Optimal. Leaf size=23 \[ e^{-1-x+\frac {e^x}{-1+2 x}} x (1+x) \]
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Rubi [F] time = 22.47, 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 {1+e^x-x-2 x^2+(-1+2 x) \log (1+x)}{-1+2 x}\right ) \left (1-3 x-x^2+8 x^3-4 x^4+e^x \left (-3 x-x^2+2 x^3\right )\right )}{1-3 x+4 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 {\exp \left (\frac {1+e^x-x-2 x^2+(-1+2 x) \log (1+x)}{-1+2 x}\right )}{(1+x) (-1+2 x)^2}-\frac {3 \exp \left (\frac {1+e^x-x-2 x^2+(-1+2 x) \log (1+x)}{-1+2 x}\right ) x}{(1+x) (-1+2 x)^2}-\frac {\exp \left (\frac {1+e^x-x-2 x^2+(-1+2 x) \log (1+x)}{-1+2 x}\right ) x^2}{(1+x) (-1+2 x)^2}+\frac {8 \exp \left (\frac {1+e^x-x-2 x^2+(-1+2 x) \log (1+x)}{-1+2 x}\right ) x^3}{(1+x) (-1+2 x)^2}-\frac {4 \exp \left (\frac {1+e^x-x-2 x^2+(-1+2 x) \log (1+x)}{-1+2 x}\right ) x^4}{(1+x) (-1+2 x)^2}+\frac {\exp \left (x+\frac {1+e^x-x-2 x^2+(-1+2 x) \log (1+x)}{-1+2 x}\right ) x (-3+2 x)}{(-1+2 x)^2}\right ) \, dx\\ &=-\left (3 \int \frac {\exp \left (\frac {1+e^x-x-2 x^2+(-1+2 x) \log (1+x)}{-1+2 x}\right ) x}{(1+x) (-1+2 x)^2} \, dx\right )-4 \int \frac {\exp \left (\frac {1+e^x-x-2 x^2+(-1+2 x) \log (1+x)}{-1+2 x}\right ) x^4}{(1+x) (-1+2 x)^2} \, dx+8 \int \frac {\exp \left (\frac {1+e^x-x-2 x^2+(-1+2 x) \log (1+x)}{-1+2 x}\right ) x^3}{(1+x) (-1+2 x)^2} \, dx+\int \frac {\exp \left (\frac {1+e^x-x-2 x^2+(-1+2 x) \log (1+x)}{-1+2 x}\right )}{(1+x) (-1+2 x)^2} \, dx-\int \frac {\exp \left (\frac {1+e^x-x-2 x^2+(-1+2 x) \log (1+x)}{-1+2 x}\right ) x^2}{(1+x) (-1+2 x)^2} \, dx+\int \frac {\exp \left (x+\frac {1+e^x-x-2 x^2+(-1+2 x) \log (1+x)}{-1+2 x}\right ) x (-3+2 x)}{(-1+2 x)^2} \, dx\\ &=-\left (3 \int \frac {e^{\frac {1+e^x-x-2 x^2}{-1+2 x}} x}{(1-2 x)^2} \, dx\right )-4 \int \frac {e^{\frac {1+e^x-x-2 x^2}{-1+2 x}} x^4}{(1-2 x)^2} \, dx+8 \int \frac {e^{\frac {1+e^x-x-2 x^2}{-1+2 x}} x^3}{(1-2 x)^2} \, dx+\int \frac {e^{\frac {1+e^x-x-2 x^2}{-1+2 x}}}{(1-2 x)^2} \, dx-\int \frac {e^{\frac {1+e^x-x-2 x^2}{-1+2 x}} x^2}{(1-2 x)^2} \, dx+\int \frac {\exp \left (\frac {1+e^x-2 x-\log (1+x)+2 x \log (1+x)}{-1+2 x}\right ) x (-3+2 x)}{(1-2 x)^2} \, dx\\ &=-\left (3 \int \left (\frac {e^{\frac {1+e^x-x-2 x^2}{-1+2 x}}}{2 (-1+2 x)^2}+\frac {e^{\frac {1+e^x-x-2 x^2}{-1+2 x}}}{2 (-1+2 x)}\right ) \, dx\right )-4 \int \left (\frac {3}{16} e^{\frac {1+e^x-x-2 x^2}{-1+2 x}}+\frac {1}{4} e^{\frac {1+e^x-x-2 x^2}{-1+2 x}} x+\frac {1}{4} e^{\frac {1+e^x-x-2 x^2}{-1+2 x}} x^2+\frac {e^{\frac {1+e^x-x-2 x^2}{-1+2 x}}}{16 (-1+2 x)^2}+\frac {e^{\frac {1+e^x-x-2 x^2}{-1+2 x}}}{4 (-1+2 x)}\right ) \, dx+8 \int \left (\frac {1}{4} e^{\frac {1+e^x-x-2 x^2}{-1+2 x}}+\frac {1}{4} e^{\frac {1+e^x-x-2 x^2}{-1+2 x}} x+\frac {e^{\frac {1+e^x-x-2 x^2}{-1+2 x}}}{8 (-1+2 x)^2}+\frac {3 e^{\frac {1+e^x-x-2 x^2}{-1+2 x}}}{8 (-1+2 x)}\right ) \, dx+\int \frac {e^{\frac {1+e^x-x-2 x^2}{-1+2 x}}}{(1-2 x)^2} \, dx-\int \left (\frac {1}{4} e^{\frac {1+e^x-x-2 x^2}{-1+2 x}}+\frac {e^{\frac {1+e^x-x-2 x^2}{-1+2 x}}}{4 (-1+2 x)^2}+\frac {e^{\frac {1+e^x-x-2 x^2}{-1+2 x}}}{2 (-1+2 x)}\right ) \, dx+\int \left (\frac {1}{2} \exp \left (\frac {1+e^x-2 x-\log (1+x)+2 x \log (1+x)}{-1+2 x}\right )-\frac {\exp \left (\frac {1+e^x-2 x-\log (1+x)+2 x \log (1+x)}{-1+2 x}\right )}{(-1+2 x)^2}-\frac {\exp \left (\frac {1+e^x-2 x-\log (1+x)+2 x \log (1+x)}{-1+2 x}\right )}{2 (-1+2 x)}\right ) \, dx\\ &=-\left (\frac {1}{4} \int e^{\frac {1+e^x-x-2 x^2}{-1+2 x}} \, dx\right )-2 \left (\frac {1}{4} \int \frac {e^{\frac {1+e^x-x-2 x^2}{-1+2 x}}}{(-1+2 x)^2} \, dx\right )+\frac {1}{2} \int \exp \left (\frac {1+e^x-2 x-\log (1+x)+2 x \log (1+x)}{-1+2 x}\right ) \, dx-\frac {1}{2} \int \frac {e^{\frac {1+e^x-x-2 x^2}{-1+2 x}}}{-1+2 x} \, dx-\frac {1}{2} \int \frac {\exp \left (\frac {1+e^x-2 x-\log (1+x)+2 x \log (1+x)}{-1+2 x}\right )}{-1+2 x} \, dx-\frac {3}{4} \int e^{\frac {1+e^x-x-2 x^2}{-1+2 x}} \, dx-\frac {3}{2} \int \frac {e^{\frac {1+e^x-x-2 x^2}{-1+2 x}}}{(-1+2 x)^2} \, dx-\frac {3}{2} \int \frac {e^{\frac {1+e^x-x-2 x^2}{-1+2 x}}}{-1+2 x} \, dx+2 \int e^{\frac {1+e^x-x-2 x^2}{-1+2 x}} \, dx+2 \int e^{\frac {1+e^x-x-2 x^2}{-1+2 x}} x \, dx+3 \int \frac {e^{\frac {1+e^x-x-2 x^2}{-1+2 x}}}{-1+2 x} \, dx+\int \frac {e^{\frac {1+e^x-x-2 x^2}{-1+2 x}}}{(1-2 x)^2} \, dx-\int e^{\frac {1+e^x-x-2 x^2}{-1+2 x}} x \, dx-\int e^{\frac {1+e^x-x-2 x^2}{-1+2 x}} x^2 \, dx+\int \frac {e^{\frac {1+e^x-x-2 x^2}{-1+2 x}}}{(-1+2 x)^2} \, dx-\int \frac {\exp \left (\frac {1+e^x-2 x-\log (1+x)+2 x \log (1+x)}{-1+2 x}\right )}{(-1+2 x)^2} \, dx-\int \frac {e^{\frac {1+e^x-x-2 x^2}{-1+2 x}}}{-1+2 x} \, dx\\ &=-\left (\frac {1}{4} \int e^{\frac {1+e^x-x-2 x^2}{-1+2 x}} \, dx\right )-2 \left (\frac {1}{4} \int \frac {e^{\frac {1+e^x-x-2 x^2}{-1+2 x}}}{(-1+2 x)^2} \, dx\right )+\frac {1}{2} \int e^{\frac {1+e^x-2 x}{-1+2 x}} (1+x) \, dx-\frac {1}{2} \int \frac {e^{\frac {1+e^x-x-2 x^2}{-1+2 x}}}{-1+2 x} \, dx-\frac {1}{2} \int \frac {\exp \left (\frac {1+e^x-2 x-\log (1+x)+2 x \log (1+x)}{-1+2 x}\right )}{-1+2 x} \, dx-\frac {3}{4} \int e^{\frac {1+e^x-x-2 x^2}{-1+2 x}} \, dx-\frac {3}{2} \int \frac {e^{\frac {1+e^x-x-2 x^2}{-1+2 x}}}{(-1+2 x)^2} \, dx-\frac {3}{2} \int \frac {e^{\frac {1+e^x-x-2 x^2}{-1+2 x}}}{-1+2 x} \, dx+2 \int e^{\frac {1+e^x-x-2 x^2}{-1+2 x}} \, dx+2 \int e^{\frac {1+e^x-x-2 x^2}{-1+2 x}} x \, dx+3 \int \frac {e^{\frac {1+e^x-x-2 x^2}{-1+2 x}}}{-1+2 x} \, dx+\int \frac {e^{\frac {1+e^x-x-2 x^2}{-1+2 x}}}{(1-2 x)^2} \, dx-\int e^{\frac {1+e^x-x-2 x^2}{-1+2 x}} x \, dx-\int e^{\frac {1+e^x-x-2 x^2}{-1+2 x}} x^2 \, dx-\int \frac {e^{\frac {1+e^x-2 x}{-1+2 x}} (1+x)}{(1-2 x)^2} \, dx+\int \frac {e^{\frac {1+e^x-x-2 x^2}{-1+2 x}}}{(-1+2 x)^2} \, dx-\int \frac {e^{\frac {1+e^x-x-2 x^2}{-1+2 x}}}{-1+2 x} \, dx\\ &=-\left (\frac {1}{4} \int e^{\frac {1+e^x-x-2 x^2}{-1+2 x}} \, dx\right )-2 \left (\frac {1}{4} \int \frac {e^{\frac {1+e^x-x-2 x^2}{-1+2 x}}}{(-1+2 x)^2} \, dx\right )-\frac {1}{2} \int \frac {e^{\frac {1+e^x-x-2 x^2}{-1+2 x}}}{-1+2 x} \, dx-\frac {1}{2} \int \frac {\exp \left (\frac {1+e^x-2 x-\log (1+x)+2 x \log (1+x)}{-1+2 x}\right )}{-1+2 x} \, dx+\frac {1}{2} \int \left (e^{\frac {1+e^x-2 x}{-1+2 x}}+e^{\frac {1+e^x-2 x}{-1+2 x}} x\right ) \, dx-\frac {3}{4} \int e^{\frac {1+e^x-x-2 x^2}{-1+2 x}} \, dx-\frac {3}{2} \int \frac {e^{\frac {1+e^x-x-2 x^2}{-1+2 x}}}{(-1+2 x)^2} \, dx-\frac {3}{2} \int \frac {e^{\frac {1+e^x-x-2 x^2}{-1+2 x}}}{-1+2 x} \, dx+2 \int e^{\frac {1+e^x-x-2 x^2}{-1+2 x}} \, dx+2 \int e^{\frac {1+e^x-x-2 x^2}{-1+2 x}} x \, dx+3 \int \frac {e^{\frac {1+e^x-x-2 x^2}{-1+2 x}}}{-1+2 x} \, dx+\int \frac {e^{\frac {1+e^x-x-2 x^2}{-1+2 x}}}{(1-2 x)^2} \, dx-\int e^{\frac {1+e^x-x-2 x^2}{-1+2 x}} x \, dx-\int e^{\frac {1+e^x-x-2 x^2}{-1+2 x}} x^2 \, dx+\int \frac {e^{\frac {1+e^x-x-2 x^2}{-1+2 x}}}{(-1+2 x)^2} \, dx-\int \frac {e^{\frac {1+e^x-x-2 x^2}{-1+2 x}}}{-1+2 x} \, dx-\int \left (\frac {3 e^{\frac {1+e^x-2 x}{-1+2 x}}}{2 (-1+2 x)^2}+\frac {e^{\frac {1+e^x-2 x}{-1+2 x}}}{2 (-1+2 x)}\right ) \, dx\\ &=-\left (\frac {1}{4} \int e^{\frac {1+e^x-x-2 x^2}{-1+2 x}} \, dx\right )-2 \left (\frac {1}{4} \int \frac {e^{\frac {1+e^x-x-2 x^2}{-1+2 x}}}{(-1+2 x)^2} \, dx\right )+\frac {1}{2} \int e^{\frac {1+e^x-2 x}{-1+2 x}} \, dx+\frac {1}{2} \int e^{\frac {1+e^x-2 x}{-1+2 x}} x \, dx-\frac {1}{2} \int \frac {e^{\frac {1+e^x-2 x}{-1+2 x}}}{-1+2 x} \, dx-\frac {1}{2} \int \frac {e^{\frac {1+e^x-x-2 x^2}{-1+2 x}}}{-1+2 x} \, dx-\frac {1}{2} \int \frac {\exp \left (\frac {1+e^x-2 x-\log (1+x)+2 x \log (1+x)}{-1+2 x}\right )}{-1+2 x} \, dx-\frac {3}{4} \int e^{\frac {1+e^x-x-2 x^2}{-1+2 x}} \, dx-\frac {3}{2} \int \frac {e^{\frac {1+e^x-2 x}{-1+2 x}}}{(-1+2 x)^2} \, dx-\frac {3}{2} \int \frac {e^{\frac {1+e^x-x-2 x^2}{-1+2 x}}}{(-1+2 x)^2} \, dx-\frac {3}{2} \int \frac {e^{\frac {1+e^x-x-2 x^2}{-1+2 x}}}{-1+2 x} \, dx+2 \int e^{\frac {1+e^x-x-2 x^2}{-1+2 x}} \, dx+2 \int e^{\frac {1+e^x-x-2 x^2}{-1+2 x}} x \, dx+3 \int \frac {e^{\frac {1+e^x-x-2 x^2}{-1+2 x}}}{-1+2 x} \, dx+\int \frac {e^{\frac {1+e^x-x-2 x^2}{-1+2 x}}}{(1-2 x)^2} \, dx-\int e^{\frac {1+e^x-x-2 x^2}{-1+2 x}} x \, dx-\int e^{\frac {1+e^x-x-2 x^2}{-1+2 x}} x^2 \, dx+\int \frac {e^{\frac {1+e^x-x-2 x^2}{-1+2 x}}}{(-1+2 x)^2} \, dx-\int \frac {e^{\frac {1+e^x-x-2 x^2}{-1+2 x}}}{-1+2 x} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 1.38, size = 23, normalized size = 1.00 \begin {gather*} e^{-1-x+\frac {e^x}{-1+2 x}} x (1+x) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.11, size = 35, normalized size = 1.52 \begin {gather*} x e^{\left (-\frac {2 \, x^{2} - {\left (2 \, x - 1\right )} \log \left (x + 1\right ) + x - e^{x} - 1}{2 \, x - 1}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.53, size = 36, normalized size = 1.57 \begin {gather*} {\left (x^{2} e^{\left (\frac {e^{x}}{2 \, x - 1}\right )} + x e^{\left (\frac {e^{x}}{2 \, x - 1}\right )}\right )} e^{\left (-x - 1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.16, size = 37, normalized size = 1.61
method | result | size |
risch | \(x \,{\mathrm e}^{\frac {2 \ln \left (x +1\right ) x -2 x^{2}+{\mathrm e}^{x}-\ln \left (x +1\right )-x +1}{2 x -1}}\) | \(37\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} -\int \frac {{\left (4 \, x^{4} - 8 \, x^{3} + x^{2} - {\left (2 \, x^{3} - x^{2} - 3 \, x\right )} e^{x} + 3 \, x - 1\right )} e^{\left (-\frac {2 \, x^{2} - {\left (2 \, x - 1\right )} \log \left (x + 1\right ) + x - e^{x} - 1}{2 \, x - 1}\right )}}{4 \, x^{3} - 3 \, x + 1}\,{d x} \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 {{\mathrm {e}}^x-x-2\,x^2+\ln \left (x+1\right )\,\left (2\,x-1\right )+1}{2\,x-1}}\,\left (3\,x+{\mathrm {e}}^x\,\left (-2\,x^3+x^2+3\,x\right )+x^2-8\,x^3+4\,x^4-1\right )}{4\,x^3-3\,x+1} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 14.31, size = 29, normalized size = 1.26 \begin {gather*} x e^{\frac {- 2 x^{2} - x + \left (2 x - 1\right ) \log {\left (x + 1 \right )} + e^{x} + 1}{2 x - 1}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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