Optimal. Leaf size=34 \[ 1+\log \left (\frac {\left (\frac {e^{e^{5-x}}}{x}+x\right ) \left (e^{2-x}-\log (x)\right )}{x^2}\right ) \]
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Rubi [F] time = 6.92, 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 {x^2+e^{2-x} \left (x^2+x^3\right )-x^2 \log (x)+e^{e^{5-x}} \left (1+e^{7-2 x} x+e^{2-x} (3+x)+\left (-3-e^{5-x} x\right ) \log (x)\right )}{-e^{2-x} x^3+x^3 \log (x)+e^{e^{5-x}} \left (-e^{2-x} x+x \log (x)\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 {e^x \left (-x^2-e^{2-x} \left (x^2+x^3\right )+x^2 \log (x)-e^{e^{5-x}} \left (1+e^{7-2 x} x+e^{2-x} (3+x)+\left (-3-e^{5-x} x\right ) \log (x)\right )\right )}{x \left (e^{e^{5-x}}+x^2\right ) \left (e^2-e^x \log (x)\right )} \, dx\\ &=\int \left (-\frac {e^{5+e^{5-x}-x}}{e^{e^{5-x}}+x^2}-\frac {3 e^{e^{5-x}}+e^{e^{5-x}} x+x^2+x^3}{x \left (e^{e^{5-x}}+x^2\right )}-\frac {e^x (1+x \log (x))}{x \left (e^2-e^x \log (x)\right )}\right ) \, dx\\ &=-\int \frac {e^{5+e^{5-x}-x}}{e^{e^{5-x}}+x^2} \, dx-\int \frac {3 e^{e^{5-x}}+e^{e^{5-x}} x+x^2+x^3}{x \left (e^{e^{5-x}}+x^2\right )} \, dx-\int \frac {e^x (1+x \log (x))}{x \left (e^2-e^x \log (x)\right )} \, dx\\ &=-\int \frac {e^{5+e^{5-x}-x}}{e^{e^{5-x}}+x^2} \, dx-\int \frac {x^2 (1+x)+e^{e^{5-x}} (3+x)}{x \left (e^{e^{5-x}}+x^2\right )} \, dx-\operatorname {Subst}\left (\int \frac {1}{e^2-x} \, dx,x,e^x \log (x)\right )\\ &=\log \left (e^2-e^x \log (x)\right )-\int \frac {e^{5+e^{5-x}-x}}{e^{e^{5-x}}+x^2} \, dx-\int \left (\frac {3+x}{x}-\frac {2 x}{e^{e^{5-x}}+x^2}\right ) \, dx\\ &=\log \left (e^2-e^x \log (x)\right )+2 \int \frac {x}{e^{e^{5-x}}+x^2} \, dx-\int \frac {3+x}{x} \, dx-\int \frac {e^{5+e^{5-x}-x}}{e^{e^{5-x}}+x^2} \, dx\\ &=\log \left (e^2-e^x \log (x)\right )+2 \int \frac {x}{e^{e^{5-x}}+x^2} \, dx-\int \left (1+\frac {3}{x}\right ) \, dx-\int \frac {e^{5+e^{5-x}-x}}{e^{e^{5-x}}+x^2} \, dx\\ &=-x-3 \log (x)+\log \left (e^2-e^x \log (x)\right )+2 \int \frac {x}{e^{e^{5-x}}+x^2} \, dx-\int \frac {e^{5+e^{5-x}-x}}{e^{e^{5-x}}+x^2} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.30, size = 34, normalized size = 1.00 \begin {gather*} -x-3 \log (x)+\log \left (e^{e^{5-x}}+x^2\right )+\log \left (e^2-e^x \log (x)\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.64, size = 32, normalized size = 0.94 \begin {gather*} \log \left (x^{2} + e^{\left (e^{\left (-x + 5\right )}\right )}\right ) + \log \left (e^{3} \log \relax (x) - e^{\left (-x + 5\right )}\right ) - 3 \, \log \relax (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 {x^{2} \log \relax (x) - x^{2} - {\left (x^{3} + x^{2}\right )} e^{\left (-x + 2\right )} - {\left ({\left (x + 3\right )} e^{\left (-x + 2\right )} + x e^{\left (-2 \, x + 7\right )} - {\left (x e^{\left (-x + 5\right )} + 3\right )} \log \relax (x) + 1\right )} e^{\left (e^{\left (-x + 5\right )}\right )}}{x^{3} e^{\left (-x + 2\right )} - x^{3} \log \relax (x) + {\left (x e^{\left (-x + 2\right )} - x \log \relax (x)\right )} e^{\left (e^{\left (-x + 5\right )}\right )}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 30, normalized size = 0.88
method | result | size |
risch | \(-3 \ln \relax (x )+\ln \left (-{\mathrm e}^{2-x}+\ln \relax (x )\right )+\ln \left (x^{2}+{\mathrm e}^{{\mathrm e}^{5-x}}\right )\) | \(30\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.42, size = 39, normalized size = 1.15 \begin {gather*} -x + \log \left (x^{2} + e^{\left (e^{\left (-x + 5\right )}\right )}\right ) - 3 \, \log \relax (x) + \log \left (\frac {e^{x} \log \relax (x) - e^{2}}{\log \relax (x)}\right ) + \log \left (\log \relax (x)\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 9.19, size = 30, normalized size = 0.88 \begin {gather*} \ln \left (\left (\ln \relax (x)-{\mathrm {e}}^{-x}\,{\mathrm {e}}^2\right )\,\left ({\mathrm {e}}^{{\mathrm {e}}^{-x}\,{\mathrm {e}}^5}+x^2\right )\right )-3\,\ln \relax (x) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.45, size = 29, normalized size = 0.85 \begin {gather*} - 3 \log {\relax (x )} + \log {\left (x^{2} + e^{e^{3} e^{2 - x}} \right )} + \log {\left (e^{2 - x} - \log {\relax (x )} \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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