Optimal. Leaf size=34 \[ \frac {e^{5-e^x+\frac {5+\frac {4}{x}}{x-x (9-\log (x))}}}{x} \]
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Rubi [F]
time = 38.88, 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 {\exp \left (\frac {4+5 x-40 x^2+8 e^x x^2+\left (13 x^2-e^x x^2\right ) \log (x)-x^2 \log ^2(x)}{-8 x^2+x^2 \log (x)}\right ) \left (60+35 x-64 x^2-64 e^x x^3+\left (-8-5 x+16 x^2+16 e^x x^3\right ) \log (x)+\left (-x^2-e^x x^3\right ) \log ^2(x)\right )}{64 x^3-16 x^3 \log (x)+x^3 \log ^2(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 {\exp \left (\frac {4+5 x-40 x^2+8 e^x x^2+\left (13 x^2-e^x x^2\right ) \log (x)-x^2 \log ^2(x)}{-8 x^2+x^2 \log (x)}\right ) \left (60+35 x-64 x^2-64 e^x x^3+\left (-8-5 x+16 x^2+16 e^x x^3\right ) \log (x)+\left (-x^2-e^x x^3\right ) \log ^2(x)\right )}{x^3 (8-\log (x))^2} \, dx\\ &=\int \left (-\exp \left (x+\frac {4+5 x-40 x^2+8 e^x x^2+\left (13 x^2-e^x x^2\right ) \log (x)-x^2 \log ^2(x)}{-8 x^2+x^2 \log (x)}\right )+\frac {60 \exp \left (\frac {4+5 x-40 x^2+8 e^x x^2+\left (13 x^2-e^x x^2\right ) \log (x)-x^2 \log ^2(x)}{-8 x^2+x^2 \log (x)}\right )}{x^3 (-8+\log (x))^2}+\frac {35 \exp \left (\frac {4+5 x-40 x^2+8 e^x x^2+\left (13 x^2-e^x x^2\right ) \log (x)-x^2 \log ^2(x)}{-8 x^2+x^2 \log (x)}\right )}{x^2 (-8+\log (x))^2}-\frac {64 \exp \left (\frac {4+5 x-40 x^2+8 e^x x^2+\left (13 x^2-e^x x^2\right ) \log (x)-x^2 \log ^2(x)}{-8 x^2+x^2 \log (x)}\right )}{x (-8+\log (x))^2}-\frac {8 \exp \left (\frac {4+5 x-40 x^2+8 e^x x^2+\left (13 x^2-e^x x^2\right ) \log (x)-x^2 \log ^2(x)}{-8 x^2+x^2 \log (x)}\right ) \log (x)}{x^3 (-8+\log (x))^2}-\frac {5 \exp \left (\frac {4+5 x-40 x^2+8 e^x x^2+\left (13 x^2-e^x x^2\right ) \log (x)-x^2 \log ^2(x)}{-8 x^2+x^2 \log (x)}\right ) \log (x)}{x^2 (-8+\log (x))^2}+\frac {16 \exp \left (\frac {4+5 x-40 x^2+8 e^x x^2+\left (13 x^2-e^x x^2\right ) \log (x)-x^2 \log ^2(x)}{-8 x^2+x^2 \log (x)}\right ) \log (x)}{x (-8+\log (x))^2}-\frac {\exp \left (\frac {4+5 x-40 x^2+8 e^x x^2+\left (13 x^2-e^x x^2\right ) \log (x)-x^2 \log ^2(x)}{-8 x^2+x^2 \log (x)}\right ) \log ^2(x)}{x (-8+\log (x))^2}\right ) \, dx\\ &=-\left (5 \int \frac {\exp \left (\frac {4+5 x-40 x^2+8 e^x x^2+\left (13 x^2-e^x x^2\right ) \log (x)-x^2 \log ^2(x)}{-8 x^2+x^2 \log (x)}\right ) \log (x)}{x^2 (-8+\log (x))^2} \, dx\right )-8 \int \frac {\exp \left (\frac {4+5 x-40 x^2+8 e^x x^2+\left (13 x^2-e^x x^2\right ) \log (x)-x^2 \log ^2(x)}{-8 x^2+x^2 \log (x)}\right ) \log (x)}{x^3 (-8+\log (x))^2} \, dx+16 \int \frac {\exp \left (\frac {4+5 x-40 x^2+8 e^x x^2+\left (13 x^2-e^x x^2\right ) \log (x)-x^2 \log ^2(x)}{-8 x^2+x^2 \log (x)}\right ) \log (x)}{x (-8+\log (x))^2} \, dx+35 \int \frac {\exp \left (\frac {4+5 x-40 x^2+8 e^x x^2+\left (13 x^2-e^x x^2\right ) \log (x)-x^2 \log ^2(x)}{-8 x^2+x^2 \log (x)}\right )}{x^2 (-8+\log (x))^2} \, dx+60 \int \frac {\exp \left (\frac {4+5 x-40 x^2+8 e^x x^2+\left (13 x^2-e^x x^2\right ) \log (x)-x^2 \log ^2(x)}{-8 x^2+x^2 \log (x)}\right )}{x^3 (-8+\log (x))^2} \, dx-64 \int \frac {\exp \left (\frac {4+5 x-40 x^2+8 e^x x^2+\left (13 x^2-e^x x^2\right ) \log (x)-x^2 \log ^2(x)}{-8 x^2+x^2 \log (x)}\right )}{x (-8+\log (x))^2} \, dx-\int \exp \left (x+\frac {4+5 x-40 x^2+8 e^x x^2+\left (13 x^2-e^x x^2\right ) \log (x)-x^2 \log ^2(x)}{-8 x^2+x^2 \log (x)}\right ) \, dx-\int \frac {\exp \left (\frac {4+5 x-40 x^2+8 e^x x^2+\left (13 x^2-e^x x^2\right ) \log (x)-x^2 \log ^2(x)}{-8 x^2+x^2 \log (x)}\right ) \log ^2(x)}{x (-8+\log (x))^2} \, dx\\ &=-\left (5 \int \left (\frac {8 e^{\frac {4+5 x-40 x^2+8 e^x x^2+\left (13 x^2-e^x x^2\right ) \log (x)-x^2 \log ^2(x)}{-8 x^2+x^2 \log (x)}}}{x^2 (-8+\log (x))^2}+\frac {e^{\frac {4+5 x-40 x^2+8 e^x x^2+\left (13 x^2-e^x x^2\right ) \log (x)-x^2 \log ^2(x)}{-8 x^2+x^2 \log (x)}}}{x^2 (-8+\log (x))}\right ) \, dx\right )-8 \int \left (\frac {8 e^{\frac {4+5 x-40 x^2+8 e^x x^2+\left (13 x^2-e^x x^2\right ) \log (x)-x^2 \log ^2(x)}{-8 x^2+x^2 \log (x)}}}{x^3 (-8+\log (x))^2}+\frac {e^{\frac {4+5 x-40 x^2+8 e^x x^2+\left (13 x^2-e^x x^2\right ) \log (x)-x^2 \log ^2(x)}{-8 x^2+x^2 \log (x)}}}{x^3 (-8+\log (x))}\right ) \, dx+16 \int \left (\frac {8 e^{\frac {4+5 x-40 x^2+8 e^x x^2+\left (13 x^2-e^x x^2\right ) \log (x)-x^2 \log ^2(x)}{-8 x^2+x^2 \log (x)}}}{x (-8+\log (x))^2}+\frac {e^{\frac {4+5 x-40 x^2+8 e^x x^2+\left (13 x^2-e^x x^2\right ) \log (x)-x^2 \log ^2(x)}{-8 x^2+x^2 \log (x)}}}{x (-8+\log (x))}\right ) \, dx+35 \int \frac {e^{\frac {4+5 x-40 x^2+8 e^x x^2+\left (13 x^2-e^x x^2\right ) \log (x)-x^2 \log ^2(x)}{-8 x^2+x^2 \log (x)}}}{x^2 (-8+\log (x))^2} \, dx+60 \int \frac {e^{\frac {4+5 x-40 x^2+8 e^x x^2+\left (13 x^2-e^x x^2\right ) \log (x)-x^2 \log ^2(x)}{-8 x^2+x^2 \log (x)}}}{x^3 (-8+\log (x))^2} \, dx-64 \int \frac {e^{\frac {4+5 x-40 x^2+8 e^x x^2+\left (13 x^2-e^x x^2\right ) \log (x)-x^2 \log ^2(x)}{-8 x^2+x^2 \log (x)}}}{x (-8+\log (x))^2} \, dx-\int e^{x+\frac {4+5 x-40 x^2+8 e^x x^2+\left (13 x^2-e^x x^2\right ) \log (x)-x^2 \log ^2(x)}{-8 x^2+x^2 \log (x)}} \, dx-\int \left (\frac {e^{\frac {4+5 x-40 x^2+8 e^x x^2+\left (13 x^2-e^x x^2\right ) \log (x)-x^2 \log ^2(x)}{-8 x^2+x^2 \log (x)}}}{x}+\frac {64 e^{\frac {4+5 x-40 x^2+8 e^x x^2+\left (13 x^2-e^x x^2\right ) \log (x)-x^2 \log ^2(x)}{-8 x^2+x^2 \log (x)}}}{x (-8+\log (x))^2}+\frac {16 e^{\frac {4+5 x-40 x^2+8 e^x x^2+\left (13 x^2-e^x x^2\right ) \log (x)-x^2 \log ^2(x)}{-8 x^2+x^2 \log (x)}}}{x (-8+\log (x))}\right ) \, dx\\ &=-\left (5 \int \frac {\exp \left (\frac {4+5 x-40 x^2+8 e^x x^2+\left (13 x^2-e^x x^2\right ) \log (x)-x^2 \log ^2(x)}{-8 x^2+x^2 \log (x)}\right )}{x^2 (-8+\log (x))} \, dx\right )-8 \int \frac {\exp \left (\frac {4+5 x-40 x^2+8 e^x x^2+\left (13 x^2-e^x x^2\right ) \log (x)-x^2 \log ^2(x)}{-8 x^2+x^2 \log (x)}\right )}{x^3 (-8+\log (x))} \, dx+35 \int \frac {\exp \left (\frac {4+5 x-40 x^2+8 e^x x^2+\left (13 x^2-e^x x^2\right ) \log (x)-x^2 \log ^2(x)}{-8 x^2+x^2 \log (x)}\right )}{x^2 (-8+\log (x))^2} \, dx-40 \int \frac {\exp \left (\frac {4+5 x-40 x^2+8 e^x x^2+\left (13 x^2-e^x x^2\right ) \log (x)-x^2 \log ^2(x)}{-8 x^2+x^2 \log (x)}\right )}{x^2 (-8+\log (x))^2} \, dx+60 \int \frac {\exp \left (\frac {4+5 x-40 x^2+8 e^x x^2+\left (13 x^2-e^x x^2\right ) \log (x)-x^2 \log ^2(x)}{-8 x^2+x^2 \log (x)}\right )}{x^3 (-8+\log (x))^2} \, dx-64 \int \frac {\exp \left (\frac {4+5 x-40 x^2+8 e^x x^2+\left (13 x^2-e^x x^2\right ) \log (x)-x^2 \log ^2(x)}{-8 x^2+x^2 \log (x)}\right )}{x^3 (-8+\log (x))^2} \, dx-2 \left (64 \int \frac {\exp \left (\frac {4+5 x-40 x^2+8 e^x x^2+\left (13 x^2-e^x x^2\right ) \log (x)-x^2 \log ^2(x)}{-8 x^2+x^2 \log (x)}\right )}{x (-8+\log (x))^2} \, dx\right )+128 \int \frac {\exp \left (\frac {4+5 x-40 x^2+8 e^x x^2+\left (13 x^2-e^x x^2\right ) \log (x)-x^2 \log ^2(x)}{-8 x^2+x^2 \log (x)}\right )}{x (-8+\log (x))^2} \, dx-\int \exp \left (x+\frac {4+5 x-40 x^2+8 e^x x^2+\left (13 x^2-e^x x^2\right ) \log (x)-x^2 \log ^2(x)}{-8 x^2+x^2 \log (x)}\right ) \, dx-\int \frac {\exp \left (\frac {4+5 x-40 x^2+8 e^x x^2+\left (13 x^2-e^x x^2\right ) \log (x)-x^2 \log ^2(x)}{-8 x^2+x^2 \log (x)}\right )}{x} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.44, size = 43, normalized size = 1.26 \begin {gather*} \frac {e^{\frac {4+5 x+8 \left (-5+e^x\right ) x^2-\left (-5+e^x\right ) x^2 \log (x)}{x^2 (-8+\log (x))}}}{x} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.19, size = 53, normalized size = 1.56
method | result | size |
risch | \({\mathrm e}^{-\frac {x^{2} \ln \left (x \right )^{2}+x^{2} {\mathrm e}^{x} \ln \left (x \right )-13 x^{2} \ln \left (x \right )-8 \,{\mathrm e}^{x} x^{2}+40 x^{2}-5 x -4}{x^{2} \left (\ln \left (x \right )-8\right )}}\) | \(53\) |
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. 82 vs.
\(2 (29) = 58\).
time = 0.45, size = 82, normalized size = 2.41 \begin {gather*} e^{\left (-\frac {e^{x} \log \left (x\right )}{\log \left (x\right ) - 8} - \frac {\log \left (x\right )^{2}}{\log \left (x\right ) - 8} + \frac {8 \, e^{x}}{\log \left (x\right ) - 8} + \frac {13 \, \log \left (x\right )}{\log \left (x\right ) - 8} + \frac {4}{x^{2} \log \left (x\right ) - 8 \, x^{2}} + \frac {5}{x \log \left (x\right ) - 8 \, x} - \frac {40}{\log \left (x\right ) - 8}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.43, size = 57, normalized size = 1.68 \begin {gather*} e^{\left (-\frac {x^{2} \log \left (x\right )^{2} - 8 \, x^{2} e^{x} + 40 \, x^{2} + {\left (x^{2} e^{x} - 13 \, x^{2}\right )} \log \left (x\right ) - 5 \, x - 4}{x^{2} \log \left (x\right ) - 8 \, x^{2}}\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. 54 vs.
\(2 (20) = 40\).
time = 0.67, size = 54, normalized size = 1.59 \begin {gather*} e^{\frac {8 x^{2} e^{x} - x^{2} \log {\left (x \right )}^{2} - 40 x^{2} + 5 x + \left (- x^{2} e^{x} + 13 x^{2}\right ) \log {\left (x \right )} + 4}{x^{2} \log {\left (x \right )} - 8 x^{2}}} \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. 142 vs.
\(2 (29) = 58\).
time = 0.63, size = 142, normalized size = 4.18 \begin {gather*} e^{\left (-\frac {x^{2} e^{x} \log \left (x\right )}{x^{2} \log \left (x\right ) - 8 \, x^{2}} - \frac {x^{2} \log \left (x\right )^{2}}{x^{2} \log \left (x\right ) - 8 \, x^{2}} + \frac {8 \, x^{2} e^{x}}{x^{2} \log \left (x\right ) - 8 \, x^{2}} + \frac {13 \, x^{2} \log \left (x\right )}{x^{2} \log \left (x\right ) - 8 \, x^{2}} - \frac {40 \, x^{2}}{x^{2} \log \left (x\right ) - 8 \, x^{2}} + \frac {5 \, x}{x^{2} \log \left (x\right ) - 8 \, x^{2}} + \frac {4}{x^{2} \log \left (x\right ) - 8 \, x^{2}}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 5.53, size = 117, normalized size = 3.44 \begin {gather*} \frac {{\mathrm {e}}^{\frac {5\,x}{x^2\,\ln \left (x\right )-8\,x^2}}\,{\mathrm {e}}^{\frac {8\,x^2\,{\mathrm {e}}^x}{x^2\,\ln \left (x\right )-8\,x^2}}\,{\mathrm {e}}^{-\frac {40\,x^2}{x^2\,\ln \left (x\right )-8\,x^2}}\,{\mathrm {e}}^{\frac {4}{x^2\,\ln \left (x\right )-8\,x^2}}\,{\mathrm {e}}^{-\frac {x^2\,{\ln \left (x\right )}^2}{x^2\,\ln \left (x\right )-8\,x^2}}}{x^{\frac {{\mathrm {e}}^x-13}{\ln \left (x\right )-8}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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