Optimal. Leaf size=31 \[ \frac {1}{9} x^4 \left (3+\left (e^2-x\right )^2 x\right )^2 \left (2+\frac {5 x}{4}+\log (x)\right ) \]
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Rubi [B] time = 0.21, antiderivative size = 278, normalized size of antiderivative = 8.97, number of steps used = 16, number of rules used = 4, integrand size = 180, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.022, Rules used = {12, 6, 2356, 2304} \begin {gather*} \frac {5 x^{11}}{36}-\frac {5 e^2 x^{10}}{9}+\frac {2 x^{10}}{9}+\frac {1}{9} x^{10} \log (x)+\frac {5 e^4 x^9}{6}-\frac {8 e^2 x^9}{9}-\frac {4}{9} e^2 x^9 \log (x)-\frac {5 e^6 x^8}{9}+\frac {4 e^4 x^8}{3}+\frac {5 x^8}{6}+\frac {2}{3} e^4 x^8 \log (x)-\frac {2}{63} \left (3-2 e^6\right ) x^7+\frac {5 e^8 x^7}{36}-\frac {20 e^6 x^7}{21}-\frac {5 e^2 x^7}{3}+\frac {10 x^7}{7}+\frac {2}{9} \left (3-2 e^6\right ) x^7 \log (x)+\frac {1}{54} e^2 \left (12-e^6\right ) x^6+\frac {13 e^8 x^6}{54}+\frac {5 e^4 x^6}{6}-\frac {26 e^2 x^6}{9}-\frac {1}{9} e^2 \left (12-e^6\right ) x^6 \log (x)+\frac {4 e^4 x^5}{3}+\frac {5 x^5}{4}+\frac {2}{3} e^4 x^5 \log (x)+2 x^4+x^4 \log (x) \end {gather*}
Antiderivative was successfully verified.
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Rule 6
Rule 12
Rule 2304
Rule 2356
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{36} \int \left (324 x^3+225 x^4+360 x^6+240 x^7+84 x^9+55 x^{10}+e^8 \left (52 x^5+35 x^6\right )+e^6 \left (-240 x^6-160 x^7\right )+e^4 \left (264 x^4+180 x^5+408 x^7+270 x^8\right )+e^2 \left (-624 x^5-420 x^6-304 x^8-200 x^9\right )+\left (144 x^3+24 e^8 x^5+168 x^6-112 e^6 x^6+40 x^9+e^4 \left (120 x^4+192 x^7\right )+e^2 \left (-288 x^5-144 x^8\right )\right ) \log (x)\right ) \, dx\\ &=\frac {9 x^4}{4}+\frac {5 x^5}{4}+\frac {10 x^7}{7}+\frac {5 x^8}{6}+\frac {7 x^{10}}{30}+\frac {5 x^{11}}{36}+\frac {1}{36} \int \left (144 x^3+24 e^8 x^5+168 x^6-112 e^6 x^6+40 x^9+e^4 \left (120 x^4+192 x^7\right )+e^2 \left (-288 x^5-144 x^8\right )\right ) \log (x) \, dx+\frac {1}{36} e^2 \int \left (-624 x^5-420 x^6-304 x^8-200 x^9\right ) \, dx+\frac {1}{36} e^4 \int \left (264 x^4+180 x^5+408 x^7+270 x^8\right ) \, dx+\frac {1}{36} e^6 \int \left (-240 x^6-160 x^7\right ) \, dx+\frac {1}{36} e^8 \int \left (52 x^5+35 x^6\right ) \, dx\\ &=\frac {9 x^4}{4}+\frac {5 x^5}{4}+\frac {22 e^4 x^5}{15}-\frac {26 e^2 x^6}{9}+\frac {5 e^4 x^6}{6}+\frac {13 e^8 x^6}{54}+\frac {10 x^7}{7}-\frac {5 e^2 x^7}{3}-\frac {20 e^6 x^7}{21}+\frac {5 e^8 x^7}{36}+\frac {5 x^8}{6}+\frac {17 e^4 x^8}{12}-\frac {5 e^6 x^8}{9}-\frac {76 e^2 x^9}{81}+\frac {5 e^4 x^9}{6}+\frac {7 x^{10}}{30}-\frac {5 e^2 x^{10}}{9}+\frac {5 x^{11}}{36}+\frac {1}{36} \int \left (144 x^3+24 e^8 x^5+\left (168-112 e^6\right ) x^6+40 x^9+e^4 \left (120 x^4+192 x^7\right )+e^2 \left (-288 x^5-144 x^8\right )\right ) \log (x) \, dx\\ &=\frac {9 x^4}{4}+\frac {5 x^5}{4}+\frac {22 e^4 x^5}{15}-\frac {26 e^2 x^6}{9}+\frac {5 e^4 x^6}{6}+\frac {13 e^8 x^6}{54}+\frac {10 x^7}{7}-\frac {5 e^2 x^7}{3}-\frac {20 e^6 x^7}{21}+\frac {5 e^8 x^7}{36}+\frac {5 x^8}{6}+\frac {17 e^4 x^8}{12}-\frac {5 e^6 x^8}{9}-\frac {76 e^2 x^9}{81}+\frac {5 e^4 x^9}{6}+\frac {7 x^{10}}{30}-\frac {5 e^2 x^{10}}{9}+\frac {5 x^{11}}{36}+\frac {1}{36} \int \left (144 x^3 \log (x)+120 e^4 x^4 \log (x)+24 e^2 \left (-12+e^6\right ) x^5 \log (x)-56 \left (-3+2 e^6\right ) x^6 \log (x)+192 e^4 x^7 \log (x)-144 e^2 x^8 \log (x)+40 x^9 \log (x)\right ) \, dx\\ &=\frac {9 x^4}{4}+\frac {5 x^5}{4}+\frac {22 e^4 x^5}{15}-\frac {26 e^2 x^6}{9}+\frac {5 e^4 x^6}{6}+\frac {13 e^8 x^6}{54}+\frac {10 x^7}{7}-\frac {5 e^2 x^7}{3}-\frac {20 e^6 x^7}{21}+\frac {5 e^8 x^7}{36}+\frac {5 x^8}{6}+\frac {17 e^4 x^8}{12}-\frac {5 e^6 x^8}{9}-\frac {76 e^2 x^9}{81}+\frac {5 e^4 x^9}{6}+\frac {7 x^{10}}{30}-\frac {5 e^2 x^{10}}{9}+\frac {5 x^{11}}{36}+\frac {10}{9} \int x^9 \log (x) \, dx+4 \int x^3 \log (x) \, dx-\left (4 e^2\right ) \int x^8 \log (x) \, dx+\frac {1}{3} \left (10 e^4\right ) \int x^4 \log (x) \, dx+\frac {1}{3} \left (16 e^4\right ) \int x^7 \log (x) \, dx+\frac {1}{9} \left (14 \left (3-2 e^6\right )\right ) \int x^6 \log (x) \, dx-\frac {1}{3} \left (2 e^2 \left (12-e^6\right )\right ) \int x^5 \log (x) \, dx\\ &=2 x^4+\frac {5 x^5}{4}+\frac {4 e^4 x^5}{3}-\frac {26 e^2 x^6}{9}+\frac {5 e^4 x^6}{6}+\frac {13 e^8 x^6}{54}+\frac {1}{54} e^2 \left (12-e^6\right ) x^6+\frac {10 x^7}{7}-\frac {5 e^2 x^7}{3}-\frac {20 e^6 x^7}{21}+\frac {5 e^8 x^7}{36}-\frac {2}{63} \left (3-2 e^6\right ) x^7+\frac {5 x^8}{6}+\frac {4 e^4 x^8}{3}-\frac {5 e^6 x^8}{9}-\frac {8 e^2 x^9}{9}+\frac {5 e^4 x^9}{6}+\frac {2 x^{10}}{9}-\frac {5 e^2 x^{10}}{9}+\frac {5 x^{11}}{36}+x^4 \log (x)+\frac {2}{3} e^4 x^5 \log (x)-\frac {1}{9} e^2 \left (12-e^6\right ) x^6 \log (x)+\frac {2}{9} \left (3-2 e^6\right ) x^7 \log (x)+\frac {2}{3} e^4 x^8 \log (x)-\frac {4}{9} e^2 x^9 \log (x)+\frac {1}{9} x^{10} \log (x)\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.05, size = 36, normalized size = 1.16 \begin {gather*} \frac {1}{36} x^4 \left (3+e^4 x-2 e^2 x^2+x^3\right )^2 (8+5 x+4 \log (x)) \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.58, size = 166, normalized size = 5.35 \begin {gather*} \frac {5}{36} \, x^{11} + \frac {2}{9} \, x^{10} + \frac {5}{6} \, x^{8} + \frac {4}{3} \, x^{7} + \frac {5}{4} \, x^{5} + 2 \, x^{4} + \frac {1}{36} \, {\left (5 \, x^{7} + 8 \, x^{6}\right )} e^{8} - \frac {1}{9} \, {\left (5 \, x^{8} + 8 \, x^{7}\right )} e^{6} + \frac {1}{6} \, {\left (5 \, x^{9} + 8 \, x^{8} + 5 \, x^{6} + 8 \, x^{5}\right )} e^{4} - \frac {1}{9} \, {\left (5 \, x^{10} + 8 \, x^{9} + 15 \, x^{7} + 24 \, x^{6}\right )} e^{2} + \frac {1}{9} \, {\left (x^{10} - 4 \, x^{7} e^{6} + 6 \, x^{7} + x^{6} e^{8} + 9 \, x^{4} + 6 \, {\left (x^{8} + x^{5}\right )} e^{4} - 4 \, {\left (x^{9} + 3 \, x^{6}\right )} e^{2}\right )} \log \relax (x) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.14, size = 227, normalized size = 7.32 \begin {gather*} \frac {5}{36} \, x^{11} + \frac {1}{9} \, x^{10} \log \relax (x) - \frac {4}{9} \, x^{9} e^{2} \log \relax (x) + \frac {2}{9} \, x^{10} + \frac {4}{81} \, x^{9} e^{2} + \frac {2}{3} \, x^{8} e^{4} \log \relax (x) - \frac {1}{12} \, x^{8} e^{4} - \frac {4}{9} \, x^{7} e^{6} \log \relax (x) + \frac {5}{6} \, x^{8} + \frac {4}{63} \, x^{7} e^{6} + \frac {2}{3} \, x^{7} \log \relax (x) + \frac {1}{9} \, x^{6} e^{8} \log \relax (x) - \frac {4}{3} \, x^{6} e^{2} \log \relax (x) + \frac {4}{3} \, x^{7} - \frac {1}{54} \, x^{6} e^{8} + \frac {2}{9} \, x^{6} e^{2} + \frac {2}{3} \, x^{5} e^{4} \log \relax (x) - \frac {2}{15} \, x^{5} e^{4} + \frac {5}{4} \, x^{5} + x^{4} \log \relax (x) + 2 \, x^{4} + \frac {1}{108} \, {\left (15 \, x^{7} + 26 \, x^{6}\right )} e^{8} - \frac {5}{63} \, {\left (7 \, x^{8} + 12 \, x^{7}\right )} e^{6} + \frac {1}{60} \, {\left (50 \, x^{9} + 85 \, x^{8} + 50 \, x^{6} + 88 \, x^{5}\right )} e^{4} - \frac {1}{81} \, {\left (45 \, x^{10} + 76 \, x^{9} + 135 \, x^{7} + 234 \, x^{6}\right )} e^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.08, size = 178, normalized size = 5.74
method | result | size |
risch | \(\frac {\left (4 \,{\mathrm e}^{8} x^{6}-16 \,{\mathrm e}^{6} x^{7}+24 x^{8} {\mathrm e}^{4}-16 \,{\mathrm e}^{2} x^{9}+4 x^{10}+24 x^{5} {\mathrm e}^{4}-48 x^{6} {\mathrm e}^{2}+24 x^{7}+36 x^{4}\right ) \ln \relax (x )}{36}+\frac {2 x^{10}}{9}-\frac {8 \,{\mathrm e}^{2} x^{9}}{9}+\frac {4 x^{8} {\mathrm e}^{4}}{3}-\frac {8 \,{\mathrm e}^{6} x^{7}}{9}+\frac {4 x^{7}}{3}+\frac {2 \,{\mathrm e}^{8} x^{6}}{9}-\frac {8 x^{6} {\mathrm e}^{2}}{3}+\frac {4 x^{5} {\mathrm e}^{4}}{3}+2 x^{4}+\frac {5 \,{\mathrm e}^{8} x^{7}}{36}-\frac {5 \,{\mathrm e}^{6} x^{8}}{9}+\frac {5 x^{9} {\mathrm e}^{4}}{6}+\frac {5 x^{6} {\mathrm e}^{4}}{6}-\frac {5 \,{\mathrm e}^{2} x^{10}}{9}-\frac {5 \,{\mathrm e}^{2} x^{7}}{3}+\frac {5 x^{11}}{36}+\frac {5 x^{8}}{6}+\frac {5 x^{5}}{4}\) | \(178\) |
norman | \(x^{4} \ln \relax (x )+\left (\frac {2}{9}-\frac {5 \,{\mathrm e}^{2}}{9}\right ) x^{10}+\left (\frac {4 \,{\mathrm e}^{4}}{3}+\frac {5}{4}\right ) x^{5}+\left (-\frac {8 \,{\mathrm e}^{2}}{9}+\frac {5 \,{\mathrm e}^{4}}{6}\right ) x^{9}+\left (\frac {4 \,{\mathrm e}^{4}}{3}+\frac {5}{6}-\frac {5 \,{\mathrm e}^{6}}{9}\right ) x^{8}+\left (\frac {2 \,{\mathrm e}^{8}}{9}-\frac {8 \,{\mathrm e}^{2}}{3}+\frac {5 \,{\mathrm e}^{4}}{6}\right ) x^{6}+\left (-\frac {8 \,{\mathrm e}^{6}}{9}+\frac {4}{3}-\frac {5 \,{\mathrm e}^{2}}{3}+\frac {5 \,{\mathrm e}^{8}}{36}\right ) x^{7}+\left (-\frac {4 \,{\mathrm e}^{6}}{9}+\frac {2}{3}\right ) x^{7} \ln \relax (x )+\left (\frac {{\mathrm e}^{8}}{9}-\frac {4 \,{\mathrm e}^{2}}{3}\right ) x^{6} \ln \relax (x )+2 x^{4}+\frac {5 x^{11}}{36}+\frac {x^{10} \ln \relax (x )}{9}-\frac {4 \,{\mathrm e}^{2} x^{9} \ln \relax (x )}{9}+\frac {2 \,{\mathrm e}^{4} x^{5} \ln \relax (x )}{3}+\frac {2 \,{\mathrm e}^{4} x^{8} \ln \relax (x )}{3}\) | \(185\) |
default | \(\frac {x^{10} \ln \relax (x )}{9}+\frac {5 x^{11}}{36}+\frac {4 x^{7}}{3}+\frac {5 x^{8}}{6}+\frac {2 x^{10}}{9}+\frac {5 x^{5}}{4}+2 x^{4}+\frac {2 x^{6} {\mathrm e}^{2}}{9}+\frac {2 x^{7} \ln \relax (x )}{3}-\frac {x^{8} {\mathrm e}^{4}}{12}+\frac {{\mathrm e}^{6} \left (-20 x^{8}-\frac {240}{7} x^{7}\right )}{36}+\frac {{\mathrm e}^{8} \left (5 x^{7}+\frac {26}{3} x^{6}\right )}{36}+\frac {{\mathrm e}^{4} \left (30 x^{9}+51 x^{8}+30 x^{6}+\frac {264}{5} x^{5}\right )}{36}+x^{4} \ln \relax (x )-\frac {2 x^{5} {\mathrm e}^{4}}{15}+\frac {4 \,{\mathrm e}^{2} x^{9}}{81}-\frac {4 \,{\mathrm e}^{2} \ln \relax (x ) x^{6}}{3}+\frac {{\mathrm e}^{8} \ln \relax (x ) x^{6}}{9}-\frac {4 \,{\mathrm e}^{6} x^{7} \ln \relax (x )}{9}-\frac {4 \,{\mathrm e}^{2} x^{9} \ln \relax (x )}{9}-\frac {{\mathrm e}^{8} x^{6}}{54}+\frac {4 \,{\mathrm e}^{6} x^{7}}{63}+\frac {{\mathrm e}^{2} \left (-20 x^{10}-\frac {304}{9} x^{9}-60 x^{7}-104 x^{6}\right )}{36}+\frac {2 \,{\mathrm e}^{4} x^{5} \ln \relax (x )}{3}+\frac {2 \,{\mathrm e}^{4} x^{8} \ln \relax (x )}{3}\) | \(250\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.35, size = 210, normalized size = 6.77 \begin {gather*} \frac {5}{36} \, x^{11} + \frac {2}{9} \, x^{10} + \frac {4}{81} \, x^{9} e^{2} - \frac {1}{12} \, x^{8} e^{4} + \frac {5}{6} \, x^{8} + \frac {2}{63} \, x^{7} {\left (2 \, e^{6} - 3\right )} + \frac {10}{7} \, x^{7} - \frac {1}{54} \, x^{6} {\left (e^{8} - 12 \, e^{2}\right )} - \frac {2}{15} \, x^{5} e^{4} + \frac {5}{4} \, x^{5} + 2 \, x^{4} + \frac {1}{108} \, {\left (15 \, x^{7} + 26 \, x^{6}\right )} e^{8} - \frac {5}{63} \, {\left (7 \, x^{8} + 12 \, x^{7}\right )} e^{6} + \frac {1}{60} \, {\left (50 \, x^{9} + 85 \, x^{8} + 50 \, x^{6} + 88 \, x^{5}\right )} e^{4} - \frac {1}{81} \, {\left (45 \, x^{10} + 76 \, x^{9} + 135 \, x^{7} + 234 \, x^{6}\right )} e^{2} + \frac {1}{9} \, {\left (x^{10} - 4 \, x^{7} e^{6} + 6 \, x^{7} + x^{6} e^{8} + 9 \, x^{4} + 6 \, {\left (x^{8} + x^{5}\right )} e^{4} - 4 \, {\left (x^{9} + 3 \, x^{6}\right )} e^{2}\right )} \log \relax (x) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.68, size = 32, normalized size = 1.03 \begin {gather*} \frac {x^4\,\left (5\,x+4\,\ln \relax (x)+8\right )\,{\left (x^3-2\,{\mathrm {e}}^2\,x^2+{\mathrm {e}}^4\,x+3\right )}^2}{36} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.33, size = 204, normalized size = 6.58 \begin {gather*} \frac {5 x^{11}}{36} + x^{10} \left (\frac {2}{9} - \frac {5 e^{2}}{9}\right ) + x^{9} \left (- \frac {8 e^{2}}{9} + \frac {5 e^{4}}{6}\right ) + x^{8} \left (- \frac {5 e^{6}}{9} + \frac {5}{6} + \frac {4 e^{4}}{3}\right ) + x^{7} \left (- \frac {8 e^{6}}{9} - \frac {5 e^{2}}{3} + \frac {4}{3} + \frac {5 e^{8}}{36}\right ) + x^{6} \left (- \frac {8 e^{2}}{3} + \frac {5 e^{4}}{6} + \frac {2 e^{8}}{9}\right ) + x^{5} \left (\frac {5}{4} + \frac {4 e^{4}}{3}\right ) + 2 x^{4} + \left (\frac {x^{10}}{9} - \frac {4 x^{9} e^{2}}{9} + \frac {2 x^{8} e^{4}}{3} - \frac {4 x^{7} e^{6}}{9} + \frac {2 x^{7}}{3} - \frac {4 x^{6} e^{2}}{3} + \frac {x^{6} e^{8}}{9} + \frac {2 x^{5} e^{4}}{3} + x^{4}\right ) \log {\relax (x )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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