Optimal. Leaf size=23 \[ \left (\frac {1}{2}-e^x-x\right ) \left (-x^2+\log (3)\right )^4 \]
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Rubi [B] time = 0.51, antiderivative size = 121, normalized size of antiderivative = 5.26, number of steps used = 57, number of rules used = 3, integrand size = 126, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.024, Rules used = {2196, 2176, 2194} \begin {gather*} -x^9-e^x x^8+\frac {x^8}{2}+4 x^7 \log (3)+4 e^x x^6 \log (3)-2 x^6 \log (3)-6 x^5 \log ^2(3)-6 e^x x^4 \log ^2(3)+3 x^4 \log ^2(3)+4 x^3 \log ^3(3)+4 e^x x^2 \log ^3(3)-2 x^2 \log ^3(3)-x \log ^4(3)-e^x \log ^4(3) \end {gather*}
Antiderivative was successfully verified.
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Rule 2176
Rule 2194
Rule 2196
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\frac {x^8}{2}-x^9-x \log ^4(3)+\log (3) \int \left (-12 x^5+28 x^6\right ) \, dx+\log ^2(3) \int \left (12 x^3-30 x^4\right ) \, dx+\log ^3(3) \int \left (-4 x+12 x^2\right ) \, dx+\int e^x \left (-8 x^7-x^8+\left (24 x^5+4 x^6\right ) \log (3)+\left (-24 x^3-6 x^4\right ) \log ^2(3)+\left (8 x+4 x^2\right ) \log ^3(3)-\log ^4(3)\right ) \, dx\\ &=\frac {x^8}{2}-x^9-2 x^6 \log (3)+4 x^7 \log (3)+3 x^4 \log ^2(3)-6 x^5 \log ^2(3)-2 x^2 \log ^3(3)+4 x^3 \log ^3(3)-x \log ^4(3)+\int \left (-8 e^x x^7-e^x x^8+4 e^x x^5 (6+x) \log (3)-6 e^x x^3 (4+x) \log ^2(3)+4 e^x x (2+x) \log ^3(3)-e^x \log ^4(3)\right ) \, dx\\ &=\frac {x^8}{2}-x^9-2 x^6 \log (3)+4 x^7 \log (3)+3 x^4 \log ^2(3)-6 x^5 \log ^2(3)-2 x^2 \log ^3(3)+4 x^3 \log ^3(3)-x \log ^4(3)-8 \int e^x x^7 \, dx+(4 \log (3)) \int e^x x^5 (6+x) \, dx-\left (6 \log ^2(3)\right ) \int e^x x^3 (4+x) \, dx+\left (4 \log ^3(3)\right ) \int e^x x (2+x) \, dx-\log ^4(3) \int e^x \, dx-\int e^x x^8 \, dx\\ &=-8 e^x x^7+\frac {x^8}{2}-e^x x^8-x^9-2 x^6 \log (3)+4 x^7 \log (3)+3 x^4 \log ^2(3)-6 x^5 \log ^2(3)-2 x^2 \log ^3(3)+4 x^3 \log ^3(3)-e^x \log ^4(3)-x \log ^4(3)+8 \int e^x x^7 \, dx+56 \int e^x x^6 \, dx+(4 \log (3)) \int \left (6 e^x x^5+e^x x^6\right ) \, dx-\left (6 \log ^2(3)\right ) \int \left (4 e^x x^3+e^x x^4\right ) \, dx+\left (4 \log ^3(3)\right ) \int \left (2 e^x x+e^x x^2\right ) \, dx\\ &=56 e^x x^6+\frac {x^8}{2}-e^x x^8-x^9-2 x^6 \log (3)+4 x^7 \log (3)+3 x^4 \log ^2(3)-6 x^5 \log ^2(3)-2 x^2 \log ^3(3)+4 x^3 \log ^3(3)-e^x \log ^4(3)-x \log ^4(3)-56 \int e^x x^6 \, dx-336 \int e^x x^5 \, dx+(4 \log (3)) \int e^x x^6 \, dx+(24 \log (3)) \int e^x x^5 \, dx-\left (6 \log ^2(3)\right ) \int e^x x^4 \, dx-\left (24 \log ^2(3)\right ) \int e^x x^3 \, dx+\left (4 \log ^3(3)\right ) \int e^x x^2 \, dx+\left (8 \log ^3(3)\right ) \int e^x x \, dx\\ &=-336 e^x x^5+\frac {x^8}{2}-e^x x^8-x^9+24 e^x x^5 \log (3)-2 x^6 \log (3)+4 e^x x^6 \log (3)+4 x^7 \log (3)-24 e^x x^3 \log ^2(3)+3 x^4 \log ^2(3)-6 e^x x^4 \log ^2(3)-6 x^5 \log ^2(3)+8 e^x x \log ^3(3)-2 x^2 \log ^3(3)+4 e^x x^2 \log ^3(3)+4 x^3 \log ^3(3)-e^x \log ^4(3)-x \log ^4(3)+336 \int e^x x^5 \, dx+1680 \int e^x x^4 \, dx-(24 \log (3)) \int e^x x^5 \, dx-(120 \log (3)) \int e^x x^4 \, dx+\left (24 \log ^2(3)\right ) \int e^x x^3 \, dx+\left (72 \log ^2(3)\right ) \int e^x x^2 \, dx-\left (8 \log ^3(3)\right ) \int e^x \, dx-\left (8 \log ^3(3)\right ) \int e^x x \, dx\\ &=1680 e^x x^4+\frac {x^8}{2}-e^x x^8-x^9-120 e^x x^4 \log (3)-2 x^6 \log (3)+4 e^x x^6 \log (3)+4 x^7 \log (3)+72 e^x x^2 \log ^2(3)+3 x^4 \log ^2(3)-6 e^x x^4 \log ^2(3)-6 x^5 \log ^2(3)-8 e^x \log ^3(3)-2 x^2 \log ^3(3)+4 e^x x^2 \log ^3(3)+4 x^3 \log ^3(3)-e^x \log ^4(3)-x \log ^4(3)-1680 \int e^x x^4 \, dx-6720 \int e^x x^3 \, dx+(120 \log (3)) \int e^x x^4 \, dx+(480 \log (3)) \int e^x x^3 \, dx-\left (72 \log ^2(3)\right ) \int e^x x^2 \, dx-\left (144 \log ^2(3)\right ) \int e^x x \, dx+\left (8 \log ^3(3)\right ) \int e^x \, dx\\ &=-6720 e^x x^3+\frac {x^8}{2}-e^x x^8-x^9+480 e^x x^3 \log (3)-2 x^6 \log (3)+4 e^x x^6 \log (3)+4 x^7 \log (3)-144 e^x x \log ^2(3)+3 x^4 \log ^2(3)-6 e^x x^4 \log ^2(3)-6 x^5 \log ^2(3)-2 x^2 \log ^3(3)+4 e^x x^2 \log ^3(3)+4 x^3 \log ^3(3)-e^x \log ^4(3)-x \log ^4(3)+6720 \int e^x x^3 \, dx+20160 \int e^x x^2 \, dx-(480 \log (3)) \int e^x x^3 \, dx-(1440 \log (3)) \int e^x x^2 \, dx+\left (144 \log ^2(3)\right ) \int e^x \, dx+\left (144 \log ^2(3)\right ) \int e^x x \, dx\\ &=20160 e^x x^2+\frac {x^8}{2}-e^x x^8-x^9-1440 e^x x^2 \log (3)-2 x^6 \log (3)+4 e^x x^6 \log (3)+4 x^7 \log (3)+144 e^x \log ^2(3)+3 x^4 \log ^2(3)-6 e^x x^4 \log ^2(3)-6 x^5 \log ^2(3)-2 x^2 \log ^3(3)+4 e^x x^2 \log ^3(3)+4 x^3 \log ^3(3)-e^x \log ^4(3)-x \log ^4(3)-20160 \int e^x x^2 \, dx-40320 \int e^x x \, dx+(1440 \log (3)) \int e^x x^2 \, dx+(2880 \log (3)) \int e^x x \, dx-\left (144 \log ^2(3)\right ) \int e^x \, dx\\ &=-40320 e^x x+\frac {x^8}{2}-e^x x^8-x^9+2880 e^x x \log (3)-2 x^6 \log (3)+4 e^x x^6 \log (3)+4 x^7 \log (3)+3 x^4 \log ^2(3)-6 e^x x^4 \log ^2(3)-6 x^5 \log ^2(3)-2 x^2 \log ^3(3)+4 e^x x^2 \log ^3(3)+4 x^3 \log ^3(3)-e^x \log ^4(3)-x \log ^4(3)+40320 \int e^x \, dx+40320 \int e^x x \, dx-(2880 \log (3)) \int e^x \, dx-(2880 \log (3)) \int e^x x \, dx\\ &=40320 e^x+\frac {x^8}{2}-e^x x^8-x^9-2880 e^x \log (3)-2 x^6 \log (3)+4 e^x x^6 \log (3)+4 x^7 \log (3)+3 x^4 \log ^2(3)-6 e^x x^4 \log ^2(3)-6 x^5 \log ^2(3)-2 x^2 \log ^3(3)+4 e^x x^2 \log ^3(3)+4 x^3 \log ^3(3)-e^x \log ^4(3)-x \log ^4(3)-40320 \int e^x \, dx+(2880 \log (3)) \int e^x \, dx\\ &=\frac {x^8}{2}-e^x x^8-x^9-2 x^6 \log (3)+4 e^x x^6 \log (3)+4 x^7 \log (3)+3 x^4 \log ^2(3)-6 e^x x^4 \log ^2(3)-6 x^5 \log ^2(3)-2 x^2 \log ^3(3)+4 e^x x^2 \log ^3(3)+4 x^3 \log ^3(3)-e^x \log ^4(3)-x \log ^4(3)\\ \end {aligned} \end {gather*}
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Mathematica [B] time = 0.16, size = 85, normalized size = 3.70 \begin {gather*} \frac {x^8}{2}-x^9-e^x \left (x^2-\log (3)\right )^4-2 x^6 \log (3)+4 x^7 \log (3)+3 x^4 \log ^2(3)-6 x^5 \log ^2(3)-2 x^2 \log ^3(3)+4 x^3 \log ^3(3)-x \log ^4(3) \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.62, size = 104, normalized size = 4.52 \begin {gather*} -x^{9} + \frac {1}{2} \, x^{8} - x \log \relax (3)^{4} + 2 \, {\left (2 \, x^{3} - x^{2}\right )} \log \relax (3)^{3} - 3 \, {\left (2 \, x^{5} - x^{4}\right )} \log \relax (3)^{2} - {\left (x^{8} - 4 \, x^{6} \log \relax (3) + 6 \, x^{4} \log \relax (3)^{2} - 4 \, x^{2} \log \relax (3)^{3} + \log \relax (3)^{4}\right )} e^{x} + 2 \, {\left (2 \, x^{7} - x^{6}\right )} \log \relax (3) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.15, size = 104, normalized size = 4.52 \begin {gather*} -x^{9} + \frac {1}{2} \, x^{8} - x \log \relax (3)^{4} + 2 \, {\left (2 \, x^{3} - x^{2}\right )} \log \relax (3)^{3} - 3 \, {\left (2 \, x^{5} - x^{4}\right )} \log \relax (3)^{2} - {\left (x^{8} - 4 \, x^{6} \log \relax (3) + 6 \, x^{4} \log \relax (3)^{2} - 4 \, x^{2} \log \relax (3)^{3} + \log \relax (3)^{4}\right )} e^{x} + 2 \, {\left (2 \, x^{7} - x^{6}\right )} \log \relax (3) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.07, size = 109, normalized size = 4.74
method | result | size |
risch | \(\left (-x^{8}+4 x^{6} \ln \relax (3)-6 x^{4} \ln \relax (3)^{2}+4 \ln \relax (3)^{3} x^{2}-\ln \relax (3)^{4}\right ) {\mathrm e}^{x}-x \ln \relax (3)^{4}+4 x^{3} \ln \relax (3)^{3}-2 \ln \relax (3)^{3} x^{2}-6 x^{5} \ln \relax (3)^{2}+3 x^{4} \ln \relax (3)^{2}+4 \ln \relax (3) x^{7}-2 x^{6} \ln \relax (3)-x^{9}+\frac {x^{8}}{2}\) | \(109\) |
default | \(-x^{8} {\mathrm e}^{x}-{\mathrm e}^{x} \ln \relax (3)^{4}-6 \,{\mathrm e}^{x} x^{4} \ln \relax (3)^{2}+4 \,{\mathrm e}^{x} \ln \relax (3) x^{6}+4 \,{\mathrm e}^{x} \ln \relax (3)^{3} x^{2}+4 x^{3} \ln \relax (3)^{3}-2 \ln \relax (3)^{3} x^{2}-6 x^{5} \ln \relax (3)^{2}+3 x^{4} \ln \relax (3)^{2}+4 \ln \relax (3) x^{7}-2 x^{6} \ln \relax (3)+\frac {x^{8}}{2}-x^{9}-x \ln \relax (3)^{4}\) | \(115\) |
norman | \(-x^{8} {\mathrm e}^{x}-{\mathrm e}^{x} \ln \relax (3)^{4}-6 \,{\mathrm e}^{x} x^{4} \ln \relax (3)^{2}+4 \,{\mathrm e}^{x} \ln \relax (3) x^{6}+4 \,{\mathrm e}^{x} \ln \relax (3)^{3} x^{2}+4 x^{3} \ln \relax (3)^{3}-2 \ln \relax (3)^{3} x^{2}-6 x^{5} \ln \relax (3)^{2}+3 x^{4} \ln \relax (3)^{2}+4 \ln \relax (3) x^{7}-2 x^{6} \ln \relax (3)+\frac {x^{8}}{2}-x^{9}-x \ln \relax (3)^{4}\) | \(115\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.45, size = 104, normalized size = 4.52 \begin {gather*} -x^{9} + \frac {1}{2} \, x^{8} - x \log \relax (3)^{4} + 2 \, {\left (2 \, x^{3} - x^{2}\right )} \log \relax (3)^{3} - 3 \, {\left (2 \, x^{5} - x^{4}\right )} \log \relax (3)^{2} - {\left (x^{8} - 4 \, x^{6} \log \relax (3) + 6 \, x^{4} \log \relax (3)^{2} - 4 \, x^{2} \log \relax (3)^{3} + \log \relax (3)^{4}\right )} e^{x} + 2 \, {\left (2 \, x^{7} - x^{6}\right )} \log \relax (3) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.36, size = 114, normalized size = 4.96 \begin {gather*} 4\,x^3\,{\ln \relax (3)}^3-2\,x^2\,{\ln \relax (3)}^3+3\,x^4\,{\ln \relax (3)}^2-6\,x^5\,{\ln \relax (3)}^2-{\mathrm {e}}^x\,{\ln \relax (3)}^4-x^8\,{\mathrm {e}}^x-x\,{\ln \relax (3)}^4-2\,x^6\,\ln \relax (3)+4\,x^7\,\ln \relax (3)+\frac {x^8}{2}-x^9+4\,x^6\,{\mathrm {e}}^x\,\ln \relax (3)+4\,x^2\,{\mathrm {e}}^x\,{\ln \relax (3)}^3-6\,x^4\,{\mathrm {e}}^x\,{\ln \relax (3)}^2 \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.18, size = 112, normalized size = 4.87 \begin {gather*} - x^{9} + \frac {x^{8}}{2} + 4 x^{7} \log {\relax (3 )} - 2 x^{6} \log {\relax (3 )} - 6 x^{5} \log {\relax (3 )}^{2} + 3 x^{4} \log {\relax (3 )}^{2} + 4 x^{3} \log {\relax (3 )}^{3} - 2 x^{2} \log {\relax (3 )}^{3} - x \log {\relax (3 )}^{4} + \left (- x^{8} + 4 x^{6} \log {\relax (3 )} - 6 x^{4} \log {\relax (3 )}^{2} + 4 x^{2} \log {\relax (3 )}^{3} - \log {\relax (3 )}^{4}\right ) e^{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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