Optimal. Leaf size=32 \[ \frac {1}{3} \left (-2+\frac {1}{2} e^{2 e^x-x} \left (2+\frac {x^4}{(-1+x)^2}\right )\right ) \]
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Rubi [F]
time = 2.77, 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^{2 e^x-x} \left (2-6 x+6 x^2-6 x^3+3 x^4-x^5+e^x \left (-4+12 x-12 x^2+4 x^3-2 x^4+2 x^5\right )\right )}{-6+18 x-18 x^2+6 x^3} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \left (\frac {e^{2 e^x-x}}{3 (-1+x)^3}-\frac {e^{2 e^x-x} x}{(-1+x)^3}+\frac {e^{2 e^x-x} x^2}{(-1+x)^3}-\frac {e^{2 e^x-x} x^3}{(-1+x)^3}+\frac {e^{2 e^x-x} x^4}{2 (-1+x)^3}-\frac {e^{2 e^x-x} x^5}{6 (-1+x)^3}+\frac {e^{2 e^x} \left (2-4 x+2 x^2+x^4\right )}{3 (-1+x)^2}\right ) \, dx\\ &=-\left (\frac {1}{6} \int \frac {e^{2 e^x-x} x^5}{(-1+x)^3} \, dx\right )+\frac {1}{3} \int \frac {e^{2 e^x-x}}{(-1+x)^3} \, dx+\frac {1}{3} \int \frac {e^{2 e^x} \left (2-4 x+2 x^2+x^4\right )}{(-1+x)^2} \, dx+\frac {1}{2} \int \frac {e^{2 e^x-x} x^4}{(-1+x)^3} \, dx-\int \frac {e^{2 e^x-x} x}{(-1+x)^3} \, dx+\int \frac {e^{2 e^x-x} x^2}{(-1+x)^3} \, dx-\int \frac {e^{2 e^x-x} x^3}{(-1+x)^3} \, dx\\ &=-\left (\frac {1}{6} \int \left (6 e^{2 e^x-x}+\frac {e^{2 e^x-x}}{(-1+x)^3}+\frac {5 e^{2 e^x-x}}{(-1+x)^2}+\frac {10 e^{2 e^x-x}}{-1+x}+3 e^{2 e^x-x} x+e^{2 e^x-x} x^2\right ) \, dx\right )+\frac {1}{3} \int \frac {e^{2 e^x-x}}{(-1+x)^3} \, dx+\frac {1}{3} \int \left (5 e^{2 e^x}+\frac {e^{2 e^x}}{(-1+x)^2}+\frac {4 e^{2 e^x}}{-1+x}+2 e^{2 e^x} x+e^{2 e^x} x^2\right ) \, dx+\frac {1}{2} \int \left (3 e^{2 e^x-x}+\frac {e^{2 e^x-x}}{(-1+x)^3}+\frac {4 e^{2 e^x-x}}{(-1+x)^2}+\frac {6 e^{2 e^x-x}}{-1+x}+e^{2 e^x-x} x\right ) \, dx-\int \left (\frac {e^{2 e^x-x}}{(-1+x)^3}+\frac {e^{2 e^x-x}}{(-1+x)^2}\right ) \, dx+\int \left (\frac {e^{2 e^x-x}}{(-1+x)^3}+\frac {2 e^{2 e^x-x}}{(-1+x)^2}+\frac {e^{2 e^x-x}}{-1+x}\right ) \, dx-\int \left (e^{2 e^x-x}+\frac {e^{2 e^x-x}}{(-1+x)^3}+\frac {3 e^{2 e^x-x}}{(-1+x)^2}+\frac {3 e^{2 e^x-x}}{-1+x}\right ) \, dx\\ &=-\left (\frac {1}{6} \int \frac {e^{2 e^x-x}}{(-1+x)^3} \, dx\right )-\frac {1}{6} \int e^{2 e^x-x} x^2 \, dx+\frac {1}{3} \int \frac {e^{2 e^x-x}}{(-1+x)^3} \, dx+\frac {1}{3} \int \frac {e^{2 e^x}}{(-1+x)^2} \, dx+\frac {1}{3} \int e^{2 e^x} x^2 \, dx+\frac {1}{2} \int \frac {e^{2 e^x-x}}{(-1+x)^3} \, dx+\frac {2}{3} \int e^{2 e^x} x \, dx-\frac {5}{6} \int \frac {e^{2 e^x-x}}{(-1+x)^2} \, dx+\frac {4}{3} \int \frac {e^{2 e^x}}{-1+x} \, dx+\frac {3}{2} \int e^{2 e^x-x} \, dx+\frac {5}{3} \int e^{2 e^x} \, dx-\frac {5}{3} \int \frac {e^{2 e^x-x}}{-1+x} \, dx+2 \left (2 \int \frac {e^{2 e^x-x}}{(-1+x)^2} \, dx\right )-3 \int \frac {e^{2 e^x-x}}{(-1+x)^2} \, dx-2 \int e^{2 e^x-x} \, dx-\int \frac {e^{2 e^x-x}}{(-1+x)^3} \, dx-\int \frac {e^{2 e^x-x}}{(-1+x)^2} \, dx+\int \frac {e^{2 e^x-x}}{-1+x} \, dx\\ &=-\left (\frac {1}{6} \int \frac {e^{2 e^x-x}}{(-1+x)^3} \, dx\right )-\frac {1}{6} \int e^{2 e^x-x} x^2 \, dx+\frac {1}{3} \int \frac {e^{2 e^x-x}}{(-1+x)^3} \, dx+\frac {1}{3} \int \frac {e^{2 e^x}}{(-1+x)^2} \, dx+\frac {1}{3} \int e^{2 e^x} x^2 \, dx+\frac {1}{2} \int \frac {e^{2 e^x-x}}{(-1+x)^3} \, dx+\frac {2}{3} \int e^{2 e^x} x \, dx-\frac {5}{6} \int \frac {e^{2 e^x-x}}{(-1+x)^2} \, dx+\frac {4}{3} \int \frac {e^{2 e^x}}{-1+x} \, dx+\frac {3}{2} \text {Subst}\left (\int \frac {e^{2 x}}{x^2} \, dx,x,e^x\right )-\frac {5}{3} \int \frac {e^{2 e^x-x}}{-1+x} \, dx+\frac {5}{3} \text {Subst}\left (\int \frac {e^{2 x}}{x} \, dx,x,e^x\right )+2 \left (2 \int \frac {e^{2 e^x-x}}{(-1+x)^2} \, dx\right )-3 \int \frac {e^{2 e^x-x}}{(-1+x)^2} \, dx-\int \frac {e^{2 e^x-x}}{(-1+x)^3} \, dx-\int \frac {e^{2 e^x-x}}{(-1+x)^2} \, dx+\int \frac {e^{2 e^x-x}}{-1+x} \, dx-2 \text {Subst}\left (\int \frac {e^{2 x}}{x^2} \, dx,x,e^x\right )\\ &=-\frac {3}{2} e^{2 e^x-x}+\frac {5 \text {Ei}\left (2 e^x\right )}{3}-\frac {1}{6} \int \frac {e^{2 e^x-x}}{(-1+x)^3} \, dx-\frac {1}{6} \int e^{2 e^x-x} x^2 \, dx+\frac {1}{3} \int \frac {e^{2 e^x-x}}{(-1+x)^3} \, dx+\frac {1}{3} \int \frac {e^{2 e^x}}{(-1+x)^2} \, dx+\frac {1}{3} \int e^{2 e^x} x^2 \, dx+\frac {1}{2} \int \frac {e^{2 e^x-x}}{(-1+x)^3} \, dx+\frac {2}{3} \int e^{2 e^x} x \, dx-\frac {5}{6} \int \frac {e^{2 e^x-x}}{(-1+x)^2} \, dx+\frac {4}{3} \int \frac {e^{2 e^x}}{-1+x} \, dx-\frac {5}{3} \int \frac {e^{2 e^x-x}}{-1+x} \, dx+2 \left (2 \int \frac {e^{2 e^x-x}}{(-1+x)^2} \, dx\right )-2 \left (-e^{2 e^x-x}+2 \text {Subst}\left (\int \frac {e^{2 x}}{x} \, dx,x,e^x\right )\right )-3 \int \frac {e^{2 e^x-x}}{(-1+x)^2} \, dx+3 \text {Subst}\left (\int \frac {e^{2 x}}{x} \, dx,x,e^x\right )-\int \frac {e^{2 e^x-x}}{(-1+x)^3} \, dx-\int \frac {e^{2 e^x-x}}{(-1+x)^2} \, dx+\int \frac {e^{2 e^x-x}}{-1+x} \, dx\\ &=-\frac {3}{2} e^{2 e^x-x}+\frac {14 \text {Ei}\left (2 e^x\right )}{3}-2 \left (-e^{2 e^x-x}+2 \text {Ei}\left (2 e^x\right )\right )-\frac {1}{6} \int \frac {e^{2 e^x-x}}{(-1+x)^3} \, dx-\frac {1}{6} \int e^{2 e^x-x} x^2 \, dx+\frac {1}{3} \int \frac {e^{2 e^x-x}}{(-1+x)^3} \, dx+\frac {1}{3} \int \frac {e^{2 e^x}}{(-1+x)^2} \, dx+\frac {1}{3} \int e^{2 e^x} x^2 \, dx+\frac {1}{2} \int \frac {e^{2 e^x-x}}{(-1+x)^3} \, dx+\frac {2}{3} \int e^{2 e^x} x \, dx-\frac {5}{6} \int \frac {e^{2 e^x-x}}{(-1+x)^2} \, dx+\frac {4}{3} \int \frac {e^{2 e^x}}{-1+x} \, dx-\frac {5}{3} \int \frac {e^{2 e^x-x}}{-1+x} \, dx+2 \left (2 \int \frac {e^{2 e^x-x}}{(-1+x)^2} \, dx\right )-3 \int \frac {e^{2 e^x-x}}{(-1+x)^2} \, dx-\int \frac {e^{2 e^x-x}}{(-1+x)^3} \, dx-\int \frac {e^{2 e^x-x}}{(-1+x)^2} \, dx+\int \frac {e^{2 e^x-x}}{-1+x} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 1.17, size = 35, normalized size = 1.09 \begin {gather*} \frac {1}{6} e^{2 e^x-x} \left (5+\frac {1}{(-1+x)^2}+\frac {4}{-1+x}+2 x+x^2\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.07, size = 30, normalized size = 0.94
method | result | size |
risch | \(\frac {\left (x^{4}+2 x^{2}-4 x +2\right ) {\mathrm e}^{2 \,{\mathrm e}^{x}-x}}{6 \left (x -1\right )^{2}}\) | \(30\) |
norman | \(\frac {-\frac {2 x \,{\mathrm e}^{2 \,{\mathrm e}^{x}-x}}{3}+\frac {x^{2} {\mathrm e}^{2 \,{\mathrm e}^{x}-x}}{3}+\frac {x^{4} {\mathrm e}^{2 \,{\mathrm e}^{x}-x}}{6}+\frac {{\mathrm e}^{2 \,{\mathrm e}^{x}-x}}{3}}{\left (x -1\right )^{2}}\) | \(59\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.30, size = 34, normalized size = 1.06 \begin {gather*} \frac {{\left (x^{4} + 2 \, x^{2} - 4 \, x + 2\right )} e^{\left (-x + 2 \, e^{x}\right )}}{6 \, {\left (x^{2} - 2 \, x + 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.33, size = 34, normalized size = 1.06 \begin {gather*} \frac {{\left (x^{4} + 2 \, x^{2} - 4 \, x + 2\right )} e^{\left (-x + 2 \, e^{x}\right )}}{6 \, {\left (x^{2} - 2 \, x + 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.11, size = 31, normalized size = 0.97 \begin {gather*} \frac {\left (x^{4} + 2 x^{2} - 4 x + 2\right ) e^{- x + 2 e^{x}}}{6 x^{2} - 12 x + 6} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 53 vs.
\(2 (24) = 48\).
time = 0.41, size = 53, normalized size = 1.66 \begin {gather*} \frac {x^{4} e^{\left (2 \, e^{x}\right )} + 2 \, x^{2} e^{\left (2 \, e^{x}\right )} - 4 \, x e^{\left (2 \, e^{x}\right )} + 2 \, e^{\left (2 \, e^{x}\right )}}{6 \, {\left (x^{2} e^{x} - 2 \, x e^{x} + e^{x}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.22, size = 35, normalized size = 1.09 \begin {gather*} \frac {{\mathrm {e}}^{2\,{\mathrm {e}}^x-x}\,\left (\frac {x^4}{6}+\frac {x^2}{3}-\frac {2\,x}{3}+\frac {1}{3}\right )}{x^2-2\,x+1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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