Optimal. Leaf size=23 \[ \left (100+\frac {1}{4} \left (x+x^2\right )^2 \log ^2\left (-\frac {12}{x}\right )\right )^2 \]
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Rubi [B] Leaf count is larger than twice the leaf count of optimal. \(176\) vs. \(2(23)=46\).
time = 0.42, antiderivative size = 176, normalized size of antiderivative = 7.65, number of steps
used = 57, number of rules used = 9, integrand size = 117, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {12, 1608, 27,
45, 2371, 14, 2403, 2342, 2341} \begin {gather*} \frac {1}{16} x^8 \log ^4\left (-\frac {12}{x}\right )+\frac {1}{4} x^7 \log ^4\left (-\frac {12}{x}\right )+\frac {3}{8} x^6 \log ^4\left (-\frac {12}{x}\right )+\frac {1}{4} x^5 \log ^4\left (-\frac {12}{x}\right )+\frac {1}{16} x^4 \log ^4\left (-\frac {12}{x}\right )+50 x^4 \log ^2\left (-\frac {12}{x}\right )+25 x^4 \log \left (-\frac {12}{x}\right )+100 x^3 \log ^2\left (-\frac {12}{x}\right )+\frac {200}{3} x^3 \log \left (-\frac {12}{x}\right )+50 x^2 \log ^2\left (-\frac {12}{x}\right )+50 x^2 \log \left (-\frac {12}{x}\right )-\frac {25}{3} \left (3 x^4+8 x^3+6 x^2\right ) \log \left (-\frac {12}{x}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 14
Rule 27
Rule 45
Rule 1608
Rule 2341
Rule 2342
Rule 2371
Rule 2403
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{4} \int \left (\left (-400 x-800 x^2-400 x^3\right ) \log \left (-\frac {12}{x}\right )+\left (400 x+1200 x^2+800 x^3\right ) \log ^2\left (-\frac {12}{x}\right )+\left (-x^3-4 x^4-6 x^5-4 x^6-x^7\right ) \log ^3\left (-\frac {12}{x}\right )+\left (x^3+5 x^4+9 x^5+7 x^6+2 x^7\right ) \log ^4\left (-\frac {12}{x}\right )\right ) \, dx\\ &=\frac {1}{4} \int \left (-400 x-800 x^2-400 x^3\right ) \log \left (-\frac {12}{x}\right ) \, dx+\frac {1}{4} \int \left (400 x+1200 x^2+800 x^3\right ) \log ^2\left (-\frac {12}{x}\right ) \, dx+\frac {1}{4} \int \left (-x^3-4 x^4-6 x^5-4 x^6-x^7\right ) \log ^3\left (-\frac {12}{x}\right ) \, dx+\frac {1}{4} \int \left (x^3+5 x^4+9 x^5+7 x^6+2 x^7\right ) \log ^4\left (-\frac {12}{x}\right ) \, dx\\ &=\frac {1}{4} \int x \left (-400-800 x-400 x^2\right ) \log \left (-\frac {12}{x}\right ) \, dx+\frac {1}{4} \int x \left (400+1200 x+800 x^2\right ) \log ^2\left (-\frac {12}{x}\right ) \, dx+\frac {1}{4} \int \left (-x^3 \log ^3\left (-\frac {12}{x}\right )-4 x^4 \log ^3\left (-\frac {12}{x}\right )-6 x^5 \log ^3\left (-\frac {12}{x}\right )-4 x^6 \log ^3\left (-\frac {12}{x}\right )-x^7 \log ^3\left (-\frac {12}{x}\right )\right ) \, dx+\frac {1}{4} \int \left (x^3 \log ^4\left (-\frac {12}{x}\right )+5 x^4 \log ^4\left (-\frac {12}{x}\right )+9 x^5 \log ^4\left (-\frac {12}{x}\right )+7 x^6 \log ^4\left (-\frac {12}{x}\right )+2 x^7 \log ^4\left (-\frac {12}{x}\right )\right ) \, dx\\ &=\frac {1}{4} \int -400 x (1+x)^2 \log \left (-\frac {12}{x}\right ) \, dx-\frac {1}{4} \int x^3 \log ^3\left (-\frac {12}{x}\right ) \, dx-\frac {1}{4} \int x^7 \log ^3\left (-\frac {12}{x}\right ) \, dx+\frac {1}{4} \int x^3 \log ^4\left (-\frac {12}{x}\right ) \, dx+\frac {1}{4} \int \left (400 x \log ^2\left (-\frac {12}{x}\right )+1200 x^2 \log ^2\left (-\frac {12}{x}\right )+800 x^3 \log ^2\left (-\frac {12}{x}\right )\right ) \, dx+\frac {1}{2} \int x^7 \log ^4\left (-\frac {12}{x}\right ) \, dx+\frac {5}{4} \int x^4 \log ^4\left (-\frac {12}{x}\right ) \, dx-\frac {3}{2} \int x^5 \log ^3\left (-\frac {12}{x}\right ) \, dx+\frac {7}{4} \int x^6 \log ^4\left (-\frac {12}{x}\right ) \, dx+\frac {9}{4} \int x^5 \log ^4\left (-\frac {12}{x}\right ) \, dx-\int x^4 \log ^3\left (-\frac {12}{x}\right ) \, dx-\int x^6 \log ^3\left (-\frac {12}{x}\right ) \, dx\\ &=-\frac {1}{16} x^4 \log ^3\left (-\frac {12}{x}\right )-\frac {1}{5} x^5 \log ^3\left (-\frac {12}{x}\right )-\frac {1}{4} x^6 \log ^3\left (-\frac {12}{x}\right )-\frac {1}{7} x^7 \log ^3\left (-\frac {12}{x}\right )-\frac {1}{32} x^8 \log ^3\left (-\frac {12}{x}\right )+\frac {1}{16} x^4 \log ^4\left (-\frac {12}{x}\right )+\frac {1}{4} x^5 \log ^4\left (-\frac {12}{x}\right )+\frac {3}{8} x^6 \log ^4\left (-\frac {12}{x}\right )+\frac {1}{4} x^7 \log ^4\left (-\frac {12}{x}\right )+\frac {1}{16} x^8 \log ^4\left (-\frac {12}{x}\right )-\frac {3}{32} \int x^7 \log ^2\left (-\frac {12}{x}\right ) \, dx-\frac {3}{16} \int x^3 \log ^2\left (-\frac {12}{x}\right ) \, dx+\frac {1}{4} \int x^3 \log ^3\left (-\frac {12}{x}\right ) \, dx+\frac {1}{4} \int x^7 \log ^3\left (-\frac {12}{x}\right ) \, dx-\frac {3}{7} \int x^6 \log ^2\left (-\frac {12}{x}\right ) \, dx-\frac {3}{5} \int x^4 \log ^2\left (-\frac {12}{x}\right ) \, dx-\frac {3}{4} \int x^5 \log ^2\left (-\frac {12}{x}\right ) \, dx+\frac {3}{2} \int x^5 \log ^3\left (-\frac {12}{x}\right ) \, dx-100 \int x (1+x)^2 \log \left (-\frac {12}{x}\right ) \, dx+100 \int x \log ^2\left (-\frac {12}{x}\right ) \, dx+200 \int x^3 \log ^2\left (-\frac {12}{x}\right ) \, dx+300 \int x^2 \log ^2\left (-\frac {12}{x}\right ) \, dx+\int x^4 \log ^3\left (-\frac {12}{x}\right ) \, dx+\int x^6 \log ^3\left (-\frac {12}{x}\right ) \, dx\\ &=-\frac {25}{3} \left (6 x^2+8 x^3+3 x^4\right ) \log \left (-\frac {12}{x}\right )+50 x^2 \log ^2\left (-\frac {12}{x}\right )+100 x^3 \log ^2\left (-\frac {12}{x}\right )+\frac {3197}{64} x^4 \log ^2\left (-\frac {12}{x}\right )-\frac {3}{25} x^5 \log ^2\left (-\frac {12}{x}\right )-\frac {1}{8} x^6 \log ^2\left (-\frac {12}{x}\right )-\frac {3}{49} x^7 \log ^2\left (-\frac {12}{x}\right )-\frac {3}{256} x^8 \log ^2\left (-\frac {12}{x}\right )+\frac {1}{16} x^4 \log ^4\left (-\frac {12}{x}\right )+\frac {1}{4} x^5 \log ^4\left (-\frac {12}{x}\right )+\frac {3}{8} x^6 \log ^4\left (-\frac {12}{x}\right )+\frac {1}{4} x^7 \log ^4\left (-\frac {12}{x}\right )+\frac {1}{16} x^8 \log ^4\left (-\frac {12}{x}\right )-\frac {3}{128} \int x^7 \log \left (-\frac {12}{x}\right ) \, dx-\frac {3}{32} \int x^3 \log \left (-\frac {12}{x}\right ) \, dx+\frac {3}{32} \int x^7 \log ^2\left (-\frac {12}{x}\right ) \, dx-\frac {6}{49} \int x^6 \log \left (-\frac {12}{x}\right ) \, dx+\frac {3}{16} \int x^3 \log ^2\left (-\frac {12}{x}\right ) \, dx-\frac {6}{25} \int x^4 \log \left (-\frac {12}{x}\right ) \, dx-\frac {1}{4} \int x^5 \log \left (-\frac {12}{x}\right ) \, dx+\frac {3}{7} \int x^6 \log ^2\left (-\frac {12}{x}\right ) \, dx+\frac {3}{5} \int x^4 \log ^2\left (-\frac {12}{x}\right ) \, dx+\frac {3}{4} \int x^5 \log ^2\left (-\frac {12}{x}\right ) \, dx-100 \int \frac {1}{12} x \left (6+8 x+3 x^2\right ) \, dx+100 \int x \log \left (-\frac {12}{x}\right ) \, dx+100 \int x^3 \log \left (-\frac {12}{x}\right ) \, dx+200 \int x^2 \log \left (-\frac {12}{x}\right ) \, dx\\ &=25 x^2+\frac {200 x^3}{9}+\frac {3197 x^4}{512}-\frac {6 x^5}{625}-\frac {x^6}{144}-\frac {6 x^7}{2401}-\frac {3 x^8}{8192}+50 x^2 \log \left (-\frac {12}{x}\right )+\frac {200}{3} x^3 \log \left (-\frac {12}{x}\right )+\frac {3197}{128} x^4 \log \left (-\frac {12}{x}\right )-\frac {6}{125} x^5 \log \left (-\frac {12}{x}\right )-\frac {1}{24} x^6 \log \left (-\frac {12}{x}\right )-\frac {6}{343} x^7 \log \left (-\frac {12}{x}\right )-\frac {3 x^8 \log \left (-\frac {12}{x}\right )}{1024}-\frac {25}{3} \left (6 x^2+8 x^3+3 x^4\right ) \log \left (-\frac {12}{x}\right )+50 x^2 \log ^2\left (-\frac {12}{x}\right )+100 x^3 \log ^2\left (-\frac {12}{x}\right )+50 x^4 \log ^2\left (-\frac {12}{x}\right )+\frac {1}{16} x^4 \log ^4\left (-\frac {12}{x}\right )+\frac {1}{4} x^5 \log ^4\left (-\frac {12}{x}\right )+\frac {3}{8} x^6 \log ^4\left (-\frac {12}{x}\right )+\frac {1}{4} x^7 \log ^4\left (-\frac {12}{x}\right )+\frac {1}{16} x^8 \log ^4\left (-\frac {12}{x}\right )+\frac {3}{128} \int x^7 \log \left (-\frac {12}{x}\right ) \, dx+\frac {3}{32} \int x^3 \log \left (-\frac {12}{x}\right ) \, dx+\frac {6}{49} \int x^6 \log \left (-\frac {12}{x}\right ) \, dx+\frac {6}{25} \int x^4 \log \left (-\frac {12}{x}\right ) \, dx+\frac {1}{4} \int x^5 \log \left (-\frac {12}{x}\right ) \, dx-\frac {25}{3} \int x \left (6+8 x+3 x^2\right ) \, dx\\ &=25 x^2+\frac {200 x^3}{9}+\frac {25 x^4}{4}+50 x^2 \log \left (-\frac {12}{x}\right )+\frac {200}{3} x^3 \log \left (-\frac {12}{x}\right )+25 x^4 \log \left (-\frac {12}{x}\right )-\frac {25}{3} \left (6 x^2+8 x^3+3 x^4\right ) \log \left (-\frac {12}{x}\right )+50 x^2 \log ^2\left (-\frac {12}{x}\right )+100 x^3 \log ^2\left (-\frac {12}{x}\right )+50 x^4 \log ^2\left (-\frac {12}{x}\right )+\frac {1}{16} x^4 \log ^4\left (-\frac {12}{x}\right )+\frac {1}{4} x^5 \log ^4\left (-\frac {12}{x}\right )+\frac {3}{8} x^6 \log ^4\left (-\frac {12}{x}\right )+\frac {1}{4} x^7 \log ^4\left (-\frac {12}{x}\right )+\frac {1}{16} x^8 \log ^4\left (-\frac {12}{x}\right )-\frac {25}{3} \int \left (6 x+8 x^2+3 x^3\right ) \, dx\\ &=50 x^2 \log \left (-\frac {12}{x}\right )+\frac {200}{3} x^3 \log \left (-\frac {12}{x}\right )+25 x^4 \log \left (-\frac {12}{x}\right )-\frac {25}{3} \left (6 x^2+8 x^3+3 x^4\right ) \log \left (-\frac {12}{x}\right )+50 x^2 \log ^2\left (-\frac {12}{x}\right )+100 x^3 \log ^2\left (-\frac {12}{x}\right )+50 x^4 \log ^2\left (-\frac {12}{x}\right )+\frac {1}{16} x^4 \log ^4\left (-\frac {12}{x}\right )+\frac {1}{4} x^5 \log ^4\left (-\frac {12}{x}\right )+\frac {3}{8} x^6 \log ^4\left (-\frac {12}{x}\right )+\frac {1}{4} x^7 \log ^4\left (-\frac {12}{x}\right )+\frac {1}{16} x^8 \log ^4\left (-\frac {12}{x}\right )\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.08, size = 43, normalized size = 1.87 \begin {gather*} \frac {1}{4} \left (200 x^2 (1+x)^2 \log ^2\left (-\frac {12}{x}\right )+\frac {1}{4} x^4 (1+x)^4 \log ^4\left (-\frac {12}{x}\right )\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. \(105\) vs.
\(2(21)=42\).
time = 0.64, size = 106, normalized size = 4.61
method | result | size |
derivativedivides | \(\frac {\ln \left (-\frac {12}{x}\right )^{4} x^{5}}{4}+100 \ln \left (-\frac {12}{x}\right )^{2} x^{3}+\frac {3 \ln \left (-\frac {12}{x}\right )^{4} x^{6}}{8}+50 \ln \left (-\frac {12}{x}\right )^{2} x^{4}+\frac {\ln \left (-\frac {12}{x}\right )^{4} x^{7}}{4}+\frac {\ln \left (-\frac {12}{x}\right )^{4} x^{8}}{16}+50 x^{2} \ln \left (-\frac {12}{x}\right )^{2}+\frac {\ln \left (-\frac {12}{x}\right )^{4} x^{4}}{16}\) | \(106\) |
default | \(\frac {\ln \left (-\frac {12}{x}\right )^{4} x^{5}}{4}+100 \ln \left (-\frac {12}{x}\right )^{2} x^{3}+\frac {3 \ln \left (-\frac {12}{x}\right )^{4} x^{6}}{8}+50 \ln \left (-\frac {12}{x}\right )^{2} x^{4}+\frac {\ln \left (-\frac {12}{x}\right )^{4} x^{7}}{4}+\frac {\ln \left (-\frac {12}{x}\right )^{4} x^{8}}{16}+50 x^{2} \ln \left (-\frac {12}{x}\right )^{2}+\frac {\ln \left (-\frac {12}{x}\right )^{4} x^{4}}{16}\) | \(106\) |
risch | \(\frac {\ln \left (-\frac {12}{x}\right )^{4} x^{5}}{4}+100 \ln \left (-\frac {12}{x}\right )^{2} x^{3}+\frac {3 \ln \left (-\frac {12}{x}\right )^{4} x^{6}}{8}+50 \ln \left (-\frac {12}{x}\right )^{2} x^{4}+\frac {\ln \left (-\frac {12}{x}\right )^{4} x^{7}}{4}+\frac {\ln \left (-\frac {12}{x}\right )^{4} x^{8}}{16}+50 x^{2} \ln \left (-\frac {12}{x}\right )^{2}+\frac {\ln \left (-\frac {12}{x}\right )^{4} x^{4}}{16}\) | \(106\) |
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. 55 vs.
\(2 (22) = 44\).
time = 0.26, size = 55, normalized size = 2.39 \begin {gather*} \frac {1}{16} \, {\left (x^{8} + 4 \, x^{7} + 6 \, x^{6} + 4 \, x^{5} + x^{4}\right )} \log \left (-\frac {12}{x}\right )^{4} + 50 \, {\left (x^{4} + 2 \, x^{3} + x^{2}\right )} \log \left (-\frac {12}{x}\right )^{2} \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. 55 vs.
\(2 (22) = 44\).
time = 0.36, size = 55, normalized size = 2.39 \begin {gather*} \frac {1}{16} \, {\left (x^{8} + 4 \, x^{7} + 6 \, x^{6} + 4 \, x^{5} + x^{4}\right )} \log \left (-\frac {12}{x}\right )^{4} + 50 \, {\left (x^{4} + 2 \, x^{3} + x^{2}\right )} \log \left (-\frac {12}{x}\right )^{2} \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. 58 vs.
\(2 (19) = 38\).
time = 0.09, size = 58, normalized size = 2.52 \begin {gather*} \left (50 x^{4} + 100 x^{3} + 50 x^{2}\right ) \log {\left (- \frac {12}{x} \right )}^{2} + \left (\frac {x^{8}}{16} + \frac {x^{7}}{4} + \frac {3 x^{6}}{8} + \frac {x^{5}}{4} + \frac {x^{4}}{16}\right ) \log {\left (- \frac {12}{x} \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. 173 vs.
\(2 (22) = 44\).
time = 0.42, size = 173, normalized size = 7.52 \begin {gather*} \frac {1}{1120} \, x^{8} {\left (\frac {160}{x} + \frac {280}{x^{2}} + \frac {224}{x^{3}} + \frac {70}{x^{4}} + 35\right )} \log \left (-\frac {12}{x}\right )^{3} + \frac {25}{3} \, x^{4} {\left (\frac {8}{x} + \frac {6}{x^{2}} + 3\right )} \log \left (-\frac {12}{x}\right ) + \frac {1}{16} \, {\left (x^{8} + 4 \, x^{7} + 6 \, x^{6} + 4 \, x^{5} + x^{4}\right )} \log \left (-\frac {12}{x}\right )^{4} - \frac {1}{1120} \, {\left (35 \, x^{8} + 160 \, x^{7} + 280 \, x^{6} + 224 \, x^{5} + 70 \, x^{4}\right )} \log \left (-\frac {12}{x}\right )^{3} + 50 \, {\left (x^{4} + 2 \, x^{3} + x^{2}\right )} \log \left (-\frac {12}{x}\right )^{2} - \frac {25}{3} \, {\left (3 \, x^{4} + 8 \, x^{3} + 6 \, x^{2}\right )} \log \left (-\frac {12}{x}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.26, size = 57, normalized size = 2.48 \begin {gather*} \frac {x^2\,{\ln \left (-\frac {12}{x}\right )}^2\,{\left (x+1\right )}^2\,\left (x^4\,{\ln \left (-\frac {12}{x}\right )}^2+2\,x^3\,{\ln \left (-\frac {12}{x}\right )}^2+x^2\,{\ln \left (-\frac {12}{x}\right )}^2+800\right )}{16} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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