Optimal. Leaf size=24 \[ \left (x-5 x^2\right ) \log ^2\left (\log \left (e^{2 (1-x) x^2}\right )\right ) \]
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Rubi [B] Leaf count is larger than twice the leaf count of optimal. \(175\) vs. \(2(24)=48\).
time = 0.41, antiderivative size = 175, normalized size of antiderivative = 7.29, number of steps
used = 47, number of rules used = 18, integrand size = 55, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.327, Rules used = {6874, 2593,
2465, 2436, 2332, 2437, 2338, 2442, 45, 2404, 2352, 2341, 78, 2584, 2458, 2372, 2463, 2441}
\begin {gather*} \frac {5 x^2}{2}-\frac {1}{20} (1-10 x)^2 \log ^2\left (2 (1-x) x^2\right )-5 x^2 \log (x-1)-\frac {1}{10} \log (1-x) \left (-\log \left (2 (1-x) x^2\right )+\log (x-1)+2 \log (x)\right )-\frac {1}{5} \log (x) \left (-\log \left (2 (1-x) x^2\right )+\log (x-1)+2 \log (x)\right )-\frac {5}{2} (1-x)^2-5 x+\frac {1}{20} \log ^2(x-1)+\frac {\log ^2(x)}{5}+5 (1-x)^2 \log (x-1)-10 (1-x) \log (x-1)+5 \log (1-x)+\frac {1}{5} \log (x-1) \log (x) \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 78
Rule 2332
Rule 2338
Rule 2341
Rule 2352
Rule 2372
Rule 2404
Rule 2436
Rule 2437
Rule 2441
Rule 2442
Rule 2458
Rule 2463
Rule 2465
Rule 2584
Rule 2593
Rule 6874
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \left (-\frac {2 (-2+3 x) (-1+5 x) \log \left (-2 (-1+x) x^2\right )}{-1+x}-(-1+10 x) \log ^2\left (-2 (-1+x) x^2\right )\right ) \, dx\\ &=-\left (2 \int \frac {(-2+3 x) (-1+5 x) \log \left (-2 (-1+x) x^2\right )}{-1+x} \, dx\right )-\int (-1+10 x) \log ^2\left (-2 (-1+x) x^2\right ) \, dx\\ &=-\frac {1}{20} (1-10 x)^2 \log ^2\left (2 (1-x) x^2\right )+\frac {1}{10} \int \frac {(-1+10 x)^2 \log \left (-2 (-1+x) x^2\right )}{-1+x} \, dx+\frac {1}{5} \int \frac {(-1+10 x)^2 \log \left (-2 (-1+x) x^2\right )}{x} \, dx-2 \int \frac {(-2+3 x) (-1+5 x) \log (-1+x)}{-1+x} \, dx-4 \int \frac {(-2+3 x) (-1+5 x) \log (x)}{-1+x} \, dx+\left (2 \left (\log (-1+x)+2 \log (x)-\log \left (-2 (-1+x) x^2\right )\right )\right ) \int \frac {(-2+3 x) (-1+5 x)}{-1+x} \, dx\\ &=-\frac {1}{20} (1-10 x)^2 \log ^2\left (2 (1-x) x^2\right )+\frac {1}{10} \int \frac {(-1+10 x)^2 \log (-1+x)}{-1+x} \, dx+\frac {1}{5} \int \frac {(-1+10 x)^2 \log (-1+x)}{x} \, dx+\frac {1}{5} \int \frac {(-1+10 x)^2 \log (x)}{-1+x} \, dx+\frac {2}{5} \int \frac {(-1+10 x)^2 \log (x)}{x} \, dx-2 \int \left (2 \log (-1+x)+\frac {4 \log (-1+x)}{-1+x}+15 x \log (-1+x)\right ) \, dx-4 \int \left (2 \log (x)+\frac {4 \log (x)}{-1+x}+15 x \log (x)\right ) \, dx+\left (2 \left (\log (-1+x)+2 \log (x)-\log \left (-2 (-1+x) x^2\right )\right )\right ) \int \left (2+\frac {4}{-1+x}+15 x\right ) \, dx+\frac {1}{10} \left (-\log (-1+x)-2 \log (x)+\log \left (-2 (-1+x) x^2\right )\right ) \int \frac {(-1+10 x)^2}{-1+x} \, dx+\frac {1}{5} \left (-\log (-1+x)-2 \log (x)+\log \left (-2 (-1+x) x^2\right )\right ) \int \frac {(-1+10 x)^2}{x} \, dx\\ &=-\frac {2}{5} \left (20 x-50 x^2-\log (x)\right ) \log (x)+4 x \left (\log (-1+x)+2 \log (x)-\log \left (2 (1-x) x^2\right )\right )+15 x^2 \left (\log (-1+x)+2 \log (x)-\log \left (2 (1-x) x^2\right )\right )+8 \log (1-x) \left (\log (-1+x)+2 \log (x)-\log \left (2 (1-x) x^2\right )\right )-\frac {1}{20} (1-10 x)^2 \log ^2\left (2 (1-x) x^2\right )+\frac {1}{10} \text {Subst}\left (\int \frac {(9+10 x)^2 \log (x)}{x} \, dx,x,-1+x\right )+\frac {1}{5} \int \left (-20 \log (-1+x)+\frac {\log (-1+x)}{x}+100 x \log (-1+x)\right ) \, dx+\frac {1}{5} \int \left (80 \log (x)+\frac {81 \log (x)}{-1+x}+100 x \log (x)\right ) \, dx-\frac {2}{5} \int \left (-20+50 x+\frac {\log (x)}{x}\right ) \, dx-4 \int \log (-1+x) \, dx-8 \int \frac {\log (-1+x)}{-1+x} \, dx-8 \int \log (x) \, dx-16 \int \frac {\log (x)}{-1+x} \, dx-30 \int x \log (-1+x) \, dx-60 \int x \log (x) \, dx+\frac {1}{10} \left (-\log (-1+x)-2 \log (x)+\log \left (-2 (-1+x) x^2\right )\right ) \int \left (80+\frac {81}{-1+x}+100 x\right ) \, dx+\frac {1}{5} \left (-\log (-1+x)-2 \log (x)+\log \left (-2 (-1+x) x^2\right )\right ) \int \left (-20+\frac {1}{x}+100 x\right ) \, dx\\ &=16 x+5 x^2-15 x^2 \log (-1+x)-\frac {1}{10} \left (180 (1-x)-50 (1-x)^2-81 \log (-1+x)\right ) \log (-1+x)-8 x \log (x)-30 x^2 \log (x)-\frac {2}{5} \left (20 x-50 x^2-\log (x)\right ) \log (x)-\frac {1}{10} \log (1-x) \left (\log (-1+x)+2 \log (x)-\log \left (2 (1-x) x^2\right )\right )-\frac {1}{5} \log (x) \left (\log (-1+x)+2 \log (x)-\log \left (2 (1-x) x^2\right )\right )-\frac {1}{20} (1-10 x)^2 \log ^2\left (2 (1-x) x^2\right )+16 \text {Li}_2(1-x)-\frac {1}{10} \text {Subst}\left (\int \left (180+50 x+\frac {81 \log (x)}{x}\right ) \, dx,x,-1+x\right )+\frac {1}{5} \int \frac {\log (-1+x)}{x} \, dx-\frac {2}{5} \int \frac {\log (x)}{x} \, dx-4 \int \log (-1+x) \, dx-4 \text {Subst}(\int \log (x) \, dx,x,-1+x)-8 \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,-1+x\right )+15 \int \frac {x^2}{-1+x} \, dx+16 \int \log (x) \, dx+\frac {81}{5} \int \frac {\log (x)}{-1+x} \, dx+20 \int x \log (-1+x) \, dx+20 \int x \log (x) \, dx\\ &=-\frac {5}{2} (1-x)^2-14 x+4 (1-x) \log (-1+x)-5 x^2 \log (-1+x)-\frac {1}{10} \left (180 (1-x)-50 (1-x)^2-81 \log (-1+x)\right ) \log (-1+x)-4 \log ^2(-1+x)+8 x \log (x)-20 x^2 \log (x)+\frac {1}{5} \log (-1+x) \log (x)-\frac {2}{5} \left (20 x-50 x^2-\log (x)\right ) \log (x)-\frac {\log ^2(x)}{5}-\frac {1}{10} \log (1-x) \left (\log (-1+x)+2 \log (x)-\log \left (2 (1-x) x^2\right )\right )-\frac {1}{5} \log (x) \left (\log (-1+x)+2 \log (x)-\log \left (2 (1-x) x^2\right )\right )-\frac {1}{20} (1-10 x)^2 \log ^2\left (2 (1-x) x^2\right )-\frac {\text {Li}_2(1-x)}{5}-\frac {1}{5} \int \frac {\log (x)}{-1+x} \, dx-4 \text {Subst}(\int \log (x) \, dx,x,-1+x)-\frac {81}{10} \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,-1+x\right )-10 \int \frac {x^2}{-1+x} \, dx+15 \int \left (1+\frac {1}{-1+x}+x\right ) \, dx\\ &=-\frac {5}{2} (1-x)^2+5 x+\frac {15 x^2}{2}+15 \log (1-x)+8 (1-x) \log (-1+x)-5 x^2 \log (-1+x)-\frac {1}{10} \left (180 (1-x)-50 (1-x)^2-81 \log (-1+x)\right ) \log (-1+x)-\frac {161}{20} \log ^2(-1+x)+8 x \log (x)-20 x^2 \log (x)+\frac {1}{5} \log (-1+x) \log (x)-\frac {2}{5} \left (20 x-50 x^2-\log (x)\right ) \log (x)-\frac {\log ^2(x)}{5}-\frac {1}{10} \log (1-x) \left (\log (-1+x)+2 \log (x)-\log \left (2 (1-x) x^2\right )\right )-\frac {1}{5} \log (x) \left (\log (-1+x)+2 \log (x)-\log \left (2 (1-x) x^2\right )\right )-\frac {1}{20} (1-10 x)^2 \log ^2\left (2 (1-x) x^2\right )-10 \int \left (1+\frac {1}{-1+x}+x\right ) \, dx\\ &=-\frac {5}{2} (1-x)^2-5 x+\frac {5 x^2}{2}+5 \log (1-x)+8 (1-x) \log (-1+x)-5 x^2 \log (-1+x)-\frac {1}{10} \left (180 (1-x)-50 (1-x)^2-81 \log (-1+x)\right ) \log (-1+x)-\frac {161}{20} \log ^2(-1+x)+8 x \log (x)-20 x^2 \log (x)+\frac {1}{5} \log (-1+x) \log (x)-\frac {2}{5} \left (20 x-50 x^2-\log (x)\right ) \log (x)-\frac {\log ^2(x)}{5}-\frac {1}{10} \log (1-x) \left (\log (-1+x)+2 \log (x)-\log \left (2 (1-x) x^2\right )\right )-\frac {1}{5} \log (x) \left (\log (-1+x)+2 \log (x)-\log \left (2 (1-x) x^2\right )\right )-\frac {1}{20} (1-10 x)^2 \log ^2\left (2 (1-x) x^2\right )\\ \end {aligned} \end {gather*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(60\) vs. \(2(24)=48\).
time = 0.10, size = 60, normalized size = 2.50 \begin {gather*} -4-4 \log (1-x)-41 \log (-1+x)-90 \log (x)+45 \log \left (-2 (-1+x) x^2\right )+x \log ^2\left (-2 (-1+x) x^2\right )-5 x^2 \log ^2\left (-2 (-1+x) x^2\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.23, size = 23, normalized size = 0.96
method | result | size |
risch | \(\left (-5 x^{2}+x \right ) \ln \left (-2 x^{3}+2 x^{2}\right )^{2}\) | \(23\) |
norman | \(\ln \left (-2 x^{3}+2 x^{2}\right )^{2} x -5 \ln \left (-2 x^{3}+2 x^{2}\right )^{2} x^{2}\) | \(37\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [C] Result contains complex when optimal does not.
time = 0.52, size = 138, normalized size = 5.75 \begin {gather*} 5 \, {\left (\pi ^{2} - 2 i \, \pi \log \left (2\right ) - \log \left (2\right )^{2}\right )} x^{2} - {\left (5 \, x^{2} - x\right )} \log \left (x - 1\right )^{2} - 4 \, {\left (5 \, x^{2} - x\right )} \log \left (x\right )^{2} - {\left (\pi ^{2} - 2 i \, \pi \log \left (2\right ) - \log \left (2\right )^{2}\right )} x + 2 \, {\left (5 \, {\left (-i \, \pi - \log \left (2\right )\right )} x^{2} + {\left (i \, \pi + \log \left (2\right )\right )} x - 2 \, {\left (5 \, x^{2} - x\right )} \log \left (x\right )\right )} \log \left (x - 1\right ) + 4 \, {\left (5 \, {\left (-i \, \pi - \log \left (2\right )\right )} x^{2} + {\left (i \, \pi + \log \left (2\right )\right )} x\right )} \log \left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.38, size = 25, normalized size = 1.04 \begin {gather*} -{\left (5 \, x^{2} - x\right )} \log \left (-2 \, x^{3} + 2 \, x^{2}\right )^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.06, size = 19, normalized size = 0.79 \begin {gather*} \left (- 5 x^{2} + x\right ) \log {\left (- 2 x^{3} + 2 x^{2} \right )}^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.43, size = 25, normalized size = 1.04 \begin {gather*} -{\left (5 \, x^{2} - x\right )} \log \left (-2 \, x^{3} + 2 \, x^{2}\right )^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.46, size = 22, normalized size = 0.92 \begin {gather*} -x\,{\ln \left (2\,x^2-2\,x^3\right )}^2\,\left (5\,x-1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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