Optimal. Leaf size=35 \[ -x+e^{-x} x^2 \log \left (\frac {e^{-2+e^3-\frac {4}{x}-x^4}}{x}\right ) \]
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Rubi [A]
time = 0.44, antiderivative size = 35, normalized size of antiderivative = 1.00, number of steps
used = 23, number of rules used = 5, integrand size = 57, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.088, Rules used = {6874, 2225,
2207, 2227, 2634} \begin {gather*} e^{-x} x^2 \log \left (\frac {e^{-x^4-\frac {4}{x}+e^3-2}}{x}\right )-x \end {gather*}
Antiderivative was successfully verified.
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Rule 2207
Rule 2225
Rule 2227
Rule 2634
Rule 6874
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \left (-1+4 e^{-x}-e^{-x} x-4 e^{-x} x^5+e^{-x} (2-x) x \log \left (\frac {e^{-2 \left (1-\frac {e^3}{2}\right )-\frac {4}{x}-x^4}}{x}\right )\right ) \, dx\\ &=-x+4 \int e^{-x} \, dx-4 \int e^{-x} x^5 \, dx-\int e^{-x} x \, dx+\int e^{-x} (2-x) x \log \left (\frac {e^{-2 \left (1-\frac {e^3}{2}\right )-\frac {4}{x}-x^4}}{x}\right ) \, dx\\ &=-4 e^{-x}-x+e^{-x} x+4 e^{-x} x^5+e^{-x} x^2 \log \left (\frac {e^{-2+e^3-\frac {4}{x}-x^4}}{x}\right )-20 \int e^{-x} x^4 \, dx-\int e^{-x} \, dx-\int e^{-x} \left (4-x-4 x^5\right ) \, dx\\ &=-3 e^{-x}-x+e^{-x} x+20 e^{-x} x^4+4 e^{-x} x^5+e^{-x} x^2 \log \left (\frac {e^{-2+e^3-\frac {4}{x}-x^4}}{x}\right )-80 \int e^{-x} x^3 \, dx-\int \left (4 e^{-x}-e^{-x} x-4 e^{-x} x^5\right ) \, dx\\ &=-3 e^{-x}-x+e^{-x} x+80 e^{-x} x^3+20 e^{-x} x^4+4 e^{-x} x^5+e^{-x} x^2 \log \left (\frac {e^{-2+e^3-\frac {4}{x}-x^4}}{x}\right )-4 \int e^{-x} \, dx+4 \int e^{-x} x^5 \, dx-240 \int e^{-x} x^2 \, dx+\int e^{-x} x \, dx\\ &=e^{-x}-x+240 e^{-x} x^2+80 e^{-x} x^3+20 e^{-x} x^4+e^{-x} x^2 \log \left (\frac {e^{-2+e^3-\frac {4}{x}-x^4}}{x}\right )+20 \int e^{-x} x^4 \, dx-480 \int e^{-x} x \, dx+\int e^{-x} \, dx\\ &=-x+480 e^{-x} x+240 e^{-x} x^2+80 e^{-x} x^3+e^{-x} x^2 \log \left (\frac {e^{-2+e^3-\frac {4}{x}-x^4}}{x}\right )+80 \int e^{-x} x^3 \, dx-480 \int e^{-x} \, dx\\ &=480 e^{-x}-x+480 e^{-x} x+240 e^{-x} x^2+e^{-x} x^2 \log \left (\frac {e^{-2+e^3-\frac {4}{x}-x^4}}{x}\right )+240 \int e^{-x} x^2 \, dx\\ &=480 e^{-x}-x+480 e^{-x} x+e^{-x} x^2 \log \left (\frac {e^{-2+e^3-\frac {4}{x}-x^4}}{x}\right )+480 \int e^{-x} x \, dx\\ &=480 e^{-x}-x+e^{-x} x^2 \log \left (\frac {e^{-2+e^3-\frac {4}{x}-x^4}}{x}\right )+480 \int e^{-x} \, dx\\ &=-x+e^{-x} x^2 \log \left (\frac {e^{-2+e^3-\frac {4}{x}-x^4}}{x}\right )\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.17, size = 33, normalized size = 0.94 \begin {gather*} x \left (-1+e^{-x} x \log \left (\frac {e^{-2+e^3-\frac {4}{x}-x^4}}{x}\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
3.
time = 1.69, size = 237, normalized size = 6.77
method | result | size |
risch | \(x^{2} {\mathrm e}^{-x} \ln \left ({\mathrm e}^{\frac {x \,{\mathrm e}^{3}-x^{5}-2 x -4}{x}}\right )-\frac {x \left (-i \pi x \,\mathrm {csgn}\left (i {\mathrm e}^{\frac {x \,{\mathrm e}^{3}-x^{5}-2 x -4}{x}}\right ) \mathrm {csgn}\left (\frac {i {\mathrm e}^{\frac {x \,{\mathrm e}^{3}-x^{5}-2 x -4}{x}}}{x}\right )^{2}+i \pi x \,\mathrm {csgn}\left (i {\mathrm e}^{\frac {x \,{\mathrm e}^{3}-x^{5}-2 x -4}{x}}\right ) \mathrm {csgn}\left (\frac {i {\mathrm e}^{\frac {x \,{\mathrm e}^{3}-x^{5}-2 x -4}{x}}}{x}\right ) \mathrm {csgn}\left (\frac {i}{x}\right )+i \pi x \mathrm {csgn}\left (\frac {i {\mathrm e}^{\frac {x \,{\mathrm e}^{3}-x^{5}-2 x -4}{x}}}{x}\right )^{3}-i \pi x \mathrm {csgn}\left (\frac {i {\mathrm e}^{\frac {x \,{\mathrm e}^{3}-x^{5}-2 x -4}{x}}}{x}\right )^{2} \mathrm {csgn}\left (\frac {i}{x}\right )+2 x \ln \left (x \right )+2 \,{\mathrm e}^{x}\right ) {\mathrm e}^{-x}}{2}\) | \(237\) |
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. 91 vs.
\(2 (32) = 64\).
time = 0.30, size = 91, normalized size = 2.60 \begin {gather*} -{\left (x^{6} + 4 \, x^{5} + 20 \, x^{4} + 80 \, x^{3} - x^{2} {\left (e^{3} - 242\right )} + x^{2} \log \left (x\right ) + 485 \, x + 477\right )} e^{\left (-x\right )} + 4 \, {\left (x^{5} + 5 \, x^{4} + 20 \, x^{3} + 60 \, x^{2} + 120 \, x + 120\right )} e^{\left (-x\right )} + {\left (x + 1\right )} e^{\left (-x\right )} - x - 4 \, e^{\left (-x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.42, size = 39, normalized size = 1.11 \begin {gather*} {\left (x^{2} \log \left (\frac {e^{\left (-\frac {x^{5} - x e^{3} + 2 \, x + 4}{x}\right )}}{x}\right ) - x e^{x}\right )} e^{\left (-x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.44, size = 50, normalized size = 1.43 \begin {gather*} -x^{6} e^{\left (-x\right )} - x^{2} e^{\left (-x\right )} \log \left (x\right ) - 2 \, x^{2} e^{\left (-x\right )} + x^{2} e^{\left (-x + 3\right )} - 4 \, x e^{\left (-x\right )} - x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 5.58, size = 121, normalized size = 3.46 \begin {gather*} 4\,{\mathrm {e}}^{-x}\,\left (x^5+5\,x^4+20\,x^3+60\,x^2+120\,x+120\right )-481\,{\mathrm {e}}^{-x}-485\,x\,{\mathrm {e}}^{-x}-x+{\mathrm {e}}^{-x}\,\left (x+1\right )-242\,x^2\,{\mathrm {e}}^{-x}-80\,x^3\,{\mathrm {e}}^{-x}-20\,x^4\,{\mathrm {e}}^{-x}-4\,x^5\,{\mathrm {e}}^{-x}-x^6\,{\mathrm {e}}^{-x}+x^2\,{\mathrm {e}}^{3-x}+x^2\,\ln \left (\frac {1}{x}\right )\,{\mathrm {e}}^{-x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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