Optimal. Leaf size=28 \[ \frac {3-x+5 e^{-x} x-\log (2)}{-4+x^2 \log (x)} \]
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Rubi [F]
time = 1.02, 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 {-20+20 x-5 x^2+e^x \left (4-3 x+x^2+x \log (2)\right )+\left (-5 x^2-5 x^3+e^x \left (-6 x+x^2+2 x \log (2)\right )\right ) \log (x)}{16 e^x-8 e^x x^2 \log (x)+e^x x^4 \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 {e^{-x} \left (-5 (-2+x)^2+e^x \left (4+x^2+x (-3+\log (2))\right )+x \left (-5 x (1+x)+e^x (-6+x+\log (4))\right ) \log (x)\right )}{\left (4-x^2 \log (x)\right )^2} \, dx\\ &=\int \left (-\frac {5 e^{-x} \left (4-4 x+x^2+x^2 \log (x)+x^3 \log (x)\right )}{\left (-4+x^2 \log (x)\right )^2}+\frac {4+x^2-3 x \left (1-\frac {\log (2)}{3}\right )+x^2 \log (x)-6 x \left (1-\frac {\log (2)}{3}\right ) \log (x)}{\left (4-x^2 \log (x)\right )^2}\right ) \, dx\\ &=-\left (5 \int \frac {e^{-x} \left (4-4 x+x^2+x^2 \log (x)+x^3 \log (x)\right )}{\left (-4+x^2 \log (x)\right )^2} \, dx\right )+\int \frac {4+x^2-3 x \left (1-\frac {\log (2)}{3}\right )+x^2 \log (x)-6 x \left (1-\frac {\log (2)}{3}\right ) \log (x)}{\left (4-x^2 \log (x)\right )^2} \, dx\\ &=-\frac {5 e^{-x} \left (4 x-x^3 \log (x)\right )}{\left (4-x^2 \log (x)\right )^2}+\int \frac {4+x^2+x (-3+\log (2))+x (-6+x+\log (4)) \log (x)}{\left (4-x^2 \log (x)\right )^2} \, dx\\ &=-\frac {5 e^{-x} \left (4 x-x^3 \log (x)\right )}{\left (4-x^2 \log (x)\right )^2}+\int \left (\frac {8 x+x^3-x^2 (3-\log (2))-4 (6-\log (4))}{x \left (4-x^2 \log (x)\right )^2}+\frac {-6+x+\log (4)}{x \left (-4+x^2 \log (x)\right )}\right ) \, dx\\ &=-\frac {5 e^{-x} \left (4 x-x^3 \log (x)\right )}{\left (4-x^2 \log (x)\right )^2}+\int \frac {8 x+x^3-x^2 (3-\log (2))-4 (6-\log (4))}{x \left (4-x^2 \log (x)\right )^2} \, dx+\int \frac {-6+x+\log (4)}{x \left (-4+x^2 \log (x)\right )} \, dx\\ &=-\frac {5 e^{-x} \left (4 x-x^3 \log (x)\right )}{\left (4-x^2 \log (x)\right )^2}+\int \left (\frac {8}{\left (-4+x^2 \log (x)\right )^2}+\frac {x^2}{\left (-4+x^2 \log (x)\right )^2}+\frac {x (-3+\log (2))}{\left (-4+x^2 \log (x)\right )^2}+\frac {4 (-6+\log (4))}{x \left (-4+x^2 \log (x)\right )^2}\right ) \, dx+\int \left (\frac {1}{-4+x^2 \log (x)}+\frac {-6+\log (4)}{x \left (-4+x^2 \log (x)\right )}\right ) \, dx\\ &=-\frac {5 e^{-x} \left (4 x-x^3 \log (x)\right )}{\left (4-x^2 \log (x)\right )^2}+8 \int \frac {1}{\left (-4+x^2 \log (x)\right )^2} \, dx+(-3+\log (2)) \int \frac {x}{\left (-4+x^2 \log (x)\right )^2} \, dx-(4 (6-\log (4))) \int \frac {1}{x \left (-4+x^2 \log (x)\right )^2} \, dx+(-6+\log (4)) \int \frac {1}{x \left (-4+x^2 \log (x)\right )} \, dx+\int \frac {x^2}{\left (-4+x^2 \log (x)\right )^2} \, dx+\int \frac {1}{-4+x^2 \log (x)} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(58\) vs. \(2(28)=56\).
time = 0.40, size = 58, normalized size = 2.07 \begin {gather*} \frac {e^{-x} \left (5 x \left (8+x^2\right )-e^x \left (8 x+x^3+x^2 (-3+\log (2))+4 (-6+\log (4))\right )\right )}{\left (8+x^2\right ) \left (-4+x^2 \log (x)\right )} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 1.96, size = 34, normalized size = 1.21
method | result | size |
risch | \(-\frac {\left ({\mathrm e}^{x} \ln \left (2\right )+{\mathrm e}^{x} x -5 x -3 \,{\mathrm e}^{x}\right ) {\mathrm e}^{-x}}{x^{2} \ln \left (x \right )-4}\) | \(34\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.52, size = 27, normalized size = 0.96 \begin {gather*} \frac {5 \, x e^{\left (-x\right )} - x - \log \left (2\right ) + 3}{x^{2} \log \left (x\right ) - 4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.39, size = 29, normalized size = 1.04 \begin {gather*} -\frac {{\left (x + \log \left (2\right ) - 3\right )} e^{x} - 5 \, x}{x^{2} e^{x} \log \left (x\right ) - 4 \, e^{x}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.11, size = 29, normalized size = 1.04 \begin {gather*} \frac {5 x e^{- x}}{x^{2} \log {\left (x \right )} - 4} + \frac {- x - \log {\left (2 \right )} + 3}{x^{2} \log {\left (x \right )} - 4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.41, size = 27, normalized size = 0.96 \begin {gather*} \frac {5 \, x e^{\left (-x\right )} - x - \log \left (2\right ) + 3}{x^{2} \log \left (x\right ) - 4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 5.58, size = 86, normalized size = 3.07 \begin {gather*} -\frac {x^2\,{\mathrm {e}}^{-x}\,\left (8\,{\mathrm {e}}^x-40\right )-x\,{\mathrm {e}}^{-x}\,\left (24\,{\mathrm {e}}^x-8\,{\mathrm {e}}^x\,\ln \left (2\right )\right )+x^4\,{\mathrm {e}}^{-x}\,\left ({\mathrm {e}}^x-5\right )-x^3\,{\mathrm {e}}^{-x}\,\left (3\,{\mathrm {e}}^x-{\mathrm {e}}^x\,\ln \left (2\right )\right )}{\left (x^3+8\,x\right )\,\left (x^2\,\ln \left (x\right )-4\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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