Optimal. Leaf size=28 \[ x^2-\left (x+\left (-5+\frac {-3+x}{x}\right ) (-3 x+\log (-1+x))\right )^2 \]
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Rubi [B] time = 0.63, antiderivative size = 86, normalized size of antiderivative = 3.07, number of steps used = 31, number of rules used = 23, integrand size = 74, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.311, Rules used = {1593, 6742, 1620, 2418, 2389, 2295, 2390, 2301, 2395, 36, 31, 29, 2394, 2315, 2416, 2398, 2411, 2347, 2344, 2317, 2391, 2314, 2397} \begin {gather*} -168 x^2-\frac {9 \log ^2(x-1)}{x^2}-234 x-40 \log ^2(x-1)-\frac {24 (1-x) \log ^2(x-1)}{x}+272 \log (1-x)-104 (1-x) \log (x-1)+\frac {18 (1-x) \log (x-1)}{x}+\frac {36 \log (x-1)}{x} \end {gather*}
Antiderivative was successfully verified.
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Rule 29
Rule 31
Rule 36
Rule 1593
Rule 1620
Rule 2295
Rule 2301
Rule 2314
Rule 2315
Rule 2317
Rule 2344
Rule 2347
Rule 2389
Rule 2390
Rule 2391
Rule 2394
Rule 2395
Rule 2397
Rule 2398
Rule 2411
Rule 2416
Rule 2418
Rule 6742
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {54 x^2+384 x^3+206 x^4-336 x^5+\left (36 x-102 x^2-136 x^3+104 x^4\right ) \log (-1+x)+\left (-18-6 x+24 x^2\right ) \log ^2(-1+x)}{(-1+x) x^3} \, dx\\ &=\int \left (-\frac {2 \left (-27-192 x-103 x^2+168 x^3\right )}{(-1+x) x}+\frac {2 \left (18-51 x-68 x^2+52 x^3\right ) \log (-1+x)}{(-1+x) x^2}+\frac {6 (3+4 x) \log ^2(-1+x)}{x^3}\right ) \, dx\\ &=-\left (2 \int \frac {-27-192 x-103 x^2+168 x^3}{(-1+x) x} \, dx\right )+2 \int \frac {\left (18-51 x-68 x^2+52 x^3\right ) \log (-1+x)}{(-1+x) x^2} \, dx+6 \int \frac {(3+4 x) \log ^2(-1+x)}{x^3} \, dx\\ &=-\left (2 \int \left (65-\frac {154}{-1+x}+\frac {27}{x}+168 x\right ) \, dx\right )+2 \int \left (52 \log (-1+x)-\frac {49 \log (-1+x)}{-1+x}-\frac {18 \log (-1+x)}{x^2}+\frac {33 \log (-1+x)}{x}\right ) \, dx+6 \int \left (\frac {3 \log ^2(-1+x)}{x^3}+\frac {4 \log ^2(-1+x)}{x^2}\right ) \, dx\\ &=-130 x-168 x^2+308 \log (1-x)-54 \log (x)+18 \int \frac {\log ^2(-1+x)}{x^3} \, dx+24 \int \frac {\log ^2(-1+x)}{x^2} \, dx-36 \int \frac {\log (-1+x)}{x^2} \, dx+66 \int \frac {\log (-1+x)}{x} \, dx-98 \int \frac {\log (-1+x)}{-1+x} \, dx+104 \int \log (-1+x) \, dx\\ &=-130 x-168 x^2+308 \log (1-x)+\frac {36 \log (-1+x)}{x}-\frac {9 \log ^2(-1+x)}{x^2}-\frac {24 (1-x) \log ^2(-1+x)}{x}-54 \log (x)+66 \log (-1+x) \log (x)+18 \int \frac {\log (-1+x)}{(-1+x) x^2} \, dx-36 \int \frac {1}{(-1+x) x} \, dx-48 \int \frac {\log (-1+x)}{x} \, dx-66 \int \frac {\log (x)}{-1+x} \, dx-98 \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,-1+x\right )+104 \operatorname {Subst}(\int \log (x) \, dx,x,-1+x)\\ &=-234 x-168 x^2+308 \log (1-x)-104 (1-x) \log (-1+x)+\frac {36 \log (-1+x)}{x}-49 \log ^2(-1+x)-\frac {9 \log ^2(-1+x)}{x^2}-\frac {24 (1-x) \log ^2(-1+x)}{x}-54 \log (x)+18 \log (-1+x) \log (x)+66 \text {Li}_2(1-x)+18 \operatorname {Subst}\left (\int \frac {\log (x)}{x (1+x)^2} \, dx,x,-1+x\right )-36 \int \frac {1}{-1+x} \, dx+36 \int \frac {1}{x} \, dx+48 \int \frac {\log (x)}{-1+x} \, dx\\ &=-234 x-168 x^2+272 \log (1-x)-104 (1-x) \log (-1+x)+\frac {36 \log (-1+x)}{x}-49 \log ^2(-1+x)-\frac {9 \log ^2(-1+x)}{x^2}-\frac {24 (1-x) \log ^2(-1+x)}{x}-18 \log (x)+18 \log (-1+x) \log (x)+18 \text {Li}_2(1-x)-18 \operatorname {Subst}\left (\int \frac {\log (x)}{(1+x)^2} \, dx,x,-1+x\right )+18 \operatorname {Subst}\left (\int \frac {\log (x)}{x (1+x)} \, dx,x,-1+x\right )\\ &=-234 x-168 x^2+272 \log (1-x)-104 (1-x) \log (-1+x)+\frac {36 \log (-1+x)}{x}+\frac {18 (1-x) \log (-1+x)}{x}-49 \log ^2(-1+x)-\frac {9 \log ^2(-1+x)}{x^2}-\frac {24 (1-x) \log ^2(-1+x)}{x}-18 \log (x)+18 \log (-1+x) \log (x)+18 \text {Li}_2(1-x)+18 \operatorname {Subst}\left (\int \frac {1}{1+x} \, dx,x,-1+x\right )+18 \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,-1+x\right )-18 \operatorname {Subst}\left (\int \frac {\log (x)}{1+x} \, dx,x,-1+x\right )\\ &=-234 x-168 x^2+272 \log (1-x)-104 (1-x) \log (-1+x)+\frac {36 \log (-1+x)}{x}+\frac {18 (1-x) \log (-1+x)}{x}-40 \log ^2(-1+x)-\frac {9 \log ^2(-1+x)}{x^2}-\frac {24 (1-x) \log ^2(-1+x)}{x}+18 \text {Li}_2(1-x)+18 \operatorname {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,-1+x\right )\\ &=-234 x-168 x^2+272 \log (1-x)-104 (1-x) \log (-1+x)+\frac {36 \log (-1+x)}{x}+\frac {18 (1-x) \log (-1+x)}{x}-40 \log ^2(-1+x)-\frac {9 \log ^2(-1+x)}{x^2}-\frac {24 (1-x) \log ^2(-1+x)}{x}\\ \end {aligned} \end {gather*}
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Mathematica [B] time = 0.12, size = 73, normalized size = 2.61 \begin {gather*} -2 \left (117 x+84 x^2+27 \log (1-x)-102 \log (-1+x)-\frac {27 \log (-1+x)}{x}-52 x \log (-1+x)+8 \log ^2(-1+x)+\frac {9 \log ^2(-1+x)}{2 x^2}+\frac {12 \log ^2(-1+x)}{x}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.61, size = 53, normalized size = 1.89 \begin {gather*} -\frac {168 \, x^{4} + 234 \, x^{3} + {\left (16 \, x^{2} + 24 \, x + 9\right )} \log \left (x - 1\right )^{2} - 2 \, {\left (52 \, x^{3} + 75 \, x^{2} + 27 \, x\right )} \log \left (x - 1\right )}{x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int -\frac {2 \, {\left (168 \, x^{5} - 103 \, x^{4} - 192 \, x^{3} - 3 \, {\left (4 \, x^{2} - x - 3\right )} \log \left (x - 1\right )^{2} - 27 \, x^{2} - {\left (52 \, x^{4} - 68 \, x^{3} - 51 \, x^{2} + 18 \, x\right )} \log \left (x - 1\right )\right )}}{x^{4} - x^{3}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.22, size = 53, normalized size = 1.89
method | result | size |
risch | \(-\frac {\left (16 x^{2}+24 x +9\right ) \ln \left (x -1\right )^{2}}{x^{2}}+\frac {2 \left (52 x^{2}+27\right ) \ln \left (x -1\right )}{x}-168 x^{2}-234 x +150 \ln \left (x -1\right )\) | \(53\) |
norman | \(\frac {150 \ln \left (x -1\right ) x^{2}-234 x^{3}-168 x^{4}-9 \ln \left (x -1\right )^{2}-16 x^{2} \ln \left (x -1\right )^{2}+54 \ln \left (x -1\right ) x +104 \ln \left (x -1\right ) x^{3}-24 \ln \left (x -1\right )^{2} x}{x^{2}}\) | \(69\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.44, size = 53, normalized size = 1.89 \begin {gather*} -\frac {168 \, x^{4} + 234 \, x^{3} + {\left (16 \, x^{2} + 24 \, x + 9\right )} \log \left (x - 1\right )^{2} - 2 \, {\left (52 \, x^{3} + 75 \, x^{2} + 27 \, x\right )} \log \left (x - 1\right )}{x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.16, size = 61, normalized size = 2.18 \begin {gather*} 150\,\ln \left (x-1\right )-234\,x+104\,x\,\ln \left (x-1\right )+\frac {54\,\ln \left (x-1\right )}{x}-16\,{\ln \left (x-1\right )}^2-168\,x^2-\frac {24\,{\ln \left (x-1\right )}^2}{x}-\frac {9\,{\ln \left (x-1\right )}^2}{x^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.22, size = 49, normalized size = 1.75 \begin {gather*} - 168 x^{2} - 234 x + 150 \log {\left (x - 1 \right )} + \frac {\left (104 x^{2} + 54\right ) \log {\left (x - 1 \right )}}{x} + \frac {\left (- 16 x^{2} - 24 x - 9\right ) \log {\left (x - 1 \right )}^{2}}{x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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