )} \log \left (x^{2} - x\right )^{2}}{x + 9}\right ) - 2 \, \log \left (\log \left (x^{2} - x\right )\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.70, size = 55, normalized size = 2.20 \begin {gather*} x + \log \left (x^{4} + 6 \, x^{2} \log \left (x^{2} - x\right ) + x \log \left (x^{2} - x\right )^{2} + 9 \, \log \left (x^{2} - x\right )^{2}\right ) - 2 \, \log \left (\log \left (x^{2} - x\right )\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.07, size = 56, normalized size = 2.24
method | result | size |
norman | \(x -2 \ln \left (\ln \left (x^{2}-x \right )\right )+\ln \left (x^{4}+\ln \left (x^{2}-x \right )^{2} x +6 x^{2} \ln \left (x^{2}-x \right )+9 \ln \left (x^{2}-x \right )^{2}\right )\) | \(56\) |
risch | \(x +\ln \left (x +9\right )-2 \ln \left (\ln \left (x^{2}-x \right )\right )+\ln \left (\ln \left (x^{2}-x \right )^{2}+\frac {6 x^{2} \ln \left (x^{2}-x \right )}{x +9}+\frac {x^{4}}{x +9}\right )\) | \(57\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.64, size = 70, normalized size = 2.80 \begin {gather*} x + \log \left (x + 9\right ) + \log \left (\frac {x^{4} + {\left (x + 9\right )} \log \left (x - 1\right )^{2} + 6 \, x^{2} \log \relax (x) + {\left (x + 9\right )} \log \relax (x)^{2} + 2 \, {\left (3 \, x^{2} + {\left (x + 9\right )} \log \relax (x)\right )} \log \left (x - 1\right )}{x + 9}\right ) - 2 \, \log \left (\log \left (x - 1\right ) + \log \relax (x)\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.79, size = 184, normalized size = 7.36 \begin {gather*} x-3\,\ln \left (x-1\right )-\ln \left (x-\frac {1}{2}\right )-2\,\ln \left (\frac {x^3\,\ln \left (x\,\left (x-1\right )\right )\,\left (9\,x^7+126\,x^6-231\,x^5+952\,x^4-1000\,x^3+1444\,x^2-1224\,x+324\right )}{{\left (x-1\right )}^2\,{\left (x+9\right )}^4}\right )+2\,\ln \left (9\,x^7+126\,x^6-231\,x^5+952\,x^4-1000\,x^3+1444\,x^2-1224\,x+324\right )+\ln \left (\frac {\left (2\,x-1\right )\,\left (x^4+6\,x^2\,\ln \left (x\,\left (x-1\right )\right )+x\,{\ln \left (x\,\left (x-1\right )\right )}^2+9\,{\ln \left (x\,\left (x-1\right )\right )}^2\right )}{x\,\left (x^2+8\,x-9\right )}\right )-\mathrm {atan}\left (\frac {x\,735002345703105317{}\mathrm {i}-177147{}\mathrm {i}}{735002345703144683\,x+177147}\right )\,14{}\mathrm {i} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: PolynomialError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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