Optimal. Leaf size=34 \[ -e^x+\frac {1}{4} x^2 (-x+(-3+x (-4-\log (x))) (x+\log (x)))^2 \]
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Rubi [B] Leaf count is larger than twice the leaf count of optimal. \(142\) vs. \(2(34)=68\).
time = 0.44, antiderivative size = 142, normalized size of antiderivative = 4.18, number of steps
used = 38, number of rules used = 6, integrand size = 116, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.052, Rules used = {12, 2225,
2403, 2341, 2342, 1608} \begin {gather*} 4 x^6+\frac {1}{4} x^6 \log ^2(x)+2 x^6 \log (x)+8 x^5+\frac {1}{2} x^5 \log ^3(x)+4 x^5 \log ^2(x)+10 x^5 \log (x)+4 x^4+\frac {1}{4} x^4 \log ^4(x)+2 x^4 \log ^3(x)+\frac {15}{2} x^4 \log ^2(x)+14 x^4 \log (x)+\frac {3}{2} x^3 \log ^3(x)+6 x^3 \log ^2(x)+6 x^3 \log (x)+\frac {9}{4} x^2 \log ^2(x)-e^x \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 1608
Rule 2225
Rule 2341
Rule 2342
Rule 2403
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{2} \int \left (-2 e^x+12 x^2+60 x^3+100 x^4+52 x^5+\left (9 x+60 x^2+142 x^3+116 x^4+25 x^5\right ) \log (x)+\left (9 x+45 x^2+72 x^3+43 x^4+3 x^5\right ) \log ^2(x)+\left (9 x^2+18 x^3+5 x^4\right ) \log ^3(x)+2 x^3 \log ^4(x)\right ) \, dx\\ &=2 x^3+\frac {15 x^4}{2}+10 x^5+\frac {13 x^6}{3}+\frac {1}{2} \int \left (9 x+60 x^2+142 x^3+116 x^4+25 x^5\right ) \log (x) \, dx+\frac {1}{2} \int \left (9 x+45 x^2+72 x^3+43 x^4+3 x^5\right ) \log ^2(x) \, dx+\frac {1}{2} \int \left (9 x^2+18 x^3+5 x^4\right ) \log ^3(x) \, dx-\int e^x \, dx+\int x^3 \log ^4(x) \, dx\\ &=-e^x+2 x^3+\frac {15 x^4}{2}+10 x^5+\frac {13 x^6}{3}+\frac {1}{4} x^4 \log ^4(x)+\frac {1}{2} \int x^2 \left (9+18 x+5 x^2\right ) \log ^3(x) \, dx+\frac {1}{2} \int \left (9 x \log (x)+60 x^2 \log (x)+142 x^3 \log (x)+116 x^4 \log (x)+25 x^5 \log (x)\right ) \, dx+\frac {1}{2} \int \left (9 x \log ^2(x)+45 x^2 \log ^2(x)+72 x^3 \log ^2(x)+43 x^4 \log ^2(x)+3 x^5 \log ^2(x)\right ) \, dx-\int x^3 \log ^3(x) \, dx\\ &=-e^x+2 x^3+\frac {15 x^4}{2}+10 x^5+\frac {13 x^6}{3}-\frac {1}{4} x^4 \log ^3(x)+\frac {1}{4} x^4 \log ^4(x)+\frac {1}{2} \int \left (9 x^2 \log ^3(x)+18 x^3 \log ^3(x)+5 x^4 \log ^3(x)\right ) \, dx+\frac {3}{4} \int x^3 \log ^2(x) \, dx+\frac {3}{2} \int x^5 \log ^2(x) \, dx+\frac {9}{2} \int x \log (x) \, dx+\frac {9}{2} \int x \log ^2(x) \, dx+\frac {25}{2} \int x^5 \log (x) \, dx+\frac {43}{2} \int x^4 \log ^2(x) \, dx+\frac {45}{2} \int x^2 \log ^2(x) \, dx+30 \int x^2 \log (x) \, dx+36 \int x^3 \log ^2(x) \, dx+58 \int x^4 \log (x) \, dx+71 \int x^3 \log (x) \, dx\\ &=-e^x-\frac {9 x^2}{8}-\frac {4 x^3}{3}+\frac {49 x^4}{16}+\frac {192 x^5}{25}+\frac {287 x^6}{72}+\frac {9}{4} x^2 \log (x)+10 x^3 \log (x)+\frac {71}{4} x^4 \log (x)+\frac {58}{5} x^5 \log (x)+\frac {25}{12} x^6 \log (x)+\frac {9}{4} x^2 \log ^2(x)+\frac {15}{2} x^3 \log ^2(x)+\frac {147}{16} x^4 \log ^2(x)+\frac {43}{10} x^5 \log ^2(x)+\frac {1}{4} x^6 \log ^2(x)-\frac {1}{4} x^4 \log ^3(x)+\frac {1}{4} x^4 \log ^4(x)-\frac {3}{8} \int x^3 \log (x) \, dx-\frac {1}{2} \int x^5 \log (x) \, dx+\frac {5}{2} \int x^4 \log ^3(x) \, dx-\frac {9}{2} \int x \log (x) \, dx+\frac {9}{2} \int x^2 \log ^3(x) \, dx-\frac {43}{5} \int x^4 \log (x) \, dx+9 \int x^3 \log ^3(x) \, dx-15 \int x^2 \log (x) \, dx-18 \int x^3 \log (x) \, dx\\ &=-e^x+\frac {x^3}{3}+\frac {539 x^4}{128}+\frac {1003 x^5}{125}+4 x^6+5 x^3 \log (x)+\frac {421}{32} x^4 \log (x)+\frac {247}{25} x^5 \log (x)+2 x^6 \log (x)+\frac {9}{4} x^2 \log ^2(x)+\frac {15}{2} x^3 \log ^2(x)+\frac {147}{16} x^4 \log ^2(x)+\frac {43}{10} x^5 \log ^2(x)+\frac {1}{4} x^6 \log ^2(x)+\frac {3}{2} x^3 \log ^3(x)+2 x^4 \log ^3(x)+\frac {1}{2} x^5 \log ^3(x)+\frac {1}{4} x^4 \log ^4(x)-\frac {3}{2} \int x^4 \log ^2(x) \, dx-\frac {9}{2} \int x^2 \log ^2(x) \, dx-\frac {27}{4} \int x^3 \log ^2(x) \, dx\\ &=-e^x+\frac {x^3}{3}+\frac {539 x^4}{128}+\frac {1003 x^5}{125}+4 x^6+5 x^3 \log (x)+\frac {421}{32} x^4 \log (x)+\frac {247}{25} x^5 \log (x)+2 x^6 \log (x)+\frac {9}{4} x^2 \log ^2(x)+6 x^3 \log ^2(x)+\frac {15}{2} x^4 \log ^2(x)+4 x^5 \log ^2(x)+\frac {1}{4} x^6 \log ^2(x)+\frac {3}{2} x^3 \log ^3(x)+2 x^4 \log ^3(x)+\frac {1}{2} x^5 \log ^3(x)+\frac {1}{4} x^4 \log ^4(x)+\frac {3}{5} \int x^4 \log (x) \, dx+3 \int x^2 \log (x) \, dx+\frac {27}{8} \int x^3 \log (x) \, dx\\ &=-e^x+4 x^4+8 x^5+4 x^6+6 x^3 \log (x)+14 x^4 \log (x)+10 x^5 \log (x)+2 x^6 \log (x)+\frac {9}{4} x^2 \log ^2(x)+6 x^3 \log ^2(x)+\frac {15}{2} x^4 \log ^2(x)+4 x^5 \log ^2(x)+\frac {1}{4} x^6 \log ^2(x)+\frac {3}{2} x^3 \log ^3(x)+2 x^4 \log ^3(x)+\frac {1}{2} x^5 \log ^3(x)+\frac {1}{4} x^4 \log ^4(x)\\ \end {aligned} \end {gather*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(89\) vs. \(2(34)=68\).
time = 0.08, size = 89, normalized size = 2.62 \begin {gather*} \frac {1}{4} \left (4 \left (-e^x+4 x^4 (1+x)^2\right )+8 x^3 (1+x)^2 (3+x) \log (x)+x^2 \left (9+24 x+30 x^2+16 x^3+x^4\right ) \log ^2(x)+2 x^3 \left (3+4 x+x^2\right ) \log ^3(x)+x^4 \log ^4(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. \(129\) vs.
\(2(31)=62\).
time = 0.43, size = 130, normalized size = 3.82
method | result | size |
default | \(\frac {x^{6} \ln \left (x \right )^{2}}{4}+2 x^{6} \ln \left (x \right )+2 x^{4} \ln \left (x \right )^{3}+\frac {x^{5} \ln \left (x \right )^{3}}{2}+\frac {3 x^{3} \ln \left (x \right )^{3}}{2}+10 x^{5} \ln \left (x \right )+\frac {15 x^{4} \ln \left (x \right )^{2}}{2}+14 x^{4} \ln \left (x \right )+\frac {x^{4} \ln \left (x \right )^{4}}{4}+4 x^{5} \ln \left (x \right )^{2}+6 x^{3} \ln \left (x \right )^{2}+\frac {9 x^{2} \ln \left (x \right )^{2}}{4}+6 x^{3} \ln \left (x \right )+4 x^{6}+8 x^{5}+4 x^{4}-{\mathrm e}^{x}\) | \(130\) |
risch | \(\frac {x^{6} \ln \left (x \right )^{2}}{4}+2 x^{6} \ln \left (x \right )+2 x^{4} \ln \left (x \right )^{3}+\frac {x^{5} \ln \left (x \right )^{3}}{2}+\frac {3 x^{3} \ln \left (x \right )^{3}}{2}+10 x^{5} \ln \left (x \right )+\frac {15 x^{4} \ln \left (x \right )^{2}}{2}+14 x^{4} \ln \left (x \right )+\frac {x^{4} \ln \left (x \right )^{4}}{4}+4 x^{5} \ln \left (x \right )^{2}+6 x^{3} \ln \left (x \right )^{2}+\frac {9 x^{2} \ln \left (x \right )^{2}}{4}+6 x^{3} \ln \left (x \right )+4 x^{6}+8 x^{5}+4 x^{4}-{\mathrm e}^{x}\) | \(130\) |
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. 243 vs.
\(2 (27) = 54\).
time = 0.29, size = 243, normalized size = 7.15 \begin {gather*} \frac {1}{72} \, {\left (18 \, \log \left (x\right )^{2} - 6 \, \log \left (x\right ) + 1\right )} x^{6} + \frac {1}{250} \, {\left (125 \, \log \left (x\right )^{3} - 75 \, \log \left (x\right )^{2} + 30 \, \log \left (x\right ) - 6\right )} x^{5} + \frac {43}{250} \, {\left (25 \, \log \left (x\right )^{2} - 10 \, \log \left (x\right ) + 2\right )} x^{5} + \frac {287}{72} \, x^{6} + \frac {1}{128} \, {\left (32 \, \log \left (x\right )^{4} - 32 \, \log \left (x\right )^{3} + 24 \, \log \left (x\right )^{2} - 12 \, \log \left (x\right ) + 3\right )} x^{4} + \frac {9}{128} \, {\left (32 \, \log \left (x\right )^{3} - 24 \, \log \left (x\right )^{2} + 12 \, \log \left (x\right ) - 3\right )} x^{4} + \frac {9}{8} \, {\left (8 \, \log \left (x\right )^{2} - 4 \, \log \left (x\right ) + 1\right )} x^{4} + \frac {192}{25} \, x^{5} + \frac {1}{6} \, {\left (9 \, \log \left (x\right )^{3} - 9 \, \log \left (x\right )^{2} + 6 \, \log \left (x\right ) - 2\right )} x^{3} + \frac {5}{6} \, {\left (9 \, \log \left (x\right )^{2} - 6 \, \log \left (x\right ) + 2\right )} x^{3} + \frac {49}{16} \, x^{4} + \frac {9}{8} \, {\left (2 \, \log \left (x\right )^{2} - 2 \, \log \left (x\right ) + 1\right )} x^{2} - \frac {4}{3} \, x^{3} - \frac {9}{8} \, x^{2} + \frac {1}{60} \, {\left (125 \, x^{6} + 696 \, x^{5} + 1065 \, x^{4} + 600 \, x^{3} + 135 \, x^{2}\right )} \log \left (x\right ) - e^{x} \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. 102 vs.
\(2 (27) = 54\).
time = 0.38, size = 102, normalized size = 3.00 \begin {gather*} \frac {1}{4} \, x^{4} \log \left (x\right )^{4} + 4 \, x^{6} + 8 \, x^{5} + 4 \, x^{4} + \frac {1}{2} \, {\left (x^{5} + 4 \, x^{4} + 3 \, x^{3}\right )} \log \left (x\right )^{3} + \frac {1}{4} \, {\left (x^{6} + 16 \, x^{5} + 30 \, x^{4} + 24 \, x^{3} + 9 \, x^{2}\right )} \log \left (x\right )^{2} + 2 \, {\left (x^{6} + 5 \, x^{5} + 7 \, x^{4} + 3 \, x^{3}\right )} \log \left (x\right ) - e^{x} \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. 107 vs.
\(2 (26) = 52\).
time = 0.15, size = 107, normalized size = 3.15 \begin {gather*} 4 x^{6} + 8 x^{5} + \frac {x^{4} \log {\left (x \right )}^{4}}{4} + 4 x^{4} + \left (\frac {x^{5}}{2} + 2 x^{4} + \frac {3 x^{3}}{2}\right ) \log {\left (x \right )}^{3} + \left (2 x^{6} + 10 x^{5} + 14 x^{4} + 6 x^{3}\right ) \log {\left (x \right )} + \left (\frac {x^{6}}{4} + 4 x^{5} + \frac {15 x^{4}}{2} + 6 x^{3} + \frac {9 x^{2}}{4}\right ) \log {\left (x \right )}^{2} - e^{x} \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. 129 vs.
\(2 (27) = 54\).
time = 0.41, size = 129, normalized size = 3.79 \begin {gather*} \frac {1}{4} \, x^{6} \log \left (x\right )^{2} + \frac {1}{2} \, x^{5} \log \left (x\right )^{3} + \frac {1}{4} \, x^{4} \log \left (x\right )^{4} + 2 \, x^{6} \log \left (x\right ) + 4 \, x^{5} \log \left (x\right )^{2} + 2 \, x^{4} \log \left (x\right )^{3} + 4 \, x^{6} + 10 \, x^{5} \log \left (x\right ) + \frac {15}{2} \, x^{4} \log \left (x\right )^{2} + \frac {3}{2} \, x^{3} \log \left (x\right )^{3} + 8 \, x^{5} + 14 \, x^{4} \log \left (x\right ) + 6 \, x^{3} \log \left (x\right )^{2} + 4 \, x^{4} + 6 \, x^{3} \log \left (x\right ) + \frac {9}{4} \, x^{2} \log \left (x\right )^{2} - e^{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 3.58, size = 105, normalized size = 3.09 \begin {gather*} {\ln \left (x\right )}^2\,\left (\frac {x^6}{4}+4\,x^5+\frac {15\,x^4}{2}+6\,x^3+\frac {9\,x^2}{4}\right )-{\mathrm {e}}^x+\frac {x^4\,{\ln \left (x\right )}^4}{4}+\ln \left (x\right )\,\left (2\,x^6+10\,x^5+14\,x^4+6\,x^3\right )+{\ln \left (x\right )}^3\,\left (\frac {x^5}{2}+2\,x^4+\frac {3\,x^3}{2}\right )+4\,x^4+8\,x^5+4\,x^6 \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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