Optimal. Leaf size=23 \[ \log \left (\frac {x}{e^{\left (x+\frac {x}{x+\log (\log (x))}\right )^2}+\log (x)}\right ) \]
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Rubi [F]
time = 16.99, 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 {-x^3 \log (x)+x^3 \log ^2(x)+\left (-3 x^2 \log (x)+3 x^2 \log ^2(x)\right ) \log (\log (x))+\left (-3 x \log (x)+3 x \log ^2(x)\right ) \log ^2(\log (x))+\left (-\log (x)+\log ^2(x)\right ) \log ^3(\log (x))+\exp \left (\frac {x^2+2 x^3+x^4+\left (2 x^2+2 x^3\right ) \log (\log (x))+x^2 \log ^2(\log (x))}{x^2+2 x \log (\log (x))+\log ^2(\log (x))}\right ) \left (2 x^2+2 x^3+\left (x^3-2 x^4-2 x^5\right ) \log (x)+\left (2 x^2+\left (x^2-6 x^3-6 x^4\right ) \log (x)\right ) \log (\log (x))+\left (3 x-4 x^2-6 x^3\right ) \log (x) \log ^2(\log (x))+\left (1-2 x^2\right ) \log (x) \log ^3(\log (x))\right )}{x^4 \log ^2(x)+3 x^3 \log ^2(x) \log (\log (x))+3 x^2 \log ^2(x) \log ^2(\log (x))+x \log ^2(x) \log ^3(\log (x))+\exp \left (\frac {x^2+2 x^3+x^4+\left (2 x^2+2 x^3\right ) \log (\log (x))+x^2 \log ^2(\log (x))}{x^2+2 x \log (\log (x))+\log ^2(\log (x))}\right ) \left (x^4 \log (x)+3 x^3 \log (x) \log (\log (x))+3 x^2 \log (x) \log ^2(\log (x))+x \log (x) \log ^3(\log (x))\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 {-x^3 \log (x)+x^3 \log ^2(x)+3 x^2 (-1+\log (x)) \log (x) \log (\log (x))+3 x (-1+\log (x)) \log (x) \log ^2(\log (x))+(-1+\log (x)) \log (x) \log ^3(\log (x))+\exp \left (\frac {x^2 (1+x+\log (\log (x)))^2}{(x+\log (\log (x)))^2}\right ) \left (2 x^2 (1+x+\log (\log (x)))+\log (x) \left (x^3-2 x^4-2 x^5+\left (x^2-6 x^3-6 x^4\right ) \log (\log (x))+x \left (3-4 x-6 x^2\right ) \log ^2(\log (x))+\left (1-2 x^2\right ) \log ^3(\log (x))\right )\right )}{x \log (x) \left (\exp \left (\frac {x^2 (1+x+\log (\log (x)))^2}{(x+\log (\log (x)))^2}\right )+\log (x)\right ) (x+\log (\log (x)))^3} \, dx\\ &=\int \left (\frac {2 x^2+2 x^3+x^3 \log (x)-2 x^4 \log (x)-2 x^5 \log (x)+2 x^2 \log (\log (x))+x^2 \log (x) \log (\log (x))-6 x^3 \log (x) \log (\log (x))-6 x^4 \log (x) \log (\log (x))+3 x \log (x) \log ^2(\log (x))-4 x^2 \log (x) \log ^2(\log (x))-6 x^3 \log (x) \log ^2(\log (x))+\log (x) \log ^3(\log (x))-2 x^2 \log (x) \log ^3(\log (x))}{x \log (x) (x+\log (\log (x)))^3}+\frac {-2 x^2-3 x^3+2 x^4 \log (x)+2 x^5 \log (x)-5 x^2 \log (\log (x))+2 x^2 \log (x) \log (\log (x))+6 x^3 \log (x) \log (\log (x))+6 x^4 \log (x) \log (\log (x))-3 x \log ^2(\log (x))+4 x^2 \log (x) \log ^2(\log (x))+6 x^3 \log (x) \log ^2(\log (x))-\log ^3(\log (x))+2 x^2 \log (x) \log ^3(\log (x))}{x \left (\exp \left (\frac {x^2 (1+x+\log (\log (x)))^2}{(x+\log (\log (x)))^2}\right )+\log (x)\right ) (x+\log (\log (x)))^3}\right ) \, dx\\ &=\int \frac {2 x^2+2 x^3+x^3 \log (x)-2 x^4 \log (x)-2 x^5 \log (x)+2 x^2 \log (\log (x))+x^2 \log (x) \log (\log (x))-6 x^3 \log (x) \log (\log (x))-6 x^4 \log (x) \log (\log (x))+3 x \log (x) \log ^2(\log (x))-4 x^2 \log (x) \log ^2(\log (x))-6 x^3 \log (x) \log ^2(\log (x))+\log (x) \log ^3(\log (x))-2 x^2 \log (x) \log ^3(\log (x))}{x \log (x) (x+\log (\log (x)))^3} \, dx+\int \frac {-2 x^2-3 x^3+2 x^4 \log (x)+2 x^5 \log (x)-5 x^2 \log (\log (x))+2 x^2 \log (x) \log (\log (x))+6 x^3 \log (x) \log (\log (x))+6 x^4 \log (x) \log (\log (x))-3 x \log ^2(\log (x))+4 x^2 \log (x) \log ^2(\log (x))+6 x^3 \log (x) \log ^2(\log (x))-\log ^3(\log (x))+2 x^2 \log (x) \log ^3(\log (x))}{x \left (\exp \left (\frac {x^2 (1+x+\log (\log (x)))^2}{(x+\log (\log (x)))^2}\right )+\log (x)\right ) (x+\log (\log (x)))^3} \, dx\\ &=\int \frac {-x^2 (2+3 x)-5 x^2 \log (\log (x))-3 x \log ^2(\log (x))-\log ^3(\log (x))+2 x^2 \log (x) \left (x^2 (1+x)+\left (1+3 x+3 x^2\right ) \log (\log (x))+(2+3 x) \log ^2(\log (x))+\log ^3(\log (x))\right )}{x \left (\exp \left (\frac {x^2 (1+x+\log (\log (x)))^2}{(x+\log (\log (x)))^2}\right )+\log (x)\right ) (x+\log (\log (x)))^3} \, dx+\int \frac {2 x^2 (1+x+\log (\log (x)))+\log (x) \left (x^3-2 x^4-2 x^5+\left (x^2-6 x^3-6 x^4\right ) \log (\log (x))+x \left (3-4 x-6 x^2\right ) \log ^2(\log (x))+\left (1-2 x^2\right ) \log ^3(\log (x))\right )}{x \log (x) (x+\log (\log (x)))^3} \, dx\\ &=\text {Rest of rules removed due to large latex content} \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.26, size = 38, normalized size = 1.65 \begin {gather*} \log (x)-\log \left (e^{x^2+\frac {x^2}{(x+\log (\log (x)))^2}+\frac {2 x^2}{x+\log (\log (x))}}+\log (x)\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(110\) vs.
\(2(22)=44\).
time = 0.08, size = 111, normalized size = 4.83
method | result | size |
risch | \(\ln \left (x \right )-x^{2}-\frac {\left (2 x +2 \ln \left (\ln \left (x \right )\right )+1\right ) x^{2}}{\left (\ln \left (\ln \left (x \right )\right )+x \right )^{2}}+\frac {x^{2} \ln \left (\ln \left (x \right )\right )^{2}+\left (2 x^{3}+2 x^{2}\right ) \ln \left (\ln \left (x \right )\right )+x^{4}+2 x^{3}+x^{2}}{\ln \left (\ln \left (x \right )\right )^{2}+2 x \ln \left (\ln \left (x \right )\right )+x^{2}}-\ln \left (\ln \left (x \right )+{\mathrm e}^{\frac {x^{2} \left (\ln \left (\ln \left (x \right )\right )+x +1\right )^{2}}{\left (\ln \left (\ln \left (x \right )\right )+x \right )^{2}}}\right )\) | \(111\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 124 vs.
\(2 (22) = 44\).
time = 0.60, size = 124, normalized size = 5.39 \begin {gather*} -\frac {x^{3} + x^{2} \log \left (\log \left (x\right )\right ) + 2 \, x^{2} + 2 \, x}{x + \log \left (\log \left (x\right )\right )} - \log \left ({\left (\log \left (x\right )^{\frac {2}{x + \log \left (\log \left (x\right )\right )}} \log \left (x\right )^{3} + e^{\left (x^{2} + 2 \, x + \frac {\log \left (\log \left (x\right )\right )^{2}}{x^{2} + 2 \, x \log \left (\log \left (x\right )\right ) + \log \left (\log \left (x\right )\right )^{2}} + \frac {2 \, \log \left (\log \left (x\right )\right )^{2}}{x + \log \left (\log \left (x\right )\right )} + 1\right )}\right )} e^{\left (-x^{2} - 2 \, x - \frac {2 \, \log \left (\log \left (x\right )\right )^{2}}{x + \log \left (\log \left (x\right )\right )} - 1\right )}\right ) + \log \left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 61 vs.
\(2 (22) = 44\).
time = 0.36, size = 61, normalized size = 2.65 \begin {gather*} \log \left (x\right ) - \log \left (e^{\left (\frac {x^{4} + x^{2} \log \left (\log \left (x\right )\right )^{2} + 2 \, x^{3} + x^{2} + 2 \, {\left (x^{3} + x^{2}\right )} \log \left (\log \left (x\right )\right )}{x^{2} + 2 \, x \log \left (\log \left (x\right )\right ) + \log \left (\log \left (x\right )\right )^{2}}\right )} + \log \left (x\right )\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 65 vs.
\(2 (19) = 38\).
time = 1.61, size = 65, normalized size = 2.83 \begin {gather*} \log {\left (x \right )} - \log {\left (e^{\frac {x^{4} + 2 x^{3} + x^{2} \log {\left (\log {\left (x \right )} \right )}^{2} + x^{2} + \left (2 x^{3} + 2 x^{2}\right ) \log {\left (\log {\left (x \right )} \right )}}{x^{2} + 2 x \log {\left (\log {\left (x \right )} \right )} + \log {\left (\log {\left (x \right )} \right )}^{2}}} + \log {\left (x \right )} \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 2.26, size = 156, normalized size = 6.78 \begin {gather*} \ln \left (x\right )-\ln \left (\ln \left (x\right )+{\mathrm {e}}^{\frac {x^2}{x^2+2\,x\,\ln \left (\ln \left (x\right )\right )+{\ln \left (\ln \left (x\right )\right )}^2}}\,{\mathrm {e}}^{\frac {x^4}{x^2+2\,x\,\ln \left (\ln \left (x\right )\right )+{\ln \left (\ln \left (x\right )\right )}^2}}\,{\mathrm {e}}^{\frac {2\,x^3}{x^2+2\,x\,\ln \left (\ln \left (x\right )\right )+{\ln \left (\ln \left (x\right )\right )}^2}}\,{\mathrm {e}}^{\frac {2\,x^2\,\ln \left (\ln \left (x\right )\right )}{x^2+2\,x\,\ln \left (\ln \left (x\right )\right )+{\ln \left (\ln \left (x\right )\right )}^2}}\,{\mathrm {e}}^{\frac {2\,x^3\,\ln \left (\ln \left (x\right )\right )}{x^2+2\,x\,\ln \left (\ln \left (x\right )\right )+{\ln \left (\ln \left (x\right )\right )}^2}}\,{\mathrm {e}}^{\frac {x^2\,{\ln \left (\ln \left (x\right )\right )}^2}{x^2+2\,x\,\ln \left (\ln \left (x\right )\right )+{\ln \left (\ln \left (x\right )\right )}^2}}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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