Optimal. Leaf size=28 \[ -3+\left (-e^3+\left (e^{e^x}-x\right ) x-\log (4)\right ) (x+\log (x)) \]
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Rubi [F]
time = 0.53, 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^3 (-1-x)-x^2-3 x^3+(-1-x) \log (4)-2 x^2 \log (x)+e^{e^x} \left (x+2 x^2+e^x x^3+\left (x+e^x x^2\right ) \log (x)\right )}{x} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-x^2-3 x^3+(-1-x) \left (e^3+\log (4)\right )-2 x^2 \log (x)+e^{e^x} \left (x+2 x^2+e^x x^3+\left (x+e^x x^2\right ) \log (x)\right )}{x} \, dx\\ &=\int \left (e^{e^x+x} x (x+\log (x))+\frac {e^{e^x} x-x^2+2 e^{e^x} x^2-3 x^3-e^3 \left (1+\frac {\log (4)}{e^3}\right )-e^3 x \left (1+\frac {\log (4)}{e^3}\right )+e^{e^x} x \log (x)-2 x^2 \log (x)}{x}\right ) \, dx\\ &=\int e^{e^x+x} x (x+\log (x)) \, dx+\int \frac {e^{e^x} x-x^2+2 e^{e^x} x^2-3 x^3-e^3 \left (1+\frac {\log (4)}{e^3}\right )-e^3 x \left (1+\frac {\log (4)}{e^3}\right )+e^{e^x} x \log (x)-2 x^2 \log (x)}{x} \, dx\\ &=\int \left (e^{e^x+x} x^2+e^{e^x+x} x \log (x)\right ) \, dx+\int \left (e^{e^x} (1+2 x+\log (x))+\frac {-x^2-3 x^3-e^3 \left (1+\frac {\log (4)}{e^3}\right )-e^3 x \left (1+\frac {\log (4)}{e^3}\right )-2 x^2 \log (x)}{x}\right ) \, dx\\ &=\int e^{e^x+x} x^2 \, dx+\int e^{e^x+x} x \log (x) \, dx+\int e^{e^x} (1+2 x+\log (x)) \, dx+\int \frac {-x^2-3 x^3-e^3 \left (1+\frac {\log (4)}{e^3}\right )-e^3 x \left (1+\frac {\log (4)}{e^3}\right )-2 x^2 \log (x)}{x} \, dx\\ &=\log (x) \int e^{e^x+x} x \, dx+\int e^{e^x+x} x^2 \, dx+\int \left (e^{e^x}+2 e^{e^x} x+e^{e^x} \log (x)\right ) \, dx+\int \left (\frac {-e^3-x^2-3 x^3-\log (4)-x \left (e^3+\log (4)\right )}{x}-2 x \log (x)\right ) \, dx-\int \frac {\int e^{e^x+x} x \, dx}{x} \, dx\\ &=2 \int e^{e^x} x \, dx-2 \int x \log (x) \, dx+\log (x) \int e^{e^x+x} x \, dx+\int e^{e^x} \, dx+\int e^{e^x+x} x^2 \, dx+\int \frac {-e^3-x^2-3 x^3-\log (4)-x \left (e^3+\log (4)\right )}{x} \, dx+\int e^{e^x} \log (x) \, dx-\int \frac {\int e^{e^x+x} x \, dx}{x} \, dx\\ &=\frac {x^2}{2}-x^2 \log (x)+\text {Ei}\left (e^x\right ) \log (x)+2 \int e^{e^x} x \, dx+\log (x) \int e^{e^x+x} x \, dx+\int e^{e^x+x} x^2 \, dx-\int \frac {\text {Ei}\left (e^x\right )}{x} \, dx+\int \left (-e^3-x-3 x^2+\frac {-e^3-\log (4)}{x}-\log (4)\right ) \, dx-\int \frac {\int e^{e^x+x} x \, dx}{x} \, dx+\text {Subst}\left (\int \frac {e^x}{x} \, dx,x,e^x\right )\\ &=-x^3+\text {Ei}\left (e^x\right )-x \left (e^3+\log (4)\right )-x^2 \log (x)+\text {Ei}\left (e^x\right ) \log (x)-\left (e^3+\log (4)\right ) \log (x)+2 \int e^{e^x} x \, dx+\log (x) \int e^{e^x+x} x \, dx+\int e^{e^x+x} x^2 \, dx-\int \frac {\text {Ei}\left (e^x\right )}{x} \, dx-\int \frac {\int e^{e^x+x} x \, dx}{x} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.08, size = 23, normalized size = 0.82 \begin {gather*} -\left (\left (e^3-e^{e^x} x+x^2+\log (4)\right ) (x+\log (x))\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.35, size = 48, normalized size = 1.71
method | result | size |
risch | \(-x^{2} \ln \left (x \right )-x^{3}-\ln \left (x \right ) {\mathrm e}^{3}-2 \ln \left (2\right ) \ln \left (x \right )-x \,{\mathrm e}^{3}-2 x \ln \left (2\right )+\left (x \ln \left (x \right )+x^{2}\right ) {\mathrm e}^{{\mathrm e}^{x}}\) | \(48\) |
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*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 42, normalized size = 1.50 \begin {gather*} -x^{3} - x e^{3} + {\left (x^{2} + x \log \left (x\right )\right )} e^{\left (e^{x}\right )} - 2 \, x \log \left (2\right ) - {\left (x^{2} + e^{3} + 2 \, \log \left (2\right )\right )} \log \left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 19.43, size = 44, normalized size = 1.57 \begin {gather*} - x^{3} - x^{2} \log {\left (x \right )} - x \left (2 \log {\left (2 \right )} + e^{3}\right ) + \left (x^{2} + x \log {\left (x \right )}\right ) e^{e^{x}} - \left (2 \log {\left (2 \right )} + e^{3}\right ) \log {\left (x \right )} \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. 69 vs.
\(2 (24) = 48\).
time = 0.41, size = 69, normalized size = 2.46 \begin {gather*} -{\left (x^{3} e^{x} + x^{2} e^{x} \log \left (x\right ) - x^{2} e^{\left (x + e^{x}\right )} + 2 \, x e^{x} \log \left (2\right ) - x e^{\left (x + e^{x}\right )} \log \left (x\right ) + 2 \, e^{x} \log \left (2\right ) \log \left (x\right ) + x e^{\left (x + 3\right )} + e^{\left (x + 3\right )} \log \left (x\right )\right )} e^{\left (-x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 3.50, size = 42, normalized size = 1.50 \begin {gather*} {\mathrm {e}}^{{\mathrm {e}}^x}\,\left (x\,\ln \left (x\right )+x^2\right )-x^2\,\ln \left (x\right )-\ln \left (x\right )\,\left ({\mathrm {e}}^3+\ln \left (4\right )\right )-x\,\left ({\mathrm {e}}^3+\ln \left (4\right )\right )-x^3 \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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