Optimal. Leaf size=32 \[ 25-\frac {x-\frac {e^x}{\left (2-e^{x^2}+x^2\right ) \log (x)}}{x^2} \]
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Rubi [F] time = 5.65, 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^{x+x^2}+e^x \left (-2-x^2\right )+\left (e^{x+x^2} \left (2-x+2 x^2\right )+e^x \left (-4+2 x-4 x^2+x^3\right )\right ) \log (x)+\left (4 x+e^{2 x^2} x+4 x^3+x^5+e^{x^2} \left (-4 x-2 x^3\right )\right ) \log ^2(x)}{\left (4 x^3+e^{2 x^2} x^3+4 x^5+x^7+e^{x^2} \left (-4 x^3-2 x^5\right )\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 (-2+e^{x^2}-x^2\right )+e^x \left (-4+2 x-4 x^2+x^3+e^{x^2} \left (2-x+2 x^2\right )\right ) \log (x)+x \left (2-e^{x^2}+x^2\right )^2 \log ^2(x)}{x^3 \left (2-e^{x^2}+x^2\right )^2 \log ^2(x)} \, dx\\ &=\int \left (\frac {1}{x^2}+\frac {2 e^x \left (1+x^2\right )}{x \left (2-e^{x^2}+x^2\right )^2 \log (x)}-\frac {e^x \left (1+2 \log (x)-x \log (x)+2 x^2 \log (x)\right )}{x^3 \left (2-e^{x^2}+x^2\right ) \log ^2(x)}\right ) \, dx\\ &=-\frac {1}{x}+2 \int \frac {e^x \left (1+x^2\right )}{x \left (2-e^{x^2}+x^2\right )^2 \log (x)} \, dx-\int \frac {e^x \left (1+2 \log (x)-x \log (x)+2 x^2 \log (x)\right )}{x^3 \left (2-e^{x^2}+x^2\right ) \log ^2(x)} \, dx\\ &=-\frac {1}{x}+2 \int \left (\frac {e^x}{x \left (2-e^{x^2}+x^2\right )^2 \log (x)}+\frac {e^x x}{\left (2-e^{x^2}+x^2\right )^2 \log (x)}\right ) \, dx-\int \left (\frac {e^x}{x^3 \left (2-e^{x^2}+x^2\right ) \log ^2(x)}+\frac {2 e^x}{x^3 \left (2-e^{x^2}+x^2\right ) \log (x)}-\frac {e^x}{x^2 \left (2-e^{x^2}+x^2\right ) \log (x)}+\frac {2 e^x}{x \left (2-e^{x^2}+x^2\right ) \log (x)}\right ) \, dx\\ &=-\frac {1}{x}+2 \int \frac {e^x}{x \left (2-e^{x^2}+x^2\right )^2 \log (x)} \, dx+2 \int \frac {e^x x}{\left (2-e^{x^2}+x^2\right )^2 \log (x)} \, dx-2 \int \frac {e^x}{x^3 \left (2-e^{x^2}+x^2\right ) \log (x)} \, dx-2 \int \frac {e^x}{x \left (2-e^{x^2}+x^2\right ) \log (x)} \, dx-\int \frac {e^x}{x^3 \left (2-e^{x^2}+x^2\right ) \log ^2(x)} \, dx+\int \frac {e^x}{x^2 \left (2-e^{x^2}+x^2\right ) \log (x)} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.10, size = 31, normalized size = 0.97 \begin {gather*} -\frac {1}{x}+\frac {e^x}{x^2 \left (2-e^{x^2}+x^2\right ) \log (x)} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.68, size = 66, normalized size = 2.06 \begin {gather*} -\frac {{\left (x e^{\left (2 \, x^{2}\right )} - {\left (x^{3} + 2 \, x\right )} e^{\left (x^{2}\right )}\right )} \log \relax (x) + e^{\left (x^{2} + x\right )}}{{\left (x^{2} e^{\left (2 \, x^{2}\right )} - {\left (x^{4} + 2 \, x^{2}\right )} e^{\left (x^{2}\right )}\right )} \log \relax (x)} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 54, normalized size = 1.69 \begin {gather*} -\frac {x^{3} \log \relax (x) - x e^{\left (x^{2}\right )} \log \relax (x) + 2 \, x \log \relax (x) - e^{x}}{x^{4} \log \relax (x) - x^{2} e^{\left (x^{2}\right )} \log \relax (x) + 2 \, x^{2} \log \relax (x)} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.18, size = 30, normalized size = 0.94
method | result | size |
risch | \(-\frac {1}{x}+\frac {{\mathrm e}^{x}}{x^{2} \left (2+x^{2}-{\mathrm e}^{x^{2}}\right ) \ln \relax (x )}\) | \(30\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.50, size = 50, normalized size = 1.56 \begin {gather*} -\frac {x e^{\left (x^{2}\right )} \log \relax (x) - {\left (x^{3} + 2 \, x\right )} \log \relax (x) + e^{x}}{x^{2} e^{\left (x^{2}\right )} \log \relax (x) - {\left (x^{4} + 2 \, x^{2}\right )} \log \relax (x)} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.32, size = 48, normalized size = 1.50 \begin {gather*} -\frac {x^3\,\ln \relax (x)-{\mathrm {e}}^x+x\,\left (2\,\ln \relax (x)-{\mathrm {e}}^{x^2}\,\ln \relax (x)\right )}{x^2\,\ln \relax (x)\,\left (x^2-{\mathrm {e}}^{x^2}+2\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.34, size = 34, normalized size = 1.06 \begin {gather*} - \frac {e^{x}}{- x^{4} \log {\relax (x )} + x^{2} e^{x^{2}} \log {\relax (x )} - 2 x^{2} \log {\relax (x )}} - \frac {1}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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