Optimal. Leaf size=31 \[ \frac {\left (2 x-x^2\right )^2}{x \left (\frac {e^x}{3}-\log (9)+\log (x)\right )^2} \]
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Rubi [F]
time = 2.48, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps
used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {}
\begin {gather*} \int \frac {-216+216 x-54 x^2+e^x \left (36-144 x+99 x^2-18 x^3\right )+\left (-108+216 x-81 x^2\right ) \log (9)+\left (108-216 x+81 x^2\right ) \log (x)}{e^{3 x}-9 e^{2 x} \log (9)+27 e^x \log ^2(9)-27 \log ^3(9)+\left (9 e^{2 x}-54 e^x \log (9)+81 \log ^2(9)\right ) \log (x)+\left (27 e^x-81 \log (9)\right ) \log ^2(x)+27 \log ^3(x)} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {9 (2-x) \left (e^x \left (2-7 x+2 x^2\right )-12 (1+\log (3))+6 x (1+\log (27))+(6-9 x) \log (x)\right )}{\left (e^x-3 \log (9)+3 \log (x)\right )^3} \, dx\\ &=9 \int \frac {(2-x) \left (e^x \left (2-7 x+2 x^2\right )-12 (1+\log (3))+6 x (1+\log (27))+(6-9 x) \log (x)\right )}{\left (e^x-3 \log (9)+3 \log (x)\right )^3} \, dx\\ &=9 \int \left (-\frac {-4+16 x-11 x^2+2 x^3}{\left (e^x-3 \log (9)+3 \log (x)\right )^2}+\frac {3 (2-x) \left (-4+2 x (1-\log (81))+x^2 \log (81)+4 x \log (x)-2 x^2 \log (x)\right )}{\left (e^x-3 \log (9)+3 \log (x)\right )^3}\right ) \, dx\\ &=-\left (9 \int \frac {-4+16 x-11 x^2+2 x^3}{\left (e^x-3 \log (9)+3 \log (x)\right )^2} \, dx\right )+27 \int \frac {(2-x) \left (-4+2 x (1-\log (81))+x^2 \log (81)+4 x \log (x)-2 x^2 \log (x)\right )}{\left (e^x-3 \log (9)+3 \log (x)\right )^3} \, dx\\ &=-\left (9 \int \left (-\frac {4}{\left (e^x-3 \log (9)+3 \log (x)\right )^2}+\frac {16 x}{\left (e^x-3 \log (9)+3 \log (x)\right )^2}-\frac {11 x^2}{\left (e^x-3 \log (9)+3 \log (x)\right )^2}+\frac {2 x^3}{\left (e^x-3 \log (9)+3 \log (x)\right )^2}\right ) \, dx\right )+27 \int \frac {(2-x)^2 (-2-x \log (81)+2 x \log (x))}{\left (e^x-3 \log (9)+3 \log (x)\right )^3} \, dx\\ &=-\left (18 \int \frac {x^3}{\left (e^x-3 \log (9)+3 \log (x)\right )^2} \, dx\right )+27 \int \left (\frac {4 (-2-x \log (81)+2 x \log (x))}{\left (e^x-3 \log (9)+3 \log (x)\right )^3}-\frac {4 x (-2-x \log (81)+2 x \log (x))}{\left (e^x-3 \log (9)+3 \log (x)\right )^3}+\frac {x^2 (-2-x \log (81)+2 x \log (x))}{\left (e^x-3 \log (9)+3 \log (x)\right )^3}\right ) \, dx+36 \int \frac {1}{\left (e^x-3 \log (9)+3 \log (x)\right )^2} \, dx+99 \int \frac {x^2}{\left (e^x-3 \log (9)+3 \log (x)\right )^2} \, dx-144 \int \frac {x}{\left (e^x-3 \log (9)+3 \log (x)\right )^2} \, dx\\ &=-\left (18 \int \frac {x^3}{\left (e^x-3 \log (9)+3 \log (x)\right )^2} \, dx\right )+27 \int \frac {x^2 (-2-x \log (81)+2 x \log (x))}{\left (e^x-3 \log (9)+3 \log (x)\right )^3} \, dx+36 \int \frac {1}{\left (e^x-3 \log (9)+3 \log (x)\right )^2} \, dx+99 \int \frac {x^2}{\left (e^x-3 \log (9)+3 \log (x)\right )^2} \, dx+108 \int \frac {-2-x \log (81)+2 x \log (x)}{\left (e^x-3 \log (9)+3 \log (x)\right )^3} \, dx-108 \int \frac {x (-2-x \log (81)+2 x \log (x))}{\left (e^x-3 \log (9)+3 \log (x)\right )^3} \, dx-144 \int \frac {x}{\left (e^x-3 \log (9)+3 \log (x)\right )^2} \, dx\\ &=-\left (18 \int \frac {x^3}{\left (e^x-3 \log (9)+3 \log (x)\right )^2} \, dx\right )+27 \int \left (-\frac {2 x^2}{\left (e^x-3 \log (9)+3 \log (x)\right )^3}-\frac {x^3 \log (81)}{\left (e^x-3 \log (9)+3 \log (x)\right )^3}+\frac {2 x^3 \log (x)}{\left (e^x-3 \log (9)+3 \log (x)\right )^3}\right ) \, dx+36 \int \frac {1}{\left (e^x-3 \log (9)+3 \log (x)\right )^2} \, dx+99 \int \frac {x^2}{\left (e^x-3 \log (9)+3 \log (x)\right )^2} \, dx+108 \int \left (-\frac {2}{\left (e^x-3 \log (9)+3 \log (x)\right )^3}-\frac {x \log (81)}{\left (e^x-3 \log (9)+3 \log (x)\right )^3}+\frac {2 x \log (x)}{\left (e^x-3 \log (9)+3 \log (x)\right )^3}\right ) \, dx-108 \int \left (-\frac {2 x}{\left (e^x-3 \log (9)+3 \log (x)\right )^3}-\frac {x^2 \log (81)}{\left (e^x-3 \log (9)+3 \log (x)\right )^3}+\frac {2 x^2 \log (x)}{\left (e^x-3 \log (9)+3 \log (x)\right )^3}\right ) \, dx-144 \int \frac {x}{\left (e^x-3 \log (9)+3 \log (x)\right )^2} \, dx\\ &=-\left (18 \int \frac {x^3}{\left (e^x-3 \log (9)+3 \log (x)\right )^2} \, dx\right )+36 \int \frac {1}{\left (e^x-3 \log (9)+3 \log (x)\right )^2} \, dx-54 \int \frac {x^2}{\left (e^x-3 \log (9)+3 \log (x)\right )^3} \, dx+54 \int \frac {x^3 \log (x)}{\left (e^x-3 \log (9)+3 \log (x)\right )^3} \, dx+99 \int \frac {x^2}{\left (e^x-3 \log (9)+3 \log (x)\right )^2} \, dx-144 \int \frac {x}{\left (e^x-3 \log (9)+3 \log (x)\right )^2} \, dx-216 \int \frac {1}{\left (e^x-3 \log (9)+3 \log (x)\right )^3} \, dx+216 \int \frac {x}{\left (e^x-3 \log (9)+3 \log (x)\right )^3} \, dx+216 \int \frac {x \log (x)}{\left (e^x-3 \log (9)+3 \log (x)\right )^3} \, dx-216 \int \frac {x^2 \log (x)}{\left (e^x-3 \log (9)+3 \log (x)\right )^3} \, dx-(27 \log (81)) \int \frac {x^3}{\left (e^x-3 \log (9)+3 \log (x)\right )^3} \, dx-(108 \log (81)) \int \frac {x}{\left (e^x-3 \log (9)+3 \log (x)\right )^3} \, dx+(108 \log (81)) \int \frac {x^2}{\left (e^x-3 \log (9)+3 \log (x)\right )^3} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.12, size = 22, normalized size = 0.71 \begin {gather*} \frac {9 (-2+x)^2 x}{\left (e^x-3 \log (9)+3 \log (x)\right )^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.12, size = 27, normalized size = 0.87
method | result | size |
risch | \(\frac {9 x \left (x^{2}-4 x +4\right )}{\left (6 \ln \left (3\right )-{\mathrm e}^{x}-3 \ln \left (x \right )\right )^{2}}\) | \(27\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.58, size = 54, normalized size = 1.74 \begin {gather*} -\frac {9 \, {\left (x^{3} - 4 \, x^{2} + 4 \, x\right )}}{6 \, {\left (2 \, \log \left (3\right ) - \log \left (x\right )\right )} e^{x} - 36 \, \log \left (3\right )^{2} + 36 \, \log \left (3\right ) \log \left (x\right ) - 9 \, \log \left (x\right )^{2} - e^{\left (2 \, x\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 52, normalized size = 1.68 \begin {gather*} -\frac {9 \, {\left (x^{3} - 4 \, x^{2} + 4 \, x\right )}}{12 \, e^{x} \log \left (3\right ) - 36 \, \log \left (3\right )^{2} - 6 \, {\left (e^{x} - 6 \, \log \left (3\right )\right )} \log \left (x\right ) - 9 \, \log \left (x\right )^{2} - e^{\left (2 \, x\right )}} \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. 53 vs.
\(2 (24) = 48\).
time = 0.11, size = 53, normalized size = 1.71 \begin {gather*} \frac {9 x^{3} - 36 x^{2} + 36 x}{\left (6 \log {\left (x \right )} - 12 \log {\left (3 \right )}\right ) e^{x} + e^{2 x} + 9 \log {\left (x \right )}^{2} - 36 \log {\left (3 \right )} \log {\left (x \right )} + 36 \log {\left (3 \right )}^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.42, size = 53, normalized size = 1.71 \begin {gather*} -\frac {9 \, {\left (x^{3} - 4 \, x^{2} + 4 \, x\right )}}{12 \, e^{x} \log \left (3\right ) - 36 \, \log \left (3\right )^{2} - 6 \, e^{x} \log \left (x\right ) + 36 \, \log \left (3\right ) \log \left (x\right ) - 9 \, \log \left (x\right )^{2} - e^{\left (2 \, x\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int -\frac {2\,\ln \left (3\right )\,\left (81\,x^2-216\,x+108\right )-216\,x-\ln \left (x\right )\,\left (81\,x^2-216\,x+108\right )+54\,x^2+{\mathrm {e}}^x\,\left (18\,x^3-99\,x^2+144\,x-36\right )+216}{27\,{\ln \left (x\right )}^3+\left (27\,{\mathrm {e}}^x-162\,\ln \left (3\right )\right )\,{\ln \left (x\right )}^2+\left (9\,{\mathrm {e}}^{2\,x}-108\,{\mathrm {e}}^x\,\ln \left (3\right )+324\,{\ln \left (3\right )}^2\right )\,\ln \left (x\right )+{\mathrm {e}}^{3\,x}-18\,{\mathrm {e}}^{2\,x}\,\ln \left (3\right )+108\,{\mathrm {e}}^x\,{\ln \left (3\right )}^2-216\,{\ln \left (3\right )}^3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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