Optimal. Leaf size=25 \[ x+\frac {1-\frac {-2+x}{\log \left (\log \left (\log \left (\frac {e^x}{2}\right )\right )\right )}}{x} \]
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Rubi [F] time = 1.26, 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 {-2 x+x^2-2 \log \left (\frac {e^x}{2}\right ) \log \left (\log \left (\frac {e^x}{2}\right )\right ) \log \left (\log \left (\log \left (\frac {e^x}{2}\right )\right )\right )+\left (-1+x^2\right ) \log \left (\frac {e^x}{2}\right ) \log \left (\log \left (\frac {e^x}{2}\right )\right ) \log ^2\left (\log \left (\log \left (\frac {e^x}{2}\right )\right )\right )}{x^2 \log \left (\frac {e^x}{2}\right ) \log \left (\log \left (\frac {e^x}{2}\right )\right ) \log ^2\left (\log \left (\log \left (\frac {e^x}{2}\right )\right )\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 {\frac {(-2+x) x}{\log \left (\frac {e^x}{2}\right ) \log \left (\log \left (\frac {e^x}{2}\right )\right )}+\log \left (\log \left (\log \left (\frac {e^x}{2}\right )\right )\right ) \left (-2+\left (-1+x^2\right ) \log \left (\log \left (\log \left (\frac {e^x}{2}\right )\right )\right )\right )}{x^2 \log ^2\left (\log \left (\log \left (\frac {e^x}{2}\right )\right )\right )} \, dx\\ &=\int \left (\frac {-1+x^2}{x^2}+\frac {-2+x}{x \log \left (\frac {e^x}{2}\right ) \log \left (\log \left (\frac {e^x}{2}\right )\right ) \log ^2\left (\log \left (\log \left (\frac {e^x}{2}\right )\right )\right )}-\frac {2}{x^2 \log \left (\log \left (\log \left (\frac {e^x}{2}\right )\right )\right )}\right ) \, dx\\ &=-\left (2 \int \frac {1}{x^2 \log \left (\log \left (\log \left (\frac {e^x}{2}\right )\right )\right )} \, dx\right )+\int \frac {-1+x^2}{x^2} \, dx+\int \frac {-2+x}{x \log \left (\frac {e^x}{2}\right ) \log \left (\log \left (\frac {e^x}{2}\right )\right ) \log ^2\left (\log \left (\log \left (\frac {e^x}{2}\right )\right )\right )} \, dx\\ &=-\left (2 \int \frac {1}{x^2 \log \left (\log \left (\log \left (\frac {e^x}{2}\right )\right )\right )} \, dx\right )+\int \left (1-\frac {1}{x^2}\right ) \, dx+\int \left (\frac {1}{\log \left (\frac {e^x}{2}\right ) \log \left (\log \left (\frac {e^x}{2}\right )\right ) \log ^2\left (\log \left (\log \left (\frac {e^x}{2}\right )\right )\right )}-\frac {2}{x \log \left (\frac {e^x}{2}\right ) \log \left (\log \left (\frac {e^x}{2}\right )\right ) \log ^2\left (\log \left (\log \left (\frac {e^x}{2}\right )\right )\right )}\right ) \, dx\\ &=\frac {1}{x}+x-2 \int \frac {1}{x \log \left (\frac {e^x}{2}\right ) \log \left (\log \left (\frac {e^x}{2}\right )\right ) \log ^2\left (\log \left (\log \left (\frac {e^x}{2}\right )\right )\right )} \, dx-2 \int \frac {1}{x^2 \log \left (\log \left (\log \left (\frac {e^x}{2}\right )\right )\right )} \, dx+\int \frac {1}{\log \left (\frac {e^x}{2}\right ) \log \left (\log \left (\frac {e^x}{2}\right )\right ) \log ^2\left (\log \left (\log \left (\frac {e^x}{2}\right )\right )\right )} \, dx\\ &=\frac {1}{x}+x-2 \int \frac {1}{x \log \left (\frac {e^x}{2}\right ) \log \left (\log \left (\frac {e^x}{2}\right )\right ) \log ^2\left (\log \left (\log \left (\frac {e^x}{2}\right )\right )\right )} \, dx-2 \int \frac {1}{x^2 \log \left (\log \left (\log \left (\frac {e^x}{2}\right )\right )\right )} \, dx+\operatorname {Subst}\left (\int \frac {1}{x \log \left (\frac {x}{2}\right ) \log \left (\log \left (\frac {x}{2}\right )\right ) \log ^2\left (\log \left (\log \left (\frac {x}{2}\right )\right )\right )} \, dx,x,e^x\right )\\ &=\frac {1}{x}+x-2 \int \frac {1}{x \log \left (\frac {e^x}{2}\right ) \log \left (\log \left (\frac {e^x}{2}\right )\right ) \log ^2\left (\log \left (\log \left (\frac {e^x}{2}\right )\right )\right )} \, dx-2 \int \frac {1}{x^2 \log \left (\log \left (\log \left (\frac {e^x}{2}\right )\right )\right )} \, dx+\operatorname {Subst}\left (\int \frac {1}{x \log (x) \log ^2(\log (x))} \, dx,x,\log \left (\frac {e^x}{2}\right )\right )\\ &=\frac {1}{x}+x-2 \int \frac {1}{x \log \left (\frac {e^x}{2}\right ) \log \left (\log \left (\frac {e^x}{2}\right )\right ) \log ^2\left (\log \left (\log \left (\frac {e^x}{2}\right )\right )\right )} \, dx-2 \int \frac {1}{x^2 \log \left (\log \left (\log \left (\frac {e^x}{2}\right )\right )\right )} \, dx+\operatorname {Subst}\left (\int \frac {1}{x \log ^2(x)} \, dx,x,\log \left (\log \left (\frac {e^x}{2}\right )\right )\right )\\ &=\frac {1}{x}+x-2 \int \frac {1}{x \log \left (\frac {e^x}{2}\right ) \log \left (\log \left (\frac {e^x}{2}\right )\right ) \log ^2\left (\log \left (\log \left (\frac {e^x}{2}\right )\right )\right )} \, dx-2 \int \frac {1}{x^2 \log \left (\log \left (\log \left (\frac {e^x}{2}\right )\right )\right )} \, dx+\operatorname {Subst}\left (\int \frac {1}{x^2} \, dx,x,\log \left (\log \left (\log \left (\frac {e^x}{2}\right )\right )\right )\right )\\ &=\frac {1}{x}+x-\frac {1}{\log \left (\log \left (\log \left (\frac {e^x}{2}\right )\right )\right )}-2 \int \frac {1}{x \log \left (\frac {e^x}{2}\right ) \log \left (\log \left (\frac {e^x}{2}\right )\right ) \log ^2\left (\log \left (\log \left (\frac {e^x}{2}\right )\right )\right )} \, dx-2 \int \frac {1}{x^2 \log \left (\log \left (\log \left (\frac {e^x}{2}\right )\right )\right )} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.08, size = 26, normalized size = 1.04 \begin {gather*} \frac {1}{x}+x+\frac {2-x}{x \log \left (\log \left (\log \left (\frac {e^x}{2}\right )\right )\right )} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.47, size = 33, normalized size = 1.32 \begin {gather*} \frac {{\left (x^{2} + 1\right )} \log \left (\log \left (x - \log \relax (2)\right )\right ) - x + 2}{x \log \left (\log \left (x - \log \relax (2)\right )\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 23, normalized size = 0.92 \begin {gather*} x + \frac {1}{x} - \frac {x - 2}{x \log \left (\log \left (x - \log \relax (2)\right )\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.37, size = 31, normalized size = 1.24
method | result | size |
risch | \(\frac {x^{2}+1}{x}-\frac {x -2}{x \ln \left (\ln \left (-\ln \relax (2)+\ln \left ({\mathrm e}^{x}\right )\right )\right )}\) | \(31\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.54, size = 31, normalized size = 1.24 \begin {gather*} x + \frac {1}{x} - \frac {1}{\log \left (\log \left (\log \left (\frac {1}{2} \, e^{x}\right )\right )\right )} + \frac {2}{x \log \left (\log \left (x - \log \relax (2)\right )\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.69, size = 32, normalized size = 1.28 \begin {gather*} x+\frac {2}{x\,\ln \left (\ln \left (x-\ln \relax (2)\right )\right )}+\frac {1}{x}-\frac {1}{\ln \left (\ln \left (x-\ln \relax (2)\right )\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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