Optimal. Leaf size=30 \[ \frac {\log (e-x)}{e^{2 x}+3 \left (-x+x^2+e^x \log (4)\right )} \]
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Rubi [F] time = 4.53, 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 {-e^{2 x}+3 x-3 x^2-3 e^x \log (4)+\left (e (3-6 x)-3 x+6 x^2+e^{2 x} (-2 e+2 x)+e^x (-3 e+3 x) \log (4)\right ) \log (e-x)}{e^{4 x} (e-x)-9 x^3+18 x^4-9 x^5+e \left (9 x^2-18 x^3+9 x^4\right )+e^{3 x} (6 e-6 x) \log (4)+e^x \left (18 x^2-18 x^3+e \left (-18 x+18 x^2\right )\right ) \log (4)+e^{2 x} \left (6 x^2-6 x^3+e \left (-6 x+6 x^2\right )+(9 e-9 x) \log ^2(4)\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-e^{2 x}-3 (-1+x) x-e^x \log (64)-(e-x) \left (-3+2 e^{2 x}+6 x+e^x \log (64)\right ) \log (e-x)}{(e-x) \left (e^{2 x}+3 (-1+x) x+e^x \log (64)\right )^2} \, dx\\ &=\int \left (\frac {\left (3-12 x+6 x^2+e^x \log (64)\right ) \log (e-x)}{\left (e^{2 x}-3 x+3 x^2+e^x \log (64)\right )^2}-\frac {1+2 e \log (e-x)-2 x \log (e-x)}{(e-x) \left (e^{2 x}-3 x+3 x^2+e^x \log (64)\right )}\right ) \, dx\\ &=\int \frac {\left (3-12 x+6 x^2+e^x \log (64)\right ) \log (e-x)}{\left (e^{2 x}-3 x+3 x^2+e^x \log (64)\right )^2} \, dx-\int \frac {1+2 e \log (e-x)-2 x \log (e-x)}{(e-x) \left (e^{2 x}-3 x+3 x^2+e^x \log (64)\right )} \, dx\\ &=(3 \log (e-x)) \int \frac {1}{\left (e^{2 x}-3 x+3 x^2+e^x \log (64)\right )^2} \, dx+(6 \log (e-x)) \int \frac {x^2}{\left (e^{2 x}-3 x+3 x^2+e^x \log (64)\right )^2} \, dx-(12 \log (e-x)) \int \frac {x}{\left (e^{2 x}-3 x+3 x^2+e^x \log (64)\right )^2} \, dx+(\log (64) \log (e-x)) \int \frac {e^x}{\left (e^{2 x}-3 x+3 x^2+e^x \log (64)\right )^2} \, dx-\int \left (\frac {1}{(e-x) \left (e^{2 x}-3 x+3 x^2+e^x \log (64)\right )}+\frac {2 e \log (e-x)}{(e-x) \left (e^{2 x}-3 x+3 x^2+e^x \log (64)\right )}-\frac {2 x \log (e-x)}{(e-x) \left (e^{2 x}-3 x+3 x^2+e^x \log (64)\right )}\right ) \, dx-\int \frac {-3 \int \frac {1}{\left (e^{2 x}+3 (-1+x) x+e^x \log (64)\right )^2} \, dx-\log (64) \int \frac {e^x}{\left (e^{2 x}+3 (-1+x) x+e^x \log (64)\right )^2} \, dx-6 \left (-2 \int \frac {x}{\left (e^{2 x}+3 (-1+x) x+e^x \log (64)\right )^2} \, dx+\int \frac {x^2}{\left (e^{2 x}+3 (-1+x) x+e^x \log (64)\right )^2} \, dx\right )}{e-x} \, dx\\ &=2 \int \frac {x \log (e-x)}{(e-x) \left (e^{2 x}-3 x+3 x^2+e^x \log (64)\right )} \, dx-(2 e) \int \frac {\log (e-x)}{(e-x) \left (e^{2 x}-3 x+3 x^2+e^x \log (64)\right )} \, dx+(3 \log (e-x)) \int \frac {1}{\left (e^{2 x}-3 x+3 x^2+e^x \log (64)\right )^2} \, dx+(6 \log (e-x)) \int \frac {x^2}{\left (e^{2 x}-3 x+3 x^2+e^x \log (64)\right )^2} \, dx-(12 \log (e-x)) \int \frac {x}{\left (e^{2 x}-3 x+3 x^2+e^x \log (64)\right )^2} \, dx+(\log (64) \log (e-x)) \int \frac {e^x}{\left (e^{2 x}-3 x+3 x^2+e^x \log (64)\right )^2} \, dx-\int \frac {1}{(e-x) \left (e^{2 x}-3 x+3 x^2+e^x \log (64)\right )} \, dx-\int \left (\frac {-3 \int \frac {1}{\left (e^{2 x}+3 (-1+x) x+e^x \log (64)\right )^2} \, dx-\log (64) \int \frac {e^x}{\left (e^{2 x}+3 (-1+x) x+e^x \log (64)\right )^2} \, dx+12 \int \frac {x}{\left (e^{2 x}+3 (-1+x) x+e^x \log (64)\right )^2} \, dx}{e-x}-\frac {6 \int \frac {x^2}{\left (e^{2 x}+3 (-1+x) x+e^x \log (64)\right )^2} \, dx}{e-x}\right ) \, dx\\ &=-\left (2 \int \frac {\int \frac {1}{e^{2 x}+3 (-1+x) x+e^x \log (64)} \, dx-e \int \frac {1}{(e-x) \left (e^{2 x}+3 (-1+x) x+e^x \log (64)\right )} \, dx}{e-x} \, dx\right )+6 \int \frac {\int \frac {x^2}{\left (e^{2 x}+3 (-1+x) x+e^x \log (64)\right )^2} \, dx}{e-x} \, dx+(2 e) \int \frac {\int \frac {1}{(e-x) \left (e^{2 x}+3 (-1+x) x+e^x \log (64)\right )} \, dx}{-e+x} \, dx-(2 \log (e-x)) \int \frac {1}{e^{2 x}-3 x+3 x^2+e^x \log (64)} \, dx+(3 \log (e-x)) \int \frac {1}{\left (e^{2 x}-3 x+3 x^2+e^x \log (64)\right )^2} \, dx+(6 \log (e-x)) \int \frac {x^2}{\left (e^{2 x}-3 x+3 x^2+e^x \log (64)\right )^2} \, dx-(12 \log (e-x)) \int \frac {x}{\left (e^{2 x}-3 x+3 x^2+e^x \log (64)\right )^2} \, dx+(\log (64) \log (e-x)) \int \frac {e^x}{\left (e^{2 x}-3 x+3 x^2+e^x \log (64)\right )^2} \, dx-\int \frac {1}{(e-x) \left (e^{2 x}-3 x+3 x^2+e^x \log (64)\right )} \, dx-\int \frac {-3 \int \frac {1}{\left (e^{2 x}+3 (-1+x) x+e^x \log (64)\right )^2} \, dx-\log (64) \int \frac {e^x}{\left (e^{2 x}+3 (-1+x) x+e^x \log (64)\right )^2} \, dx+12 \int \frac {x}{\left (e^{2 x}+3 (-1+x) x+e^x \log (64)\right )^2} \, dx}{e-x} \, dx\\ &=-\left (2 \int \left (\frac {\int \frac {1}{e^{2 x}+3 (-1+x) x+e^x \log (64)} \, dx}{e-x}-\frac {e \int \frac {1}{(e-x) \left (e^{2 x}+3 (-1+x) x+e^x \log (64)\right )} \, dx}{e-x}\right ) \, dx\right )+6 \int \frac {\int \frac {x^2}{\left (e^{2 x}+3 (-1+x) x+e^x \log (64)\right )^2} \, dx}{e-x} \, dx+(2 e) \int \frac {\int \frac {1}{(e-x) \left (e^{2 x}+3 (-1+x) x+e^x \log (64)\right )} \, dx}{-e+x} \, dx-(2 \log (e-x)) \int \frac {1}{e^{2 x}-3 x+3 x^2+e^x \log (64)} \, dx+(3 \log (e-x)) \int \frac {1}{\left (e^{2 x}-3 x+3 x^2+e^x \log (64)\right )^2} \, dx+(6 \log (e-x)) \int \frac {x^2}{\left (e^{2 x}-3 x+3 x^2+e^x \log (64)\right )^2} \, dx-(12 \log (e-x)) \int \frac {x}{\left (e^{2 x}-3 x+3 x^2+e^x \log (64)\right )^2} \, dx+(\log (64) \log (e-x)) \int \frac {e^x}{\left (e^{2 x}-3 x+3 x^2+e^x \log (64)\right )^2} \, dx-\int \frac {1}{(e-x) \left (e^{2 x}-3 x+3 x^2+e^x \log (64)\right )} \, dx-\int \left (\frac {-3 \int \frac {1}{\left (e^{2 x}+3 (-1+x) x+e^x \log (64)\right )^2} \, dx-\log (64) \int \frac {e^x}{\left (e^{2 x}+3 (-1+x) x+e^x \log (64)\right )^2} \, dx}{e-x}+\frac {12 \int \frac {x}{\left (e^{2 x}+3 (-1+x) x+e^x \log (64)\right )^2} \, dx}{e-x}\right ) \, dx\\ &=-\left (2 \int \frac {\int \frac {1}{e^{2 x}+3 (-1+x) x+e^x \log (64)} \, dx}{e-x} \, dx\right )+6 \int \frac {\int \frac {x^2}{\left (e^{2 x}+3 (-1+x) x+e^x \log (64)\right )^2} \, dx}{e-x} \, dx-12 \int \frac {\int \frac {x}{\left (e^{2 x}+3 (-1+x) x+e^x \log (64)\right )^2} \, dx}{e-x} \, dx+(2 e) \int \frac {\int \frac {1}{(e-x) \left (e^{2 x}+3 (-1+x) x+e^x \log (64)\right )} \, dx}{e-x} \, dx+(2 e) \int \frac {\int \frac {1}{(e-x) \left (e^{2 x}+3 (-1+x) x+e^x \log (64)\right )} \, dx}{-e+x} \, dx-(2 \log (e-x)) \int \frac {1}{e^{2 x}-3 x+3 x^2+e^x \log (64)} \, dx+(3 \log (e-x)) \int \frac {1}{\left (e^{2 x}-3 x+3 x^2+e^x \log (64)\right )^2} \, dx+(6 \log (e-x)) \int \frac {x^2}{\left (e^{2 x}-3 x+3 x^2+e^x \log (64)\right )^2} \, dx-(12 \log (e-x)) \int \frac {x}{\left (e^{2 x}-3 x+3 x^2+e^x \log (64)\right )^2} \, dx+(\log (64) \log (e-x)) \int \frac {e^x}{\left (e^{2 x}-3 x+3 x^2+e^x \log (64)\right )^2} \, dx-\int \frac {1}{(e-x) \left (e^{2 x}-3 x+3 x^2+e^x \log (64)\right )} \, dx-\int \frac {-3 \int \frac {1}{\left (e^{2 x}+3 (-1+x) x+e^x \log (64)\right )^2} \, dx-\log (64) \int \frac {e^x}{\left (e^{2 x}+3 (-1+x) x+e^x \log (64)\right )^2} \, dx}{e-x} \, dx\\ &=-\left (2 \int \frac {\int \frac {1}{e^{2 x}+3 (-1+x) x+e^x \log (64)} \, dx}{e-x} \, dx\right )+6 \int \frac {\int \frac {x^2}{\left (e^{2 x}+3 (-1+x) x+e^x \log (64)\right )^2} \, dx}{e-x} \, dx-12 \int \frac {\int \frac {x}{\left (e^{2 x}+3 (-1+x) x+e^x \log (64)\right )^2} \, dx}{e-x} \, dx+(2 e) \int \frac {\int \frac {1}{(e-x) \left (e^{2 x}+3 (-1+x) x+e^x \log (64)\right )} \, dx}{e-x} \, dx+(2 e) \int \frac {\int \frac {1}{(e-x) \left (e^{2 x}+3 (-1+x) x+e^x \log (64)\right )} \, dx}{-e+x} \, dx-(2 \log (e-x)) \int \frac {1}{e^{2 x}-3 x+3 x^2+e^x \log (64)} \, dx+(3 \log (e-x)) \int \frac {1}{\left (e^{2 x}-3 x+3 x^2+e^x \log (64)\right )^2} \, dx+(6 \log (e-x)) \int \frac {x^2}{\left (e^{2 x}-3 x+3 x^2+e^x \log (64)\right )^2} \, dx-(12 \log (e-x)) \int \frac {x}{\left (e^{2 x}-3 x+3 x^2+e^x \log (64)\right )^2} \, dx+(\log (64) \log (e-x)) \int \frac {e^x}{\left (e^{2 x}-3 x+3 x^2+e^x \log (64)\right )^2} \, dx-\int \frac {1}{(e-x) \left (e^{2 x}-3 x+3 x^2+e^x \log (64)\right )} \, dx-\int \left (-\frac {3 \int \frac {1}{\left (e^{2 x}+3 (-1+x) x+e^x \log (64)\right )^2} \, dx}{e-x}-\frac {\log (64) \int \frac {e^x}{\left (e^{2 x}+3 (-1+x) x+e^x \log (64)\right )^2} \, dx}{e-x}\right ) \, dx\\ &=-\left (2 \int \frac {\int \frac {1}{e^{2 x}+3 (-1+x) x+e^x \log (64)} \, dx}{e-x} \, dx\right )+3 \int \frac {\int \frac {1}{\left (e^{2 x}+3 (-1+x) x+e^x \log (64)\right )^2} \, dx}{e-x} \, dx+6 \int \frac {\int \frac {x^2}{\left (e^{2 x}+3 (-1+x) x+e^x \log (64)\right )^2} \, dx}{e-x} \, dx-12 \int \frac {\int \frac {x}{\left (e^{2 x}+3 (-1+x) x+e^x \log (64)\right )^2} \, dx}{e-x} \, dx+(2 e) \int \frac {\int \frac {1}{(e-x) \left (e^{2 x}+3 (-1+x) x+e^x \log (64)\right )} \, dx}{e-x} \, dx+(2 e) \int \frac {\int \frac {1}{(e-x) \left (e^{2 x}+3 (-1+x) x+e^x \log (64)\right )} \, dx}{-e+x} \, dx+\log (64) \int \frac {\int \frac {e^x}{\left (e^{2 x}+3 (-1+x) x+e^x \log (64)\right )^2} \, dx}{e-x} \, dx-(2 \log (e-x)) \int \frac {1}{e^{2 x}-3 x+3 x^2+e^x \log (64)} \, dx+(3 \log (e-x)) \int \frac {1}{\left (e^{2 x}-3 x+3 x^2+e^x \log (64)\right )^2} \, dx+(6 \log (e-x)) \int \frac {x^2}{\left (e^{2 x}-3 x+3 x^2+e^x \log (64)\right )^2} \, dx-(12 \log (e-x)) \int \frac {x}{\left (e^{2 x}-3 x+3 x^2+e^x \log (64)\right )^2} \, dx+(\log (64) \log (e-x)) \int \frac {e^x}{\left (e^{2 x}-3 x+3 x^2+e^x \log (64)\right )^2} \, dx-\int \frac {1}{(e-x) \left (e^{2 x}-3 x+3 x^2+e^x \log (64)\right )} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.88, size = 29, normalized size = 0.97 \begin {gather*} \frac {\log (e-x)}{e^{2 x}-3 x+3 x^2+e^x \log (64)} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.84, size = 29, normalized size = 0.97 \begin {gather*} \frac {\log \left (-x + e\right )}{3 \, x^{2} + 6 \, e^{x} \log \relax (2) - 3 \, x + e^{\left (2 \, x\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.57, size = 29, normalized size = 0.97 \begin {gather*} \frac {\log \left (-x + e\right )}{3 \, x^{2} + 6 \, e^{x} \log \relax (2) - 3 \, x + e^{\left (2 \, x\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 30, normalized size = 1.00
method | result | size |
risch | \(\frac {\ln \left ({\mathrm e}-x \right )}{{\mathrm e}^{2 x}+3 x^{2}+6 \,{\mathrm e}^{x} \ln \relax (2)-3 x}\) | \(30\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.68, size = 29, normalized size = 0.97 \begin {gather*} \frac {\log \left (-x + e\right )}{3 \, x^{2} + 6 \, e^{x} \log \relax (2) - 3 \, x + e^{\left (2 \, x\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.17, size = 29, normalized size = 0.97 \begin {gather*} \frac {\ln \left (\mathrm {e}-x\right )}{{\mathrm {e}}^{2\,x}-3\,x+6\,{\mathrm {e}}^x\,\ln \relax (2)+3\,x^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.50, size = 27, normalized size = 0.90 \begin {gather*} \frac {\log {\left (e - x \right )}}{3 x^{2} - 3 x + e^{2 x} + 6 e^{x} \log {\relax (2 )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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