Optimal. Leaf size=17 \[ 5 x+\frac {x}{e^2+\log (1+\log (x))} \]
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Rubi [F] time = 0.47, 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+e^2+5 e^4+\left (e^2+5 e^4\right ) \log (x)+\left (1+10 e^2+\left (1+10 e^2\right ) \log (x)\right ) \log (1+\log (x))+(5+5 \log (x)) \log ^2(1+\log (x))}{e^4+e^4 \log (x)+\left (2 e^2+2 e^2 \log (x)\right ) \log (1+\log (x))+(1+\log (x)) \log ^2(1+\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 \frac {-1+e^2 \left (1+5 e^2\right )+\left (e^2+5 e^4\right ) \log (x)+\left (1+10 e^2+\left (1+10 e^2\right ) \log (x)\right ) \log (1+\log (x))+(5+5 \log (x)) \log ^2(1+\log (x))}{(1+\log (x)) \left (e^2+\log (1+\log (x))\right )^2} \, dx\\ &=\int \left (5-\frac {1}{(1+\log (x)) \left (e^2+\log (1+\log (x))\right )^2}+\frac {1}{e^2+\log (1+\log (x))}\right ) \, dx\\ &=5 x-\int \frac {1}{(1+\log (x)) \left (e^2+\log (1+\log (x))\right )^2} \, dx+\int \frac {1}{e^2+\log (1+\log (x))} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.05, size = 17, normalized size = 1.00 \begin {gather*} 5 x+\frac {x}{e^2+\log (1+\log (x))} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.74, size = 26, normalized size = 1.53 \begin {gather*} \frac {5 \, x e^{2} + 5 \, x \log \left (\log \relax (x) + 1\right ) + x}{e^{2} + \log \left (\log \relax (x) + 1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.34, size = 46, normalized size = 2.71 \begin {gather*} \frac {5 \, x e^{2}}{e^{2} + \log \left (\log \relax (x) + 1\right )} + \frac {5 \, x \log \left (\log \relax (x) + 1\right )}{e^{2} + \log \left (\log \relax (x) + 1\right )} + \frac {x}{e^{2} + \log \left (\log \relax (x) + 1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.12, size = 17, normalized size = 1.00
method | result | size |
risch | \(\frac {x}{{\mathrm e}^{2}+\ln \left (\ln \relax (x )+1\right )}+5 x\) | \(17\) |
norman | \(\frac {\left (5 \,{\mathrm e}^{2}+1\right ) x +5 x \ln \left (\ln \relax (x )+1\right )}{{\mathrm e}^{2}+\ln \left (\ln \relax (x )+1\right )}\) | \(29\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.56, size = 28, normalized size = 1.65 \begin {gather*} \frac {x {\left (5 \, e^{2} + 1\right )} + 5 \, x \log \left (\log \relax (x) + 1\right )}{e^{2} + \log \left (\log \relax (x) + 1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.57, size = 25, normalized size = 1.47 \begin {gather*} \frac {x\,\left (5\,{\mathrm {e}}^2+5\,\ln \left (\ln \relax (x)+1\right )+1\right )}{{\mathrm {e}}^2+\ln \left (\ln \relax (x)+1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.28, size = 14, normalized size = 0.82 \begin {gather*} 5 x + \frac {x}{\log {\left (\log {\relax (x )} + 1 \right )} + e^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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