Optimal. Leaf size=24 \[ e^{-\frac {8 x^2}{\left (e^x-x\right )^2}} \log (-x+\log (x)) \]
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Rubi [B] Leaf count is larger than twice the leaf count of optimal. \(194\) vs. \(2(24)=48\).
time = 0.40, antiderivative size = 194, normalized size of antiderivative = 8.08, number of steps
used = 1, number of rules used = 1, integrand size = 184, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.005, Rules used = {2326}
\begin {gather*} \frac {e^{-\frac {8 x^2}{x^2-2 e^x x+e^{2 x}}} \left (e^x \left (x^3-x^4\right )-e^x \left (x^2-x^3\right ) \log (x)\right ) \log (\log (x)-x)}{\left (\frac {\left (e^x x-x+e^x-e^{2 x}\right ) x^2}{\left (x^2-2 e^x x+e^{2 x}\right )^2}+\frac {x}{x^2-2 e^x x+e^{2 x}}\right ) \left (-x^5+3 e^x x^4-3 e^{2 x} x^3+e^{3 x} x^2-\left (-x^4+3 e^x x^3-3 e^{2 x} x^2+e^{3 x} x\right ) \log (x)\right )} \end {gather*}
Antiderivative was successfully verified.
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Rule 2326
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\frac {e^{-\frac {8 x^2}{e^{2 x}-2 e^x x+x^2}} \left (e^x \left (x^3-x^4\right )-e^x \left (x^2-x^3\right ) \log (x)\right ) \log (-x+\log (x))}{\left (\frac {x^2 \left (e^x-e^{2 x}-x+e^x x\right )}{\left (e^{2 x}-2 e^x x+x^2\right )^2}+\frac {x}{e^{2 x}-2 e^x x+x^2}\right ) \left (e^{3 x} x^2-3 e^{2 x} x^3+3 e^x x^4-x^5-\left (e^{3 x} x-3 e^{2 x} x^2+3 e^x x^3-x^4\right ) \log (x)\right )}\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.13, size = 24, normalized size = 1.00 \begin {gather*} e^{-\frac {8 x^2}{\left (e^x-x\right )^2}} \log (-x+\log (x)) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.15, size = 34, normalized size = 1.42
method | result | size |
risch | \({\mathrm e}^{\frac {8 x^{2}}{-{\mathrm e}^{2 x}+2 \,{\mathrm e}^{x} x -x^{2}}} \ln \left (\ln \left (x \right )-x \right )\) | \(34\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.58, size = 44, normalized size = 1.83 \begin {gather*} e^{\left (-\frac {8 \, e^{\left (2 \, x\right )}}{x^{2} - 2 \, x e^{x} + e^{\left (2 \, x\right )}} - \frac {16 \, e^{x}}{x - e^{x}} - 8\right )} \log \left (-x + \log \left (x\right )\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 29, normalized size = 1.21 \begin {gather*} e^{\left (-\frac {8 \, x^{2}}{x^{2} - 2 \, x e^{x} + e^{\left (2 \, x\right )}}\right )} \log \left (-x + \log \left (x\right )\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 51.06, size = 27, normalized size = 1.12 \begin {gather*} e^{- \frac {8 x^{2}}{x^{2} - 2 x e^{x} + e^{2 x}}} \log {\left (- x + \log {\left (x \right )} \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \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 = -0.04 \begin {gather*} \int -\frac {{\mathrm {e}}^{-\frac {8\,x^2}{{\mathrm {e}}^{2\,x}-2\,x\,{\mathrm {e}}^x+x^2}}\,\left ({\mathrm {e}}^{2\,x}\,\left (3\,x-3\,x^2\right )-{\mathrm {e}}^x\,\left (3\,x^2-3\,x^3\right )-\ln \left (\ln \left (x\right )-x\right )\,\left ({\mathrm {e}}^x\,\left (16\,x^3-16\,x^4\right )-{\mathrm {e}}^x\,\ln \left (x\right )\,\left (16\,x^2-16\,x^3\right )\right )+{\mathrm {e}}^{3\,x}\,\left (x-1\right )+x^3-x^4\right )}{3\,x^3\,{\mathrm {e}}^{2\,x}-x^2\,{\mathrm {e}}^{3\,x}-3\,x^4\,{\mathrm {e}}^x+\ln \left (x\right )\,\left (x\,{\mathrm {e}}^{3\,x}+3\,x^3\,{\mathrm {e}}^x-3\,x^2\,{\mathrm {e}}^{2\,x}-x^4\right )+x^5} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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