Optimal. Leaf size=27 \[ \frac {5-e^3-e^{\left (-e^x+x+x^2\right )^x}}{x} \]
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Rubi [F] time = 9.13, 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^x \left (-5+e^3\right )+5 x+5 x^2+e^3 \left (-x-x^2\right )+e^{\left (-e^x+x+x^2\right )^x} \left (e^x-x-x^2+\left (-e^x+x+x^2\right )^x \left (x^2-e^x x^2+2 x^3+\left (-e^x x+x^2+x^3\right ) \log \left (-e^x+x+x^2\right )\right )\right )}{e^x x^2-x^3-x^4} \, dx \end {gather*}
Verification is not applicable to the result.
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\begin {gather*} \begin {aligned} \text {integral} &=\int \left (\frac {5}{e^x-x-x^2}-\frac {e^{\left (-e^x+x+x^2\right )^x}}{e^x-x-x^2}+\frac {e^{x+\left (-e^x+x+x^2\right )^x}}{x^2 \left (e^x-x-x^2\right )}+\frac {e^x \left (-5+e^3\right )}{x^2 \left (e^x-x-x^2\right )}-\frac {5}{x \left (-e^x+x+x^2\right )}+\frac {e^{\left (-e^x+x+x^2\right )^x}}{x \left (-e^x+x+x^2\right )}+\frac {e^3 (1+x)}{x \left (-e^x+x+x^2\right )}-\frac {e^{\left (-e^x+x+x^2\right )^x} \left (-e^x+x+x^2\right )^{-1+x} \left (x-e^x x+2 x^2-e^x \log \left (-e^x+x+x^2\right )+x \log \left (-e^x+x+x^2\right )+x^2 \log \left (-e^x+x+x^2\right )\right )}{x}\right ) \, dx\\ &=5 \int \frac {1}{e^x-x-x^2} \, dx-5 \int \frac {1}{x \left (-e^x+x+x^2\right )} \, dx+e^3 \int \frac {1+x}{x \left (-e^x+x+x^2\right )} \, dx+\left (-5+e^3\right ) \int \frac {e^x}{x^2 \left (e^x-x-x^2\right )} \, dx-\int \frac {e^{\left (-e^x+x+x^2\right )^x}}{e^x-x-x^2} \, dx+\int \frac {e^{x+\left (-e^x+x+x^2\right )^x}}{x^2 \left (e^x-x-x^2\right )} \, dx+\int \frac {e^{\left (-e^x+x+x^2\right )^x}}{x \left (-e^x+x+x^2\right )} \, dx-\int \frac {e^{\left (-e^x+x+x^2\right )^x} \left (-e^x+x+x^2\right )^{-1+x} \left (x-e^x x+2 x^2-e^x \log \left (-e^x+x+x^2\right )+x \log \left (-e^x+x+x^2\right )+x^2 \log \left (-e^x+x+x^2\right )\right )}{x} \, dx\\ &=5 \int \frac {1}{e^x-x-x^2} \, dx-5 \int \frac {1}{x \left (-e^x+x+x^2\right )} \, dx+e^3 \int \left (-\frac {1}{e^x-x-x^2}+\frac {1}{x \left (-e^x+x+x^2\right )}\right ) \, dx+\left (-5+e^3\right ) \int \frac {e^x}{x^2 \left (e^x-x-x^2\right )} \, dx-\int \frac {e^{\left (-e^x+x+x^2\right )^x}}{e^x-x-x^2} \, dx+\int \frac {e^{x+\left (-e^x+x+x^2\right )^x}}{x^2 \left (e^x-x-x^2\right )} \, dx+\int \frac {e^{\left (-e^x+x+x^2\right )^x}}{x \left (-e^x+x+x^2\right )} \, dx-\int \frac {e^{\left (-e^x+x+x^2\right )^x} \left (-e^x+x+x^2\right )^{-1+x} \left (x \left (1-e^x+2 x\right )+\left (-e^x+x+x^2\right ) \log \left (-e^x+x+x^2\right )\right )}{x} \, dx\\ &=5 \int \frac {1}{e^x-x-x^2} \, dx-5 \int \frac {1}{x \left (-e^x+x+x^2\right )} \, dx-e^3 \int \frac {1}{e^x-x-x^2} \, dx+e^3 \int \frac {1}{x \left (-e^x+x+x^2\right )} \, dx+\left (-5+e^3\right ) \int \frac {e^x}{x^2 \left (e^x-x-x^2\right )} \, dx-\int \frac {e^{\left (-e^x+x+x^2\right )^x}}{e^x-x-x^2} \, dx+\int \frac {e^{x+\left (-e^x+x+x^2\right )^x}}{x^2 \left (e^x-x-x^2\right )} \, dx+\int \frac {e^{\left (-e^x+x+x^2\right )^x}}{x \left (-e^x+x+x^2\right )} \, dx-\int \left (e^{\left (-e^x+x+x^2\right )^x} \left (-e^x+x+x^2\right )^{-1+x}+2 e^{\left (-e^x+x+x^2\right )^x} x \left (-e^x+x+x^2\right )^{-1+x}+e^{\left (-e^x+x+x^2\right )^x} \left (-e^x+x+x^2\right )^{-1+x} \log \left (-e^x+x+x^2\right )+e^{\left (-e^x+x+x^2\right )^x} x \left (-e^x+x+x^2\right )^{-1+x} \log \left (-e^x+x+x^2\right )-\frac {e^{x+\left (-e^x+x+x^2\right )^x} \left (-e^x+x+x^2\right )^{-1+x} \left (x+\log \left (-e^x+x+x^2\right )\right )}{x}\right ) \, dx\\ &=-\left (2 \int e^{\left (-e^x+x+x^2\right )^x} x \left (-e^x+x+x^2\right )^{-1+x} \, dx\right )+5 \int \frac {1}{e^x-x-x^2} \, dx-5 \int \frac {1}{x \left (-e^x+x+x^2\right )} \, dx-e^3 \int \frac {1}{e^x-x-x^2} \, dx+e^3 \int \frac {1}{x \left (-e^x+x+x^2\right )} \, dx+\left (-5+e^3\right ) \int \frac {e^x}{x^2 \left (e^x-x-x^2\right )} \, dx-\int \frac {e^{\left (-e^x+x+x^2\right )^x}}{e^x-x-x^2} \, dx+\int \frac {e^{x+\left (-e^x+x+x^2\right )^x}}{x^2 \left (e^x-x-x^2\right )} \, dx+\int \frac {e^{\left (-e^x+x+x^2\right )^x}}{x \left (-e^x+x+x^2\right )} \, dx-\int e^{\left (-e^x+x+x^2\right )^x} \left (-e^x+x+x^2\right )^{-1+x} \, dx-\int e^{\left (-e^x+x+x^2\right )^x} \left (-e^x+x+x^2\right )^{-1+x} \log \left (-e^x+x+x^2\right ) \, dx-\int e^{\left (-e^x+x+x^2\right )^x} x \left (-e^x+x+x^2\right )^{-1+x} \log \left (-e^x+x+x^2\right ) \, dx+\int \frac {e^{x+\left (-e^x+x+x^2\right )^x} \left (-e^x+x+x^2\right )^{-1+x} \left (x+\log \left (-e^x+x+x^2\right )\right )}{x} \, dx\\ &=-\left (2 \int e^{\left (-e^x+x+x^2\right )^x} x \left (-e^x+x+x^2\right )^{-1+x} \, dx\right )+5 \int \frac {1}{e^x-x-x^2} \, dx-5 \int \frac {1}{x \left (-e^x+x+x^2\right )} \, dx-e^3 \int \frac {1}{e^x-x-x^2} \, dx+e^3 \int \frac {1}{x \left (-e^x+x+x^2\right )} \, dx+\left (-5+e^3\right ) \int \frac {e^x}{x^2 \left (e^x-x-x^2\right )} \, dx-\log \left (-e^x+x+x^2\right ) \int e^{\left (-e^x+x+x^2\right )^x} \left (-e^x+x+x^2\right )^{-1+x} \, dx-\log \left (-e^x+x+x^2\right ) \int e^{\left (-e^x+x+x^2\right )^x} x \left (-e^x+x+x^2\right )^{-1+x} \, dx-\int \frac {e^{\left (-e^x+x+x^2\right )^x}}{e^x-x-x^2} \, dx+\int \frac {e^{x+\left (-e^x+x+x^2\right )^x}}{x^2 \left (e^x-x-x^2\right )} \, dx+\int \frac {e^{\left (-e^x+x+x^2\right )^x}}{x \left (-e^x+x+x^2\right )} \, dx-\int e^{\left (-e^x+x+x^2\right )^x} \left (-e^x+x+x^2\right )^{-1+x} \, dx+\int \left (e^{x+\left (-e^x+x+x^2\right )^x} \left (-e^x+x+x^2\right )^{-1+x}+\frac {e^{x+\left (-e^x+x+x^2\right )^x} \left (-e^x+x+x^2\right )^{-1+x} \log \left (-e^x+x+x^2\right )}{x}\right ) \, dx+\int \frac {\left (-1+e^x-2 x\right ) \int e^{\left (-e^x+x+x^2\right )^x} \left (-e^x+x+x^2\right )^{-1+x} \, dx}{e^x-x (1+x)} \, dx+\int \frac {\left (-1+e^x-2 x\right ) \int e^{\left (-e^x+x+x^2\right )^x} x \left (-e^x+x+x^2\right )^{-1+x} \, dx}{e^x-x (1+x)} \, dx\\ &=-\left (2 \int e^{\left (-e^x+x+x^2\right )^x} x \left (-e^x+x+x^2\right )^{-1+x} \, dx\right )+5 \int \frac {1}{e^x-x-x^2} \, dx-5 \int \frac {1}{x \left (-e^x+x+x^2\right )} \, dx-e^3 \int \frac {1}{e^x-x-x^2} \, dx+e^3 \int \frac {1}{x \left (-e^x+x+x^2\right )} \, dx+\left (-5+e^3\right ) \int \frac {e^x}{x^2 \left (e^x-x-x^2\right )} \, dx-\log \left (-e^x+x+x^2\right ) \int e^{\left (-e^x+x+x^2\right )^x} \left (-e^x+x+x^2\right )^{-1+x} \, dx-\log \left (-e^x+x+x^2\right ) \int e^{\left (-e^x+x+x^2\right )^x} x \left (-e^x+x+x^2\right )^{-1+x} \, dx-\int \frac {e^{\left (-e^x+x+x^2\right )^x}}{e^x-x-x^2} \, dx+\int \frac {e^{x+\left (-e^x+x+x^2\right )^x}}{x^2 \left (e^x-x-x^2\right )} \, dx+\int \frac {e^{\left (-e^x+x+x^2\right )^x}}{x \left (-e^x+x+x^2\right )} \, dx-\int e^{\left (-e^x+x+x^2\right )^x} \left (-e^x+x+x^2\right )^{-1+x} \, dx+\int e^{x+\left (-e^x+x+x^2\right )^x} \left (-e^x+x+x^2\right )^{-1+x} \, dx+\int \frac {e^{x+\left (-e^x+x+x^2\right )^x} \left (-e^x+x+x^2\right )^{-1+x} \log \left (-e^x+x+x^2\right )}{x} \, dx+\int \left (\int e^{\left (-e^x+x+x^2\right )^x} \left (-e^x+x+x^2\right )^{-1+x} \, dx-\frac {\left (-1-x+x^2\right ) \int e^{\left (-e^x+x+x^2\right )^x} \left (-e^x+x+x^2\right )^{-1+x} \, dx}{-e^x+x+x^2}\right ) \, dx+\int \left (\int e^{\left (-e^x+x+x^2\right )^x} x \left (-e^x+x+x^2\right )^{-1+x} \, dx-\frac {\left (-1-x+x^2\right ) \int e^{\left (-e^x+x+x^2\right )^x} x \left (-e^x+x+x^2\right )^{-1+x} \, dx}{-e^x+x+x^2}\right ) \, dx\\ &=-\left (2 \int e^{\left (-e^x+x+x^2\right )^x} x \left (-e^x+x+x^2\right )^{-1+x} \, dx\right )+5 \int \frac {1}{e^x-x-x^2} \, dx-5 \int \frac {1}{x \left (-e^x+x+x^2\right )} \, dx-e^3 \int \frac {1}{e^x-x-x^2} \, dx+e^3 \int \frac {1}{x \left (-e^x+x+x^2\right )} \, dx+\left (-5+e^3\right ) \int \frac {e^x}{x^2 \left (e^x-x-x^2\right )} \, dx-\log \left (-e^x+x+x^2\right ) \int e^{\left (-e^x+x+x^2\right )^x} \left (-e^x+x+x^2\right )^{-1+x} \, dx+\log \left (-e^x+x+x^2\right ) \int \frac {e^{x+\left (-e^x+x+x^2\right )^x} \left (-e^x+x+x^2\right )^{-1+x}}{x} \, dx-\log \left (-e^x+x+x^2\right ) \int e^{\left (-e^x+x+x^2\right )^x} x \left (-e^x+x+x^2\right )^{-1+x} \, dx-\int \frac {e^{\left (-e^x+x+x^2\right )^x}}{e^x-x-x^2} \, dx+\int \frac {e^{x+\left (-e^x+x+x^2\right )^x}}{x^2 \left (e^x-x-x^2\right )} \, dx+\int \frac {e^{\left (-e^x+x+x^2\right )^x}}{x \left (-e^x+x+x^2\right )} \, dx-\int e^{\left (-e^x+x+x^2\right )^x} \left (-e^x+x+x^2\right )^{-1+x} \, dx+\int e^{x+\left (-e^x+x+x^2\right )^x} \left (-e^x+x+x^2\right )^{-1+x} \, dx+\int \left (\int e^{\left (-e^x+x+x^2\right )^x} \left (-e^x+x+x^2\right )^{-1+x} \, dx\right ) \, dx-\int \frac {\left (-1-x+x^2\right ) \int e^{\left (-e^x+x+x^2\right )^x} \left (-e^x+x+x^2\right )^{-1+x} \, dx}{-e^x+x+x^2} \, dx-\int \frac {\left (-1+e^x-2 x\right ) \int \frac {e^{x+\left (-e^x+x+x^2\right )^x} \left (-e^x+x+x^2\right )^{-1+x}}{x} \, dx}{e^x-x (1+x)} \, dx+\int \left (\int e^{\left (-e^x+x+x^2\right )^x} x \left (-e^x+x+x^2\right )^{-1+x} \, dx\right ) \, dx-\int \frac {\left (-1-x+x^2\right ) \int e^{\left (-e^x+x+x^2\right )^x} x \left (-e^x+x+x^2\right )^{-1+x} \, dx}{-e^x+x+x^2} \, dx\\ &=\text {Rest of rules removed due to large latex content} \end {aligned} \end {gather*}
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Mathematica [A] time = 0.18, size = 24, normalized size = 0.89 \begin {gather*} -\frac {-5+e^3+e^{\left (-e^x+x+x^2\right )^x}}{x} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.56, size = 21, normalized size = 0.78 \begin {gather*} -\frac {e^{3} + e^{\left ({\left (x^{2} + x - e^{x}\right )}^{x}\right )} - 5}{x} \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*} \int -\frac {5 \, x^{2} - {\left (x^{2} + x\right )} e^{3} + {\left ({\left (2 \, x^{3} - x^{2} e^{x} + x^{2} + {\left (x^{3} + x^{2} - x e^{x}\right )} \log \left (x^{2} + x - e^{x}\right )\right )} {\left (x^{2} + x - e^{x}\right )}^{x} - x^{2} - x + e^{x}\right )} e^{\left ({\left (x^{2} + x - e^{x}\right )}^{x}\right )} + {\left (e^{3} - 5\right )} e^{x} + 5 \, x}{x^{4} + x^{3} - x^{2} e^{x}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 31, normalized size = 1.15
method | result | size |
risch | \(-\frac {{\mathrm e}^{3}}{x}+\frac {5}{x}-\frac {{\mathrm e}^{\left (-{\mathrm e}^{x}+x^{2}+x \right )^{x}}}{x}\) | \(31\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.45, size = 21, normalized size = 0.78 \begin {gather*} -\frac {e^{3} + e^{\left ({\left (x^{2} + x - e^{x}\right )}^{x}\right )} - 5}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.62, size = 21, normalized size = 0.78 \begin {gather*} -\frac {{\mathrm {e}}^{{\left (x-{\mathrm {e}}^x+x^2\right )}^x}+{\mathrm {e}}^3-5}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] 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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