Optimal. Leaf size=29 \[ \log \left (\frac {x}{1+\left (-e^{5-x^2}-x+x^2\right )^2+\log (x)}\right ) \]
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Rubi [F] time = 54.36, 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^{10} \left (1+4 x^2\right )+e^{5+x^2} \left (2 x^2+4 x^3-4 x^4\right )+e^{2 x^2} \left (-x^2+4 x^3-3 x^4\right )+e^{2 x^2} \log (x)}{e^{10} x+e^{5+x^2} \left (2 x^2-2 x^3\right )+e^{2 x^2} \left (x+x^3-2 x^4+x^5\right )+e^{2 x^2} x \log (x)} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \left (\frac {-x^2+4 x^3-3 x^4+\log (x)}{x \left (1+x^2-2 x^3+x^4+\log (x)\right )}+\frac {e^5 \left (e^5+6 e^5 x^2+2 e^{x^2} x^2-6 e^5 x^3+6 e^{x^2} x^3+8 e^5 x^4-12 e^{x^2} x^4-8 e^5 x^5+14 e^{x^2} x^5+4 e^5 x^6-16 e^{x^2} x^6+12 e^{x^2} x^7-4 e^{x^2} x^8-2 e^{x^2} x \log (x)+4 e^5 x^2 \log (x)+4 e^{x^2} x^2 \log (x)+4 e^{x^2} x^3 \log (x)-4 e^{x^2} x^4 \log (x)\right )}{x \left (1+x^2-2 x^3+x^4+\log (x)\right ) \left (e^{10}+e^{2 x^2}+2 e^{5+x^2} x+e^{2 x^2} x^2-2 e^{5+x^2} x^2-2 e^{2 x^2} x^3+e^{2 x^2} x^4+e^{2 x^2} \log (x)\right )}\right ) \, dx\\ &=e^5 \int \frac {e^5+6 e^5 x^2+2 e^{x^2} x^2-6 e^5 x^3+6 e^{x^2} x^3+8 e^5 x^4-12 e^{x^2} x^4-8 e^5 x^5+14 e^{x^2} x^5+4 e^5 x^6-16 e^{x^2} x^6+12 e^{x^2} x^7-4 e^{x^2} x^8-2 e^{x^2} x \log (x)+4 e^5 x^2 \log (x)+4 e^{x^2} x^2 \log (x)+4 e^{x^2} x^3 \log (x)-4 e^{x^2} x^4 \log (x)}{x \left (1+x^2-2 x^3+x^4+\log (x)\right ) \left (e^{10}+e^{2 x^2}+2 e^{5+x^2} x+e^{2 x^2} x^2-2 e^{5+x^2} x^2-2 e^{2 x^2} x^3+e^{2 x^2} x^4+e^{2 x^2} \log (x)\right )} \, dx+\int \frac {-x^2+4 x^3-3 x^4+\log (x)}{x \left (1+x^2-2 x^3+x^4+\log (x)\right )} \, dx\\ &=e^5 \int \frac {-2 e^{x^2} x^2 \left (-1-3 x+6 x^2-7 x^3+8 x^4-6 x^5+2 x^6\right )+e^5 \left (1+6 x^2-6 x^3+8 x^4-8 x^5+4 x^6\right )-2 x \left (-2 e^5 x+e^{x^2} \left (1-2 x-2 x^2+2 x^3\right )\right ) \log (x)}{x \left (1+x^2-2 x^3+x^4+\log (x)\right ) \left (e^{10}-2 e^{5+x^2} (-1+x) x+e^{2 x^2} \left (1+x^2-2 x^3+x^4\right )+e^{2 x^2} \log (x)\right )} \, dx+\int \left (\frac {1}{x}+\frac {-1-2 x^2+6 x^3-4 x^4}{x \left (1+x^2-2 x^3+x^4+\log (x)\right )}\right ) \, dx\\ &=\text {Rest of rules removed due to large latex content} \end {aligned} \end {gather*}
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Mathematica [B] time = 0.16, size = 88, normalized size = 3.03 \begin {gather*} 2 x^2+\log (x)-\log \left (e^{10}+e^{2 x^2}+2 e^{5+x^2} x+e^{2 x^2} x^2-2 e^{5+x^2} x^2-2 e^{2 x^2} x^3+e^{2 x^2} x^4+e^{2 x^2} \log (x)\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.52, size = 66, normalized size = 2.28 \begin {gather*} -\log \left ({\left ({\left (x^{4} - 2 \, x^{3} + x^{2} + 1\right )} e^{\left (2 \, x^{2} + 10\right )} - 2 \, {\left (x^{2} - x\right )} e^{\left (x^{2} + 15\right )} + e^{\left (2 \, x^{2} + 10\right )} \log \relax (x) + e^{20}\right )} e^{\left (-2 \, x^{2} - 10\right )}\right ) + \log \relax (x) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.00, size = 120, normalized size = 4.14 \begin {gather*} 2 \, x^{2} - \log \left (x^{4} e^{\left (2 \, x^{2}\right )} - 2 \, x^{3} e^{\left (2 \, x^{2}\right )} + x^{2} e^{\left (2 \, x^{2}\right )} - 2 \, x^{2} e^{\left (x^{2} + 5\right )} + 2 \, x e^{\left (x^{2} + 5\right )} + e^{\left (2 \, x^{2}\right )} \log \relax (x) + e^{10} + e^{\left (2 \, x^{2}\right )}\right ) + \log \left (x^{4} - 2 \, x^{3} + x^{2} + \log \relax (x) + 1\right ) - \log \left (-x^{4} + 2 \, x^{3} - x^{2} - \log \relax (x) - 1\right ) + \log \relax (x) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.04, size = 77, normalized size = 2.66
method | result | size |
risch | \(\ln \relax (x )-\ln \left (\ln \relax (x )+\left (x^{4} {\mathrm e}^{2 x^{2}}-2 \,{\mathrm e}^{2 x^{2}} x^{3}-2 \,{\mathrm e}^{x^{2}+5} x^{2}+{\mathrm e}^{2 x^{2}} x^{2}+2 \,{\mathrm e}^{x^{2}+5} x +{\mathrm e}^{10}+{\mathrm e}^{2 x^{2}}\right ) {\mathrm e}^{-2 x^{2}}\right )\) | \(77\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.72, size = 90, normalized size = 3.10 \begin {gather*} 2 \, x^{2} - \log \left (x^{4} - 2 \, x^{3} + x^{2} + \log \relax (x) + 1\right ) + \log \relax (x) - \log \left (\frac {{\left (x^{4} - 2 \, x^{3} + x^{2} + \log \relax (x) + 1\right )} e^{\left (2 \, x^{2}\right )} - 2 \, {\left (x^{2} e^{5} - x e^{5}\right )} e^{\left (x^{2}\right )} + e^{10}}{x^{4} - 2 \, x^{3} + x^{2} + \log \relax (x) + 1}\right ) \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 = -0.03 \begin {gather*} \int \frac {{\mathrm {e}}^{10}\,\left (4\,x^2+1\right )-{\mathrm {e}}^{2\,x^2}\,\left (3\,x^4-4\,x^3+x^2\right )+{\mathrm {e}}^{2\,x^2}\,\ln \relax (x)+{\mathrm {e}}^{x^2+5}\,\left (-4\,x^4+4\,x^3+2\,x^2\right )}{x\,{\mathrm {e}}^{10}+{\mathrm {e}}^{2\,x^2}\,\left (x^5-2\,x^4+x^3+x\right )+{\mathrm {e}}^{x^2+5}\,\left (2\,x^2-2\,x^3\right )+x\,{\mathrm {e}}^{2\,x^2}\,\ln \relax (x)} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 9.63, size = 92, normalized size = 3.17 \begin {gather*} 2 x^{2} + \log {\relax (x )} - \log {\left (\frac {\left (- 2 x^{2} e^{5} + 2 x e^{5}\right ) e^{x^{2}}}{x^{4} - 2 x^{3} + x^{2} + \log {\relax (x )} + 1} + e^{2 x^{2}} + \frac {e^{10}}{x^{4} - 2 x^{3} + x^{2} + \log {\relax (x )} + 1} \right )} - \log {\left (x^{4} - 2 x^{3} + x^{2} + \log {\relax (x )} + 1 \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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