Optimal. Leaf size=22 \[ \frac {1+e^x}{1+e^{x^2}-x-\log (x)} \]
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Rubi [F] time = 3.79, 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 {1+x+e^x \left (1+2 x-x^2\right )+e^{x^2} \left (-2 x^2+e^x \left (x-2 x^2\right )\right )-e^x x \log (x)}{x+e^{2 x^2} x-2 x^2+x^3+e^{x^2} \left (2 x-2 x^2\right )+\left (-2 x-2 e^{x^2} x+2 x^2\right ) \log (x)+x \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 {1+x-2 e^{x^2} x^2+e^{x+x^2} \left (x-2 x^2\right )+e^x \left (1+2 x-x^2\right )-e^x x \log (x)}{x \left (1+e^{x^2}-x-\log (x)\right )^2} \, dx\\ &=\int \left (\frac {-e^x+2 x+2 e^x x}{-1-e^{x^2}+x+\log (x)}-\frac {\left (1+e^x\right ) \left (-1-x-2 x^2+2 x^3+2 x^2 \log (x)\right )}{x \left (-1-e^{x^2}+x+\log (x)\right )^2}\right ) \, dx\\ &=\int \frac {-e^x+2 x+2 e^x x}{-1-e^{x^2}+x+\log (x)} \, dx-\int \frac {\left (1+e^x\right ) \left (-1-x-2 x^2+2 x^3+2 x^2 \log (x)\right )}{x \left (-1-e^{x^2}+x+\log (x)\right )^2} \, dx\\ &=-\int \left (-\frac {1}{\left (1+e^{x^2}-x-\log (x)\right )^2}-\frac {e^x}{\left (1+e^{x^2}-x-\log (x)\right )^2}-\frac {1}{x \left (-1-e^{x^2}+x+\log (x)\right )^2}-\frac {e^x}{x \left (-1-e^{x^2}+x+\log (x)\right )^2}-\frac {2 x}{\left (-1-e^{x^2}+x+\log (x)\right )^2}-\frac {2 e^x x}{\left (-1-e^{x^2}+x+\log (x)\right )^2}+\frac {2 x^2}{\left (-1-e^{x^2}+x+\log (x)\right )^2}+\frac {2 e^x x^2}{\left (-1-e^{x^2}+x+\log (x)\right )^2}+\frac {2 x \log (x)}{\left (-1-e^{x^2}+x+\log (x)\right )^2}+\frac {2 e^x x \log (x)}{\left (-1-e^{x^2}+x+\log (x)\right )^2}\right ) \, dx+\int \left (\frac {e^x}{1+e^{x^2}-x-\log (x)}+\frac {2 x}{-1-e^{x^2}+x+\log (x)}+\frac {2 e^x x}{-1-e^{x^2}+x+\log (x)}\right ) \, dx\\ &=2 \int \frac {x}{\left (-1-e^{x^2}+x+\log (x)\right )^2} \, dx+2 \int \frac {e^x x}{\left (-1-e^{x^2}+x+\log (x)\right )^2} \, dx-2 \int \frac {x^2}{\left (-1-e^{x^2}+x+\log (x)\right )^2} \, dx-2 \int \frac {e^x x^2}{\left (-1-e^{x^2}+x+\log (x)\right )^2} \, dx-2 \int \frac {x \log (x)}{\left (-1-e^{x^2}+x+\log (x)\right )^2} \, dx-2 \int \frac {e^x x \log (x)}{\left (-1-e^{x^2}+x+\log (x)\right )^2} \, dx+2 \int \frac {x}{-1-e^{x^2}+x+\log (x)} \, dx+2 \int \frac {e^x x}{-1-e^{x^2}+x+\log (x)} \, dx+\int \frac {1}{\left (1+e^{x^2}-x-\log (x)\right )^2} \, dx+\int \frac {e^x}{\left (1+e^{x^2}-x-\log (x)\right )^2} \, dx+\int \frac {e^x}{1+e^{x^2}-x-\log (x)} \, dx+\int \frac {1}{x \left (-1-e^{x^2}+x+\log (x)\right )^2} \, dx+\int \frac {e^x}{x \left (-1-e^{x^2}+x+\log (x)\right )^2} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.69, size = 22, normalized size = 1.00 \begin {gather*} \frac {1+e^x}{1+e^{x^2}-x-\log (x)} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.52, size = 19, normalized size = 0.86 \begin {gather*} -\frac {e^{x} + 1}{x - e^{\left (x^{2}\right )} + \log \relax (x) - 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.28, size = 19, normalized size = 0.86 \begin {gather*} -\frac {e^{x} + 1}{x - e^{\left (x^{2}\right )} + \log \relax (x) - 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 20, normalized size = 0.91
method | result | size |
risch | \(-\frac {{\mathrm e}^{x}+1}{x -1+\ln \relax (x )-{\mathrm e}^{x^{2}}}\) | \(20\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.41, size = 19, normalized size = 0.86 \begin {gather*} -\frac {e^{x} + 1}{x - e^{\left (x^{2}\right )} + \log \relax (x) - 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.42, size = 85, normalized size = 3.86 \begin {gather*} -\frac {x\,\left ({\mathrm {e}}^x+1\right )+x^2\,\left ({\mathrm {e}}^x+1\right )+x^3\,\left (2\,{\mathrm {e}}^x-\ln \relax (x)\,\left (2\,{\mathrm {e}}^x+2\right )+2\right )-x^4\,\left (2\,{\mathrm {e}}^x+2\right )}{\left (x-{\mathrm {e}}^{x^2}+\ln \relax (x)-1\right )\,\left (x-2\,x^3\,\ln \relax (x)+x^2+2\,x^3-2\,x^4\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.31, size = 15, normalized size = 0.68 \begin {gather*} \frac {e^{x} + 1}{- x + e^{x^{2}} - \log {\relax (x )} + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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