Optimal. Leaf size=21 \[ \frac {e^x}{x \log \left (3 x \left (e^x+x+x^2\right )\right )} \]
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Rubi [F] time = 6.39, 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 x} (-1-x)+e^x \left (-2 x-3 x^2\right )+\left (e^{2 x} (-1+x)+e^x \left (-x+x^3\right )\right ) \log \left (3 e^x x+3 x^2+3 x^3\right )}{\left (e^x x^2+x^3+x^4\right ) \log ^2\left (3 e^x x+3 x^2+3 x^3\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 (-e^x (1+x)-x (2+3 x)+(-1+x) \left (e^x+x+x^2\right ) \log \left (3 x \left (e^x+x+x^2\right )\right )\right )}{x^2 \left (e^x+x+x^2\right ) \log ^2\left (3 x \left (e^x+x+x^2\right )\right )} \, dx\\ &=\int \left (\frac {e^x \left (-1-x+x^2\right )}{x \left (e^x+x+x^2\right ) \log ^2\left (3 x \left (e^x+x+x^2\right )\right )}+\frac {e^x \left (-1-x-\log \left (3 x \left (e^x+x+x^2\right )\right )+x \log \left (3 x \left (e^x+x+x^2\right )\right )\right )}{x^2 \log ^2\left (3 x \left (e^x+x+x^2\right )\right )}\right ) \, dx\\ &=\int \frac {e^x \left (-1-x+x^2\right )}{x \left (e^x+x+x^2\right ) \log ^2\left (3 x \left (e^x+x+x^2\right )\right )} \, dx+\int \frac {e^x \left (-1-x-\log \left (3 x \left (e^x+x+x^2\right )\right )+x \log \left (3 x \left (e^x+x+x^2\right )\right )\right )}{x^2 \log ^2\left (3 x \left (e^x+x+x^2\right )\right )} \, dx\\ &=\int \left (-\frac {e^x}{\left (e^x+x+x^2\right ) \log ^2\left (3 x \left (e^x+x+x^2\right )\right )}-\frac {e^x}{x \left (e^x+x+x^2\right ) \log ^2\left (3 x \left (e^x+x+x^2\right )\right )}+\frac {e^x x}{\left (e^x+x+x^2\right ) \log ^2\left (3 x \left (e^x+x+x^2\right )\right )}\right ) \, dx+\int \frac {e^x \left (-1-x+(-1+x) \log \left (3 x \left (e^x+x+x^2\right )\right )\right )}{x^2 \log ^2\left (3 x \left (e^x+x+x^2\right )\right )} \, dx\\ &=\int \left (\frac {e^x (-1-x)}{x^2 \log ^2\left (3 x \left (e^x+x+x^2\right )\right )}+\frac {e^x (-1+x)}{x^2 \log \left (3 x \left (e^x+x+x^2\right )\right )}\right ) \, dx-\int \frac {e^x}{\left (e^x+x+x^2\right ) \log ^2\left (3 x \left (e^x+x+x^2\right )\right )} \, dx-\int \frac {e^x}{x \left (e^x+x+x^2\right ) \log ^2\left (3 x \left (e^x+x+x^2\right )\right )} \, dx+\int \frac {e^x x}{\left (e^x+x+x^2\right ) \log ^2\left (3 x \left (e^x+x+x^2\right )\right )} \, dx\\ &=\int \frac {e^x (-1-x)}{x^2 \log ^2\left (3 x \left (e^x+x+x^2\right )\right )} \, dx-\int \frac {e^x}{\left (e^x+x+x^2\right ) \log ^2\left (3 x \left (e^x+x+x^2\right )\right )} \, dx-\int \frac {e^x}{x \left (e^x+x+x^2\right ) \log ^2\left (3 x \left (e^x+x+x^2\right )\right )} \, dx+\int \frac {e^x x}{\left (e^x+x+x^2\right ) \log ^2\left (3 x \left (e^x+x+x^2\right )\right )} \, dx+\int \frac {e^x (-1+x)}{x^2 \log \left (3 x \left (e^x+x+x^2\right )\right )} \, dx\\ &=\int \left (-\frac {e^x}{x^2 \log ^2\left (3 x \left (e^x+x+x^2\right )\right )}-\frac {e^x}{x \log ^2\left (3 x \left (e^x+x+x^2\right )\right )}\right ) \, dx+\int \left (-\frac {e^x}{x^2 \log \left (3 x \left (e^x+x+x^2\right )\right )}+\frac {e^x}{x \log \left (3 x \left (e^x+x+x^2\right )\right )}\right ) \, dx-\int \frac {e^x}{\left (e^x+x+x^2\right ) \log ^2\left (3 x \left (e^x+x+x^2\right )\right )} \, dx-\int \frac {e^x}{x \left (e^x+x+x^2\right ) \log ^2\left (3 x \left (e^x+x+x^2\right )\right )} \, dx+\int \frac {e^x x}{\left (e^x+x+x^2\right ) \log ^2\left (3 x \left (e^x+x+x^2\right )\right )} \, dx\\ &=-\int \frac {e^x}{x^2 \log ^2\left (3 x \left (e^x+x+x^2\right )\right )} \, dx-\int \frac {e^x}{x \log ^2\left (3 x \left (e^x+x+x^2\right )\right )} \, dx-\int \frac {e^x}{\left (e^x+x+x^2\right ) \log ^2\left (3 x \left (e^x+x+x^2\right )\right )} \, dx-\int \frac {e^x}{x \left (e^x+x+x^2\right ) \log ^2\left (3 x \left (e^x+x+x^2\right )\right )} \, dx+\int \frac {e^x x}{\left (e^x+x+x^2\right ) \log ^2\left (3 x \left (e^x+x+x^2\right )\right )} \, dx-\int \frac {e^x}{x^2 \log \left (3 x \left (e^x+x+x^2\right )\right )} \, dx+\int \frac {e^x}{x \log \left (3 x \left (e^x+x+x^2\right )\right )} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 1.48, size = 21, normalized size = 1.00 \begin {gather*} \frac {e^x}{x \log \left (3 x \left (e^x+x+x^2\right )\right )} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.65, size = 25, normalized size = 1.19 \begin {gather*} \frac {e^{x}}{x \log \left (3 \, x^{3} + 3 \, x^{2} + 3 \, x e^{x}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.43, size = 25, normalized size = 1.19 \begin {gather*} \frac {e^{x}}{x \log \left (3 \, x^{3} + 3 \, x^{2} + 3 \, x e^{x}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.11, size = 129, normalized size = 6.14
method | result | size |
risch | \(\frac {2 i {\mathrm e}^{x}}{x \left (\pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i \left ({\mathrm e}^{x}+x^{2}+x \right )\right ) \mathrm {csgn}\left (i x \left ({\mathrm e}^{x}+x^{2}+x \right )\right )-\pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x \left ({\mathrm e}^{x}+x^{2}+x \right )\right )^{2}-\pi \,\mathrm {csgn}\left (i \left ({\mathrm e}^{x}+x^{2}+x \right )\right ) \mathrm {csgn}\left (i x \left ({\mathrm e}^{x}+x^{2}+x \right )\right )^{2}+\pi \mathrm {csgn}\left (i x \left ({\mathrm e}^{x}+x^{2}+x \right )\right )^{3}+2 i \ln \relax (3)+2 i \ln \relax (x )+2 i \ln \left ({\mathrm e}^{x}+x^{2}+x \right )\right )}\) | \(129\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.59, size = 24, normalized size = 1.14 \begin {gather*} \frac {e^{x}}{x \log \relax (3) + x \log \left (x^{2} + x + e^{x}\right ) + x \log \relax (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 = -0.05 \begin {gather*} \int -\frac {\ln \left (3\,x\,{\mathrm {e}}^x+3\,x^2+3\,x^3\right )\,\left ({\mathrm {e}}^x\,\left (x-x^3\right )-{\mathrm {e}}^{2\,x}\,\left (x-1\right )\right )+{\mathrm {e}}^{2\,x}\,\left (x+1\right )+{\mathrm {e}}^x\,\left (3\,x^2+2\,x\right )}{{\ln \left (3\,x\,{\mathrm {e}}^x+3\,x^2+3\,x^3\right )}^2\,\left (x^2\,{\mathrm {e}}^x+x^3+x^4\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.21, size = 22, normalized size = 1.05 \begin {gather*} \frac {e^{x}}{x \log {\left (3 x^{3} + 3 x^{2} + 3 x e^{x} \right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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