Optimal. Leaf size=25 \[ \frac {x \log \left (x^2\right )}{\left (x-3 \left (-e^{-x}+x\right )\right ) \log (x)} \]
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Rubi [F] time = 3.87, 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 {\left (6 e^x-4 e^{2 x} x\right ) \log (x)+\left (-3 e^x+2 e^{2 x} x+e^x (3+3 x) \log (x)\right ) \log \left (x^2\right )}{\left (9-12 e^x x+4 e^{2 x} x^2\right ) \log ^2(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 {e^x \left (\left (-3+2 e^x x\right ) \log \left (x^2\right )+\log (x) \left (6-4 e^x x+3 (1+x) \log \left (x^2\right )\right )\right )}{\left (3-2 e^x x\right )^2 \log ^2(x)} \, dx\\ &=\int \left (-\frac {e^x \left (2 \log (x)-\log \left (x^2\right )\right )}{\left (-3+2 e^x x\right ) \log ^2(x)}+\frac {3 e^x (1+x) \log \left (x^2\right )}{\left (-3+2 e^x x\right )^2 \log (x)}\right ) \, dx\\ &=3 \int \frac {e^x (1+x) \log \left (x^2\right )}{\left (-3+2 e^x x\right )^2 \log (x)} \, dx-\int \frac {e^x \left (2 \log (x)-\log \left (x^2\right )\right )}{\left (-3+2 e^x x\right ) \log ^2(x)} \, dx\\ &=3 \int \left (\frac {e^x \log \left (x^2\right )}{\left (-3+2 e^x x\right )^2 \log (x)}+\frac {e^x x \log \left (x^2\right )}{\left (-3+2 e^x x\right )^2 \log (x)}\right ) \, dx-\int \left (\frac {2 e^x}{\left (-3+2 e^x x\right ) \log (x)}-\frac {e^x \log \left (x^2\right )}{\left (-3+2 e^x x\right ) \log ^2(x)}\right ) \, dx\\ &=-\left (2 \int \frac {e^x}{\left (-3+2 e^x x\right ) \log (x)} \, dx\right )+3 \int \frac {e^x \log \left (x^2\right )}{\left (-3+2 e^x x\right )^2 \log (x)} \, dx+3 \int \frac {e^x x \log \left (x^2\right )}{\left (-3+2 e^x x\right )^2 \log (x)} \, dx+\int \frac {e^x \log \left (x^2\right )}{\left (-3+2 e^x x\right ) \log ^2(x)} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.55, size = 25, normalized size = 1.00 \begin {gather*} 1+\frac {e^x x \log \left (x^2\right )}{\left (3-2 e^x x\right ) \log (x)} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.77, size = 11, normalized size = 0.44 \begin {gather*} -\frac {3}{2 \, x e^{x} - 3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.27, size = 11, normalized size = 0.44 \begin {gather*} -\frac {3}{2 \, x e^{x} - 3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.12, size = 71, normalized size = 2.84
method | result | size |
risch | \(-\frac {3}{2 \,{\mathrm e}^{x} x -3}+\frac {i x \pi \,\mathrm {csgn}\left (i x^{2}\right ) {\mathrm e}^{x} \left (\mathrm {csgn}\left (i x \right )^{2}-2 \,\mathrm {csgn}\left (i x^{2}\right ) \mathrm {csgn}\left (i x \right )+\mathrm {csgn}\left (i x^{2}\right )^{2}\right )}{2 \left (2 \,{\mathrm e}^{x} x -3\right ) \ln \relax (x )}\) | \(71\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.54, size = 11, normalized size = 0.44 \begin {gather*} -\frac {3}{2 \, x e^{x} - 3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.46, size = 35, normalized size = 1.40 \begin {gather*} -\frac {3\,\ln \relax (x)-2\,x\,{\mathrm {e}}^x\,\ln \relax (x)+x\,\ln \left (x^2\right )\,{\mathrm {e}}^x}{\ln \relax (x)\,\left (2\,x\,{\mathrm {e}}^x-3\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.27, size = 10, normalized size = 0.40 \begin {gather*} - \frac {3}{2 x e^{x} - 3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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