Optimal. Leaf size=33 \[ x-x^2+3 e^{-\frac {e^x (5+x)}{3+x-\log ^2(2)}} (2+x) \]
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Rubi [F]
time = 79.84, 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^{-\frac {e^x (-5-x)}{-3-x+\log ^2(2)}} \left (27+18 x+3 x^2+(-18-6 x) \log ^2(2)+3 \log ^4(2)+e^x \left (-78-87 x-30 x^2-3 x^3+\left (36+24 x+3 x^2\right ) \log ^2(2)\right )+e^{\frac {e^x (-5-x)}{-3-x+\log ^2(2)}} \left (9-12 x-11 x^2-2 x^3+\left (-6+10 x+4 x^2\right ) \log ^2(2)+(1-2 x) \log ^4(2)\right )\right )}{9+6 x+x^2+(-6-2 x) \log ^2(2)+\log ^4(2)} \, 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^{-\frac {e^x (-5-x)}{-3-x+\log ^2(2)}} \left (18 x+3 x^2+(-18-6 x) \log ^2(2)+e^x \left (-78-87 x-30 x^2-3 x^3+\left (36+24 x+3 x^2\right ) \log ^2(2)\right )+27 \left (1+\frac {\log ^4(2)}{9}\right )+e^{\frac {e^x (-5-x)}{-3-x+\log ^2(2)}} \left (9-12 x-11 x^2-2 x^3+\left (-6+10 x+4 x^2\right ) \log ^2(2)+(1-2 x) \log ^4(2)\right )\right )}{x^2+2 x \left (3-\log ^2(2)\right )+\left (-3+\log ^2(2)\right )^2} \, dx\\ &=\int \frac {e^{-\frac {e^x (-5-x)}{-3-x+\log ^2(2)}} \left (18 x+3 x^2+(-18-6 x) \log ^2(2)+e^x \left (-78-87 x-30 x^2-3 x^3+\left (36+24 x+3 x^2\right ) \log ^2(2)\right )+27 \left (1+\frac {\log ^4(2)}{9}\right )+e^{\frac {e^x (-5-x)}{-3-x+\log ^2(2)}} \left (9-12 x-11 x^2-2 x^3+\left (-6+10 x+4 x^2\right ) \log ^2(2)+(1-2 x) \log ^4(2)\right )\right )}{\left (3+x-\log ^2(2)\right )^2} \, dx\\ &=\int \left (-\exp \left (\frac {e^x (5+x)}{3+x-\log ^2(2)}-\frac {e^x (-5-x)}{-3-x+\log ^2(2)}\right ) (-1+2 x)+\frac {18 e^{-\frac {e^x (-5-x)}{-3-x+\log ^2(2)}} x}{\left (3+x-\log ^2(2)\right )^2}+\frac {3 e^{-\frac {e^x (-5-x)}{-3-x+\log ^2(2)}} x^2}{\left (3+x-\log ^2(2)\right )^2}-\frac {6 e^{-\frac {e^x (-5-x)}{-3-x+\log ^2(2)}} (3+x) \log ^2(2)}{\left (3+x-\log ^2(2)\right )^2}+\frac {3 e^{-\frac {e^x (-5-x)}{-3-x+\log ^2(2)}} \left (9+\log ^4(2)\right )}{\left (3+x-\log ^2(2)\right )^2}+\frac {3 e^{x-\frac {e^x (-5-x)}{-3-x+\log ^2(2)}} (2+x) \left (-13-x^2+6 \log ^2(2)-x \left (8-\log ^2(2)\right )\right )}{\left (3+x-\log ^2(2)\right )^2}\right ) \, dx\\ &=3 \int \frac {e^{-\frac {e^x (-5-x)}{-3-x+\log ^2(2)}} x^2}{\left (3+x-\log ^2(2)\right )^2} \, dx+3 \int \frac {e^{x-\frac {e^x (-5-x)}{-3-x+\log ^2(2)}} (2+x) \left (-13-x^2+6 \log ^2(2)-x \left (8-\log ^2(2)\right )\right )}{\left (3+x-\log ^2(2)\right )^2} \, dx+18 \int \frac {e^{-\frac {e^x (-5-x)}{-3-x+\log ^2(2)}} x}{\left (3+x-\log ^2(2)\right )^2} \, dx-\left (6 \log ^2(2)\right ) \int \frac {e^{-\frac {e^x (-5-x)}{-3-x+\log ^2(2)}} (3+x)}{\left (3+x-\log ^2(2)\right )^2} \, dx+\left (3 \left (9+\log ^4(2)\right )\right ) \int \frac {e^{-\frac {e^x (-5-x)}{-3-x+\log ^2(2)}}}{\left (3+x-\log ^2(2)\right )^2} \, dx-\int \exp \left (\frac {e^x (5+x)}{3+x-\log ^2(2)}-\frac {e^x (-5-x)}{-3-x+\log ^2(2)}\right ) (-1+2 x) \, dx\\ &=-\frac {1}{4} (1-2 x)^2+3 \int \left (e^{-\frac {e^x (-5-x)}{-3-x+\log ^2(2)}}+\frac {2 e^{-\frac {e^x (-5-x)}{-3-x+\log ^2(2)}} \left (-3+\log ^2(2)\right )}{3+x-\log ^2(2)}+\frac {e^{-\frac {e^x (-5-x)}{-3-x+\log ^2(2)}} \left (-3+\log ^2(2)\right )^2}{\left (3+x-\log ^2(2)\right )^2}\right ) \, dx+3 \int \left (-e^{x-\frac {e^x (-5-x)}{-3-x+\log ^2(2)}} x+e^{x-\frac {e^x (-5-x)}{-3-x+\log ^2(2)}} \left (-4-\log ^2(2)\right )+\frac {e^{x-\frac {e^x (-5-x)}{-3-x+\log ^2(2)}} \left (4-\log ^4(2)\right )}{3+x-\log ^2(2)}+\frac {e^{x-\frac {e^x (-5-x)}{-3-x+\log ^2(2)}} \left (-2+\log ^2(2)+\log ^4(2)\right )}{\left (3+x-\log ^2(2)\right )^2}\right ) \, dx+18 \int \left (\frac {e^{-\frac {e^x (-5-x)}{-3-x+\log ^2(2)}}}{3+x-\log ^2(2)}+\frac {e^{-\frac {e^x (-5-x)}{-3-x+\log ^2(2)}} \left (-3+\log ^2(2)\right )}{\left (3+x-\log ^2(2)\right )^2}\right ) \, dx-\left (6 \log ^2(2)\right ) \int \left (\frac {e^{-\frac {e^x (-5-x)}{-3-x+\log ^2(2)}}}{3+x-\log ^2(2)}+\frac {e^{-\frac {e^x (-5-x)}{-3-x+\log ^2(2)}} \log ^2(2)}{\left (-3-x+\log ^2(2)\right )^2}\right ) \, dx+\left (3 \left (9+\log ^4(2)\right )\right ) \int \frac {e^{-\frac {e^x (-5-x)}{-3-x+\log ^2(2)}}}{\left (3+x-\log ^2(2)\right )^2} \, dx\\ &=-\frac {1}{4} (1-2 x)^2+3 \int e^{-\frac {e^x (-5-x)}{-3-x+\log ^2(2)}} \, dx-3 \int e^{x-\frac {e^x (-5-x)}{-3-x+\log ^2(2)}} x \, dx+18 \int \frac {e^{-\frac {e^x (-5-x)}{-3-x+\log ^2(2)}}}{3+x-\log ^2(2)} \, dx-\left (6 \log ^2(2)\right ) \int \frac {e^{-\frac {e^x (-5-x)}{-3-x+\log ^2(2)}}}{3+x-\log ^2(2)} \, dx-\left (6 \log ^4(2)\right ) \int \frac {e^{-\frac {e^x (-5-x)}{-3-x+\log ^2(2)}}}{\left (-3-x+\log ^2(2)\right )^2} \, dx-\left (6 \left (3-\log ^2(2)\right )\right ) \int \frac {e^{-\frac {e^x (-5-x)}{-3-x+\log ^2(2)}}}{3+x-\log ^2(2)} \, dx-\left (18 \left (3-\log ^2(2)\right )\right ) \int \frac {e^{-\frac {e^x (-5-x)}{-3-x+\log ^2(2)}}}{\left (3+x-\log ^2(2)\right )^2} \, dx+\left (3 \left (3-\log ^2(2)\right )^2\right ) \int \frac {e^{-\frac {e^x (-5-x)}{-3-x+\log ^2(2)}}}{\left (3+x-\log ^2(2)\right )^2} \, dx-\left (3 \left (4+\log ^2(2)\right )\right ) \int e^{x-\frac {e^x (-5-x)}{-3-x+\log ^2(2)}} \, dx+\left (3 \left (4-\log ^4(2)\right )\right ) \int \frac {e^{x-\frac {e^x (-5-x)}{-3-x+\log ^2(2)}}}{3+x-\log ^2(2)} \, dx-\left (3 \left (2-\log ^2(2)-\log ^4(2)\right )\right ) \int \frac {e^{x-\frac {e^x (-5-x)}{-3-x+\log ^2(2)}}}{\left (3+x-\log ^2(2)\right )^2} \, dx+\left (3 \left (9+\log ^4(2)\right )\right ) \int \frac {e^{-\frac {e^x (-5-x)}{-3-x+\log ^2(2)}}}{\left (3+x-\log ^2(2)\right )^2} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.28, size = 34, normalized size = 1.03 \begin {gather*} x-x^2+e^{-\frac {e^x (5+x)}{3+x-\log ^2(2)}} (6+3 x) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 2.78, size = 32, normalized size = 0.97
method | result | size |
risch | \(-x^{2}+x +\left (6+3 x \right ) {\mathrm e}^{\frac {\left (5+x \right ) {\mathrm e}^{x}}{\ln \left (2\right )^{2}-3-x}}\) | \(32\) |
norman | \(\frac {\left (x^{3} {\mathrm e}^{\frac {\left (-x -5\right ) {\mathrm e}^{x}}{\ln \left (2\right )^{2}-3-x}}+\left (-15+3 \ln \left (2\right )^{2}\right ) x +\left (\ln \left (2\right )^{4}-6 \ln \left (2\right )^{2}+9\right ) {\mathrm e}^{\frac {\left (-x -5\right ) {\mathrm e}^{x}}{\ln \left (2\right )^{2}-3-x}}+\left (-\ln \left (2\right )^{2}+2\right ) x^{2} {\mathrm e}^{\frac {\left (-x -5\right ) {\mathrm e}^{x}}{\ln \left (2\right )^{2}-3-x}}-3 x^{2}+6 \ln \left (2\right )^{2}-18\right ) {\mathrm e}^{-\frac {\left (-x -5\right ) {\mathrm e}^{x}}{\ln \left (2\right )^{2}-3-x}}}{\ln \left (2\right )^{2}-3-x}\) | \(147\) |
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. 357 vs.
\(2 (30) = 60\).
time = 0.62, size = 357, normalized size = 10.82 \begin {gather*} -2 \, {\left (\frac {\log \left (2\right )^{2} - 3}{\log \left (2\right )^{2} - x - 3} + \log \left (-\log \left (2\right )^{2} + x + 3\right )\right )} \log \left (2\right )^{4} + 4 \, {\left (2 \, {\left (\log \left (2\right )^{2} - 3\right )} \log \left (-\log \left (2\right )^{2} + x + 3\right ) + x + \frac {\log \left (2\right )^{4} - 6 \, \log \left (2\right )^{2} + 9}{\log \left (2\right )^{2} - x - 3}\right )} \log \left (2\right )^{2} + 10 \, {\left (\frac {\log \left (2\right )^{2} - 3}{\log \left (2\right )^{2} - x - 3} + \log \left (-\log \left (2\right )^{2} + x + 3\right )\right )} \log \left (2\right )^{2} + \frac {\log \left (2\right )^{4}}{\log \left (2\right )^{2} - x - 3} - 4 \, {\left (\log \left (2\right )^{2} - 3\right )} x - x^{2} + 3 \, {\left (x + 2\right )} e^{\left (\frac {e^{x} \log \left (2\right )^{2}}{\log \left (2\right )^{2} - x - 3} + \frac {2 \, e^{x}}{\log \left (2\right )^{2} - x - 3} - e^{x}\right )} - 6 \, {\left (\log \left (2\right )^{4} - 6 \, \log \left (2\right )^{2} + 9\right )} \log \left (-\log \left (2\right )^{2} + x + 3\right ) - 22 \, {\left (\log \left (2\right )^{2} - 3\right )} \log \left (-\log \left (2\right )^{2} + x + 3\right ) - 11 \, x - \frac {6 \, \log \left (2\right )^{2}}{\log \left (2\right )^{2} - x - 3} - \frac {2 \, {\left (\log \left (2\right )^{6} - 9 \, \log \left (2\right )^{4} + 27 \, \log \left (2\right )^{2} - 27\right )}}{\log \left (2\right )^{2} - x - 3} - \frac {11 \, {\left (\log \left (2\right )^{4} - 6 \, \log \left (2\right )^{2} + 9\right )}}{\log \left (2\right )^{2} - x - 3} - \frac {12 \, {\left (\log \left (2\right )^{2} - 3\right )}}{\log \left (2\right )^{2} - x - 3} + \frac {9}{\log \left (2\right )^{2} - x - 3} - 12 \, \log \left (-\log \left (2\right )^{2} + x + 3\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 30, normalized size = 0.91 \begin {gather*} -x^{2} + 3 \, {\left (x + 2\right )} e^{\left (\frac {{\left (x + 5\right )} e^{x}}{\log \left (2\right )^{2} - x - 3}\right )} + x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 13.07, size = 27, normalized size = 0.82 \begin {gather*} - x^{2} + x + \left (3 x + 6\right ) e^{- \frac {\left (- x - 5\right ) e^{x}}{- x - 3 + \log {\left (2 \right )}^{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 4.75, size = 46, normalized size = 1.39 \begin {gather*} x+{\mathrm {e}}^{-\frac {5\,{\mathrm {e}}^x}{x-{\ln \left (2\right )}^2+3}-\frac {x\,{\mathrm {e}}^x}{x-{\ln \left (2\right )}^2+3}}\,\left (3\,x+6\right )-x^2 \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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