Optimal. Leaf size=27 \[ \left (4+\log \left (e^{\frac {2 x^2}{\left (1-\frac {1}{x}\right )^2 \log ^2(-1+x)}}\right )\right )^2 \]
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Rubi [B] Leaf count is larger than twice the leaf count of optimal. \(503\) vs. \(2(27)=54\).
time = 2.13, antiderivative size = 503, normalized size of antiderivative = 18.63, number of steps
used = 266, number of rules used = 17, integrand size = 105, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.162, Rules used = {6820, 12,
6874, 2458, 2395, 2334, 2335, 2343, 2346, 2209, 2339, 30, 2465, 2436, 2437, 2447, 2446}
\begin {gather*} \frac {16 x^3 (1-x)}{3 \log ^3(x-1)}+\frac {32 x^3 (1-x)}{3 \log ^2(x-1)}+\frac {128 x^3 (1-x)}{3 \log (x-1)}+\frac {16 x^2 (1-x)}{\log ^3(x-1)}+\frac {16 x^2 (1-x)}{\log ^2(x-1)}+\frac {16 x^2 (1-x)}{\log (x-1)}+\frac {4 (1-x)^4}{\log ^4(x-1)}-\frac {32 (1-x)^3}{\log ^4(x-1)}+\frac {112 (1-x)^2}{\log ^4(x-1)}-\frac {224 (1-x)}{\log ^4(x-1)}+\frac {280}{\log ^4(x-1)}-\frac {224}{(1-x) \log ^4(x-1)}+\frac {112}{(1-x)^2 \log ^4(x-1)}-\frac {32}{(1-x)^3 \log ^4(x-1)}+\frac {4}{(1-x)^4 \log ^4(x-1)}+\frac {16 (1-x)^4}{3 \log ^3(x-1)}-\frac {32 (1-x)^3}{\log ^3(x-1)}+\frac {224 (1-x)^2}{3 \log ^3(x-1)}+\frac {80 x (1-x)}{3 \log ^3(x-1)}-\frac {48 (1-x)}{\log ^3(x-1)}+\frac {32 (1-x)^4}{3 \log ^2(x-1)}-\frac {48 (1-x)^3}{\log ^2(x-1)}+\frac {272 (1-x)^2}{3 \log ^2(x-1)}+\frac {32 x (1-x)}{3 \log ^2(x-1)}-\frac {304 (1-x)}{3 \log ^2(x-1)}+\frac {96}{\log ^2(x-1)}-\frac {64}{(1-x) \log ^2(x-1)}+\frac {16}{(1-x)^2 \log ^2(x-1)}+\frac {128 (1-x)^4}{3 \log (x-1)}-\frac {144 (1-x)^3}{\log (x-1)}+\frac {544 (1-x)^2}{3 \log (x-1)}+\frac {64 x (1-x)}{3 \log (x-1)}-\frac {80 (1-x)}{\log (x-1)} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 30
Rule 2209
Rule 2334
Rule 2335
Rule 2339
Rule 2343
Rule 2346
Rule 2395
Rule 2436
Rule 2437
Rule 2446
Rule 2447
Rule 2458
Rule 2465
Rule 6820
Rule 6874
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {16 x^3 \left (x^5-(-2+x) x^4 \log (-1+x)+2 (-1+x)^2 x \log ^2(-1+x)-2 (-2+x) (-1+x)^2 \log ^3(-1+x)\right )}{(1-x)^5 \log ^5(-1+x)} \, dx\\ &=16 \int \frac {x^3 \left (x^5-(-2+x) x^4 \log (-1+x)+2 (-1+x)^2 x \log ^2(-1+x)-2 (-2+x) (-1+x)^2 \log ^3(-1+x)\right )}{(1-x)^5 \log ^5(-1+x)} \, dx\\ &=16 \int \left (-\frac {x^8}{(-1+x)^5 \log ^5(-1+x)}+\frac {(-2+x) x^7}{(-1+x)^5 \log ^4(-1+x)}-\frac {2 x^4}{(-1+x)^3 \log ^3(-1+x)}+\frac {2 (-2+x) x^3}{(-1+x)^3 \log ^2(-1+x)}\right ) \, dx\\ &=-\left (16 \int \frac {x^8}{(-1+x)^5 \log ^5(-1+x)} \, dx\right )+16 \int \frac {(-2+x) x^7}{(-1+x)^5 \log ^4(-1+x)} \, dx-32 \int \frac {x^4}{(-1+x)^3 \log ^3(-1+x)} \, dx+32 \int \frac {(-2+x) x^3}{(-1+x)^3 \log ^2(-1+x)} \, dx\\ &=16 \int \left (\frac {5}{\log ^4(-1+x)}-\frac {1}{(-1+x)^5 \log ^4(-1+x)}-\frac {6}{(-1+x)^4 \log ^4(-1+x)}-\frac {14}{(-1+x)^3 \log ^4(-1+x)}-\frac {14}{(-1+x)^2 \log ^4(-1+x)}+\frac {5 x}{\log ^4(-1+x)}+\frac {3 x^2}{\log ^4(-1+x)}+\frac {x^3}{\log ^4(-1+x)}\right ) \, dx-16 \text {Subst}\left (\int \frac {(1+x)^8}{x^5 \log ^5(x)} \, dx,x,-1+x\right )+32 \int \left (\frac {1}{\log ^2(-1+x)}-\frac {1}{(-1+x)^3 \log ^2(-1+x)}-\frac {2}{(-1+x)^2 \log ^2(-1+x)}+\frac {x}{\log ^2(-1+x)}\right ) \, dx-32 \text {Subst}\left (\int \frac {(1+x)^4}{x^3 \log ^3(x)} \, dx,x,-1+x\right )\\ &=-\left (16 \int \frac {1}{(-1+x)^5 \log ^4(-1+x)} \, dx\right )+16 \int \frac {x^3}{\log ^4(-1+x)} \, dx-16 \text {Subst}\left (\int \left (\frac {56}{\log ^5(x)}+\frac {1}{x^5 \log ^5(x)}+\frac {8}{x^4 \log ^5(x)}+\frac {28}{x^3 \log ^5(x)}+\frac {56}{x^2 \log ^5(x)}+\frac {70}{x \log ^5(x)}+\frac {28 x}{\log ^5(x)}+\frac {8 x^2}{\log ^5(x)}+\frac {x^3}{\log ^5(x)}\right ) \, dx,x,-1+x\right )+32 \int \frac {1}{\log ^2(-1+x)} \, dx-32 \int \frac {1}{(-1+x)^3 \log ^2(-1+x)} \, dx+32 \int \frac {x}{\log ^2(-1+x)} \, dx-32 \text {Subst}\left (\int \left (\frac {4}{\log ^3(x)}+\frac {1}{x^3 \log ^3(x)}+\frac {4}{x^2 \log ^3(x)}+\frac {6}{x \log ^3(x)}+\frac {x}{\log ^3(x)}\right ) \, dx,x,-1+x\right )+48 \int \frac {x^2}{\log ^4(-1+x)} \, dx-64 \int \frac {1}{(-1+x)^2 \log ^2(-1+x)} \, dx+80 \int \frac {1}{\log ^4(-1+x)} \, dx+80 \int \frac {x}{\log ^4(-1+x)} \, dx-96 \int \frac {1}{(-1+x)^4 \log ^4(-1+x)} \, dx-224 \int \frac {1}{(-1+x)^3 \log ^4(-1+x)} \, dx-224 \int \frac {1}{(-1+x)^2 \log ^4(-1+x)} \, dx\\ &=\text {Rest of rules removed due to large latex content} \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.11, size = 36, normalized size = 1.33 \begin {gather*} 16 \left (\frac {x^8}{4 (-1+x)^4 \log ^4(-1+x)}+\frac {x^4}{(-1+x)^2 \log ^2(-1+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. \(169\) vs.
\(2(27)=54\).
time = 0.12, size = 170, normalized size = 6.30
method | result | size |
risch | \(\frac {4 \left (x^{4}+4 \ln \left (x -1\right )^{2} x^{2}-8 \ln \left (x -1\right )^{2} x +4 \ln \left (x -1\right )^{2}\right ) x^{4}}{\left (x^{4}-4 x^{3}+6 x^{2}-4 x +1\right ) \ln \left (x -1\right )^{4}}\) | \(64\) |
derivativedivides | \(\frac {16 \left (x -1\right )^{2}}{\ln \left (x -1\right )^{2}}+\frac {4 \left (x -1\right )^{4}}{\ln \left (x -1\right )^{4}}+\frac {64}{\left (x -1\right ) \ln \left (x -1\right )^{2}}+\frac {224}{\left (x -1\right ) \ln \left (x -1\right )^{4}}+\frac {64 x -64}{\ln \left (x -1\right )^{2}}+\frac {32 \left (x -1\right )^{3}}{\ln \left (x -1\right )^{4}}+\frac {96}{\ln \left (x -1\right )^{2}}+\frac {112 \left (x -1\right )^{2}}{\ln \left (x -1\right )^{4}}+\frac {224 x -224}{\ln \left (x -1\right )^{4}}+\frac {16}{\left (x -1\right )^{2} \ln \left (x -1\right )^{2}}+\frac {280}{\ln \left (x -1\right )^{4}}+\frac {112}{\left (x -1\right )^{2} \ln \left (x -1\right )^{4}}+\frac {32}{\left (x -1\right )^{3} \ln \left (x -1\right )^{4}}+\frac {4}{\left (x -1\right )^{4} \ln \left (x -1\right )^{4}}\) | \(170\) |
default | \(\frac {16 \left (x -1\right )^{2}}{\ln \left (x -1\right )^{2}}+\frac {4 \left (x -1\right )^{4}}{\ln \left (x -1\right )^{4}}+\frac {64}{\left (x -1\right ) \ln \left (x -1\right )^{2}}+\frac {224}{\left (x -1\right ) \ln \left (x -1\right )^{4}}+\frac {64 x -64}{\ln \left (x -1\right )^{2}}+\frac {32 \left (x -1\right )^{3}}{\ln \left (x -1\right )^{4}}+\frac {96}{\ln \left (x -1\right )^{2}}+\frac {112 \left (x -1\right )^{2}}{\ln \left (x -1\right )^{4}}+\frac {224 x -224}{\ln \left (x -1\right )^{4}}+\frac {16}{\left (x -1\right )^{2} \ln \left (x -1\right )^{2}}+\frac {280}{\ln \left (x -1\right )^{4}}+\frac {112}{\left (x -1\right )^{2} \ln \left (x -1\right )^{4}}+\frac {32}{\left (x -1\right )^{3} \ln \left (x -1\right )^{4}}+\frac {4}{\left (x -1\right )^{4} \ln \left (x -1\right )^{4}}\) | \(170\) |
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. 52 vs.
\(2 (24) = 48\).
time = 0.32, size = 52, normalized size = 1.93 \begin {gather*} \frac {4 \, {\left (x^{8} + 4 \, {\left (x^{6} - 2 \, x^{5} + x^{4}\right )} \log \left (x - 1\right )^{2}\right )}}{{\left (x^{4} - 4 \, x^{3} + 6 \, x^{2} - 4 \, x + 1\right )} \log \left (x - 1\right )^{4}} \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. 52 vs.
\(2 (24) = 48\).
time = 0.33, size = 52, normalized size = 1.93 \begin {gather*} \frac {4 \, {\left (x^{8} + 4 \, {\left (x^{6} - 2 \, x^{5} + x^{4}\right )} \log \left (x - 1\right )^{2}\right )}}{{\left (x^{4} - 4 \, x^{3} + 6 \, x^{2} - 4 \, x + 1\right )} \log \left (x - 1\right )^{4}} \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. 51 vs.
\(2 (24) = 48\).
time = 0.09, size = 51, normalized size = 1.89 \begin {gather*} \frac {4 x^{8} + \left (16 x^{6} - 32 x^{5} + 16 x^{4}\right ) \log {\left (x - 1 \right )}^{2}}{\left (x^{4} - 4 x^{3} + 6 x^{2} - 4 x + 1\right ) \log {\left (x - 1 \right )}^{4}} \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. 89 vs.
\(2 (24) = 48\).
time = 0.41, size = 89, normalized size = 3.30 \begin {gather*} \frac {4 \, {\left (x^{8} + 4 \, x^{6} \log \left (x - 1\right )^{2} - 8 \, x^{5} \log \left (x - 1\right )^{2} + 4 \, x^{4} \log \left (x - 1\right )^{2}\right )}}{x^{4} \log \left (x - 1\right )^{4} - 4 \, x^{3} \log \left (x - 1\right )^{4} + 6 \, x^{2} \log \left (x - 1\right )^{4} - 4 \, x \log \left (x - 1\right )^{4} + \log \left (x - 1\right )^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.17, size = 33, normalized size = 1.22 \begin {gather*} \frac {16\,x^4}{{\ln \left (x-1\right )}^2\,{\left (x-1\right )}^2}+\frac {4\,x^8}{{\ln \left (x-1\right )}^4\,{\left (x-1\right )}^4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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