Optimal. Leaf size=27 \[ \frac {4}{15} e^{-x \left (2+\frac {e^x}{10+x-\log \left (\frac {5}{2}\right )}\right )} x \]
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Rubi [F]
time = 31.36, 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 {-20 x-e^x x-2 x^2+2 x \log \left (\frac {5}{2}\right )}{-10-x+\log \left (\frac {5}{2}\right )}\right ) \left (400-720 x-156 x^2-8 x^3+\left (-80+152 x+16 x^2\right ) \log \left (\frac {5}{2}\right )+(4-8 x) \log ^2\left (\frac {5}{2}\right )+e^x \left (-40 x-40 x^2-4 x^3+\left (4 x+4 x^2\right ) \log \left (\frac {5}{2}\right )\right )\right )}{1500+300 x+15 x^2+(-300-30 x) \log \left (\frac {5}{2}\right )+15 \log ^2\left (\frac {5}{2}\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 {\exp \left (-\frac {-e^x x-2 x^2-20 x \left (1-\frac {1}{10} \log \left (\frac {5}{2}\right )\right )}{-10-x+\log \left (\frac {5}{2}\right )}\right ) \left (400-720 x-156 x^2-8 x^3+\left (-80+152 x+16 x^2\right ) \log \left (\frac {5}{2}\right )+(4-8 x) \log ^2\left (\frac {5}{2}\right )+e^x \left (-40 x-40 x^2-4 x^3+\left (4 x+4 x^2\right ) \log \left (\frac {5}{2}\right )\right )\right )}{15 x^2+30 x \left (10-\log \left (\frac {5}{2}\right )\right )+15 \left (10-\log \left (\frac {5}{2}\right )\right )^2} \, dx\\ &=\int \frac {\exp \left (-\frac {-e^x x-2 x^2-20 x \left (1-\frac {1}{10} \log \left (\frac {5}{2}\right )\right )}{-10-x+\log \left (\frac {5}{2}\right )}\right ) \left (400-720 x-156 x^2-8 x^3+\left (-80+152 x+16 x^2\right ) \log \left (\frac {5}{2}\right )+(4-8 x) \log ^2\left (\frac {5}{2}\right )+e^x \left (-40 x-40 x^2-4 x^3+\left (4 x+4 x^2\right ) \log \left (\frac {5}{2}\right )\right )\right )}{15 \left (10+x-\log \left (\frac {5}{2}\right )\right )^2} \, dx\\ &=\frac {1}{15} \int \frac {\exp \left (-\frac {-e^x x-2 x^2-20 x \left (1-\frac {1}{10} \log \left (\frac {5}{2}\right )\right )}{-10-x+\log \left (\frac {5}{2}\right )}\right ) \left (400-720 x-156 x^2-8 x^3+\left (-80+152 x+16 x^2\right ) \log \left (\frac {5}{2}\right )+(4-8 x) \log ^2\left (\frac {5}{2}\right )+e^x \left (-40 x-40 x^2-4 x^3+\left (4 x+4 x^2\right ) \log \left (\frac {5}{2}\right )\right )\right )}{\left (10+x-\log \left (\frac {5}{2}\right )\right )^2} \, dx\\ &=\frac {1}{15} \int \left (\frac {400 \exp \left (-\frac {-e^x x-2 x^2-20 x \left (1-\frac {1}{10} \log \left (\frac {5}{2}\right )\right )}{-10-x+\log \left (\frac {5}{2}\right )}\right )}{\left (10+x-\log \left (\frac {5}{2}\right )\right )^2}-\frac {720 \exp \left (-\frac {-e^x x-2 x^2-20 x \left (1-\frac {1}{10} \log \left (\frac {5}{2}\right )\right )}{-10-x+\log \left (\frac {5}{2}\right )}\right ) x}{\left (10+x-\log \left (\frac {5}{2}\right )\right )^2}-\frac {156 \exp \left (-\frac {-e^x x-2 x^2-20 x \left (1-\frac {1}{10} \log \left (\frac {5}{2}\right )\right )}{-10-x+\log \left (\frac {5}{2}\right )}\right ) x^2}{\left (10+x-\log \left (\frac {5}{2}\right )\right )^2}-\frac {8 \exp \left (-\frac {-e^x x-2 x^2-20 x \left (1-\frac {1}{10} \log \left (\frac {5}{2}\right )\right )}{-10-x+\log \left (\frac {5}{2}\right )}\right ) x^3}{\left (10+x-\log \left (\frac {5}{2}\right )\right )^2}+\frac {8 \exp \left (-\frac {-e^x x-2 x^2-20 x \left (1-\frac {1}{10} \log \left (\frac {5}{2}\right )\right )}{-10-x+\log \left (\frac {5}{2}\right )}\right ) (10+x) (-1+2 x) \log \left (\frac {5}{2}\right )}{\left (10+x-\log \left (\frac {5}{2}\right )\right )^2}-\frac {4 \exp \left (-\frac {-e^x x-2 x^2-20 x \left (1-\frac {1}{10} \log \left (\frac {5}{2}\right )\right )}{-10-x+\log \left (\frac {5}{2}\right )}\right ) (-1+2 x) \log ^2\left (\frac {5}{2}\right )}{\left (10+x-\log \left (\frac {5}{2}\right )\right )^2}+\frac {4 \exp \left (x-\frac {-e^x x-2 x^2-20 x \left (1-\frac {1}{10} \log \left (\frac {5}{2}\right )\right )}{-10-x+\log \left (\frac {5}{2}\right )}\right ) x \left (-10-x^2-x \left (10-\log \left (\frac {5}{2}\right )\right )+\log \left (\frac {5}{2}\right )\right )}{\left (10+x-\log \left (\frac {5}{2}\right )\right )^2}\right ) \, dx\\ &=\frac {4}{15} \int \frac {\exp \left (x-\frac {-e^x x-2 x^2-20 x \left (1-\frac {1}{10} \log \left (\frac {5}{2}\right )\right )}{-10-x+\log \left (\frac {5}{2}\right )}\right ) x \left (-10-x^2-x \left (10-\log \left (\frac {5}{2}\right )\right )+\log \left (\frac {5}{2}\right )\right )}{\left (10+x-\log \left (\frac {5}{2}\right )\right )^2} \, dx-\frac {8}{15} \int \frac {\exp \left (-\frac {-e^x x-2 x^2-20 x \left (1-\frac {1}{10} \log \left (\frac {5}{2}\right )\right )}{-10-x+\log \left (\frac {5}{2}\right )}\right ) x^3}{\left (10+x-\log \left (\frac {5}{2}\right )\right )^2} \, dx-\frac {52}{5} \int \frac {\exp \left (-\frac {-e^x x-2 x^2-20 x \left (1-\frac {1}{10} \log \left (\frac {5}{2}\right )\right )}{-10-x+\log \left (\frac {5}{2}\right )}\right ) x^2}{\left (10+x-\log \left (\frac {5}{2}\right )\right )^2} \, dx+\frac {80}{3} \int \frac {\exp \left (-\frac {-e^x x-2 x^2-20 x \left (1-\frac {1}{10} \log \left (\frac {5}{2}\right )\right )}{-10-x+\log \left (\frac {5}{2}\right )}\right )}{\left (10+x-\log \left (\frac {5}{2}\right )\right )^2} \, dx-48 \int \frac {\exp \left (-\frac {-e^x x-2 x^2-20 x \left (1-\frac {1}{10} \log \left (\frac {5}{2}\right )\right )}{-10-x+\log \left (\frac {5}{2}\right )}\right ) x}{\left (10+x-\log \left (\frac {5}{2}\right )\right )^2} \, dx+\frac {1}{15} \left (8 \log \left (\frac {5}{2}\right )\right ) \int \frac {\exp \left (-\frac {-e^x x-2 x^2-20 x \left (1-\frac {1}{10} \log \left (\frac {5}{2}\right )\right )}{-10-x+\log \left (\frac {5}{2}\right )}\right ) (10+x) (-1+2 x)}{\left (10+x-\log \left (\frac {5}{2}\right )\right )^2} \, dx-\frac {1}{15} \left (4 \log ^2\left (\frac {5}{2}\right )\right ) \int \frac {\exp \left (-\frac {-e^x x-2 x^2-20 x \left (1-\frac {1}{10} \log \left (\frac {5}{2}\right )\right )}{-10-x+\log \left (\frac {5}{2}\right )}\right ) (-1+2 x)}{\left (10+x-\log \left (\frac {5}{2}\right )\right )^2} \, dx\\ &=\text {Rest of rules removed due to large latex content} \end {aligned} \end {gather*}
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Mathematica [F]
time = 4.48, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {e^{-\frac {-20 x-e^x x-2 x^2+2 x \log \left (\frac {5}{2}\right )}{-10-x+\log \left (\frac {5}{2}\right )}} \left (400-720 x-156 x^2-8 x^3+\left (-80+152 x+16 x^2\right ) \log \left (\frac {5}{2}\right )+(4-8 x) \log ^2\left (\frac {5}{2}\right )+e^x \left (-40 x-40 x^2-4 x^3+\left (4 x+4 x^2\right ) \log \left (\frac {5}{2}\right )\right )\right )}{1500+300 x+15 x^2+(-300-30 x) \log \left (\frac {5}{2}\right )+15 \log ^2\left (\frac {5}{2}\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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Maple [A]
time = 2.18, size = 34, normalized size = 1.26
method | result | size |
risch | \(\frac {4 x \,{\mathrm e}^{-\frac {x \left ({\mathrm e}^{x}-2 \ln \left (5\right )+2 \ln \left (2\right )+2 x +20\right )}{-\ln \left (5\right )+\ln \left (2\right )+x +10}}}{15}\) | \(34\) |
norman | \(\frac {\left (\left (-\frac {8}{3}+\frac {4 \ln \left (5\right )}{15}-\frac {4 \ln \left (2\right )}{15}\right ) x -\frac {4 x^{2}}{15}\right ) {\mathrm e}^{-\frac {-{\mathrm e}^{x} x +2 x \ln \left (\frac {5}{2}\right )-2 x^{2}-20 x}{\ln \left (\frac {5}{2}\right )-x -10}}}{\ln \left (\frac {5}{2}\right )-x -10}\) | \(61\) |
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. 60 vs.
\(2 (21) = 42\).
time = 0.68, size = 60, normalized size = 2.22 \begin {gather*} \frac {4}{15} \, x e^{\left (-2 \, x - \frac {e^{x} \log \left (5\right )}{x - \log \left (5\right ) + \log \left (2\right ) + 10} + \frac {e^{x} \log \left (2\right )}{x - \log \left (5\right ) + \log \left (2\right ) + 10} + \frac {10 \, e^{x}}{x - \log \left (5\right ) + \log \left (2\right ) + 10} - e^{x}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.44, size = 33, normalized size = 1.22 \begin {gather*} \frac {4}{15} \, x e^{\left (-\frac {2 \, x^{2} + x e^{x} - 2 \, x \log \left (\frac {5}{2}\right ) + 20 \, x}{x - \log \left (\frac {5}{2}\right ) + 10}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 45.52, size = 36, normalized size = 1.33 \begin {gather*} \frac {4 x e^{- \frac {- 2 x^{2} - x e^{x} - 20 x + 2 x \log {\left (\frac {5}{2} \right )}}{- x - 10 + \log {\left (\frac {5}{2} \right )}}}}{15} \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 [F(-1)]
time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \text {Hanged} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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