Optimal. Leaf size=26 \[ \frac {\log \left (\frac {x}{2}\right )}{\log \left (\frac {4 \left (-1-\log \left (\frac {x}{e^4}\right )\right )}{\log (x)}\right )} \]
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Rubi [F]
time = 0.91, 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 {-\log \left (\frac {x}{2}\right ) \log (x)+\log \left (\frac {x}{2}\right ) \left (1+\log \left (\frac {x}{e^4}\right )\right )+\log (x) \left (1+\log \left (\frac {x}{e^4}\right )\right ) \log \left (\frac {-4-4 \log \left (\frac {x}{e^4}\right )}{\log (x)}\right )}{\log (x) \left (x+x \log \left (\frac {x}{e^4}\right )\right ) \log ^2\left (\frac {-4-4 \log \left (\frac {x}{e^4}\right )}{\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 {-\log \left (\frac {x}{2}\right ) \log (x)+\log \left (\frac {x}{2}\right ) \left (1+\log \left (\frac {x}{e^4}\right )\right )+\log (x) \left (1+\log \left (\frac {x}{e^4}\right )\right ) \log \left (\frac {-4-4 \log \left (\frac {x}{e^4}\right )}{\log (x)}\right )}{x \log (x) \left (1+\log \left (\frac {x}{e^4}\right )\right ) \log ^2\left (\frac {-4-4 \log \left (\frac {x}{e^4}\right )}{\log (x)}\right )} \, dx\\ &=\int \frac {-\log (8)-\log ^2(x) \log \left (-4+\frac {12}{\log (x)}\right )+3 \log (x) \left (1+\log \left (-4+\frac {12}{\log (x)}\right )\right )}{x (3-\log (x)) \log (x) \log ^2\left (-\frac {4 (-3+\log (x))}{\log (x)}\right )} \, dx\\ &=\text {Subst}\left (\int \frac {-3 x+\log (8)-3 x \log \left (-4+\frac {12}{x}\right )+x^2 \log \left (-4+\frac {12}{x}\right )}{(-3+x) x \log ^2\left (-\frac {4 (-3+x)}{x}\right )} \, dx,x,\log (x)\right )\\ &=\text {Subst}\left (\int \frac {3 x-\log (8)+3 x \log \left (-4+\frac {12}{x}\right )-x^2 \log \left (-4+\frac {12}{x}\right )}{(3-x) x \log ^2\left (-4+\frac {12}{x}\right )} \, dx,x,\log (x)\right )\\ &=\text {Subst}\left (\int \frac {3 x-\log (8)-(-3+x) x \log \left (-4+\frac {12}{x}\right )}{(3-x) x \log ^2\left (-4+\frac {12}{x}\right )} \, dx,x,\log (x)\right )\\ &=\text {Subst}\left (\int \left (\frac {-3 x+\log (8)}{(-3+x) x \log ^2\left (-4+\frac {12}{x}\right )}+\frac {1}{\log \left (-4+\frac {12}{x}\right )}\right ) \, dx,x,\log (x)\right )\\ &=\text {Subst}\left (\int \frac {-3 x+\log (8)}{(-3+x) x \log ^2\left (-4+\frac {12}{x}\right )} \, dx,x,\log (x)\right )+\text {Subst}\left (\int \frac {1}{\log \left (-4+\frac {12}{x}\right )} \, dx,x,\log (x)\right )\\ &=\text {Subst}\left (\int \frac {1}{\log \left (-4+\frac {12}{x}\right )} \, dx,x,\log (x)\right )+\text {Subst}\left (\int \frac {-3 x+\log (8)}{(-3+x) x \log ^2\left (\frac {12-4 x}{x}\right )} \, dx,x,\log (x)\right )\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.07, size = 24, normalized size = 0.92 \begin {gather*} \frac {-\log (8)+3 \log (x)}{3 \log \left (-4+\frac {12}{\log (x)}\right )} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
3.
time = 30.75, size = 275, normalized size = 10.58
method | result | size |
risch | \(-\frac {2 \ln \left (2\right )-2 \ln \left (x \right )}{2 \ln \left (2\right )+i \pi -2 \ln \left (\ln \left (x \right )\right )+2 \ln \left (-6 i+2 i \ln \left (x \right )\right )-i \pi \,\mathrm {csgn}\left (\frac {i}{\ln \left (x \right )}\right ) \mathrm {csgn}\left (i \left (-6 i+2 i \ln \left (x \right )\right )\right ) \mathrm {csgn}\left (\frac {i \left (-6 i+2 i \ln \left (x \right )\right )}{\ln \left (x \right )}\right )+i \pi \,\mathrm {csgn}\left (\frac {i}{\ln \left (x \right )}\right ) \mathrm {csgn}\left (\frac {i \left (-6 i+2 i \ln \left (x \right )\right )}{\ln \left (x \right )}\right )^{2}+i \pi \,\mathrm {csgn}\left (i \left (-6 i+2 i \ln \left (x \right )\right )\right ) \mathrm {csgn}\left (\frac {i \left (-6 i+2 i \ln \left (x \right )\right )}{\ln \left (x \right )}\right )^{2}-i \pi \mathrm {csgn}\left (\frac {i \left (-6 i+2 i \ln \left (x \right )\right )}{\ln \left (x \right )}\right )^{3}+i \pi \,\mathrm {csgn}\left (\frac {i \left (-6 i+2 i \ln \left (x \right )\right )}{\ln \left (x \right )}\right ) \mathrm {csgn}\left (\frac {-6 i+2 i \ln \left (x \right )}{\ln \left (x \right )}\right )^{2}-i \pi \,\mathrm {csgn}\left (\frac {i \left (-6 i+2 i \ln \left (x \right )\right )}{\ln \left (x \right )}\right ) \mathrm {csgn}\left (\frac {-6 i+2 i \ln \left (x \right )}{\ln \left (x \right )}\right )+i \pi \mathrm {csgn}\left (\frac {-6 i+2 i \ln \left (x \right )}{\ln \left (x \right )}\right )^{3}-i \pi \mathrm {csgn}\left (\frac {-6 i+2 i \ln \left (x \right )}{\ln \left (x \right )}\right )^{2}}\) | \(275\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [C] Result contains complex when optimal does not.
time = 0.55, size = 29, normalized size = 1.12 \begin {gather*} -\frac {\log \left (2\right ) - \log \left (x\right )}{i \, \pi + 2 \, \log \left (2\right ) + \log \left (\log \left (x\right ) - 3\right ) - \log \left (\log \left (x\right )\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.35, size = 34, normalized size = 1.31 \begin {gather*} -\frac {\log \left (2\right ) - \log \left (x e^{\left (-4\right )}\right ) - 4}{\log \left (-\frac {4 \, {\left (\log \left (x e^{\left (-4\right )}\right ) + 1\right )}}{\log \left (x e^{\left (-4\right )}\right ) + 4}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.08, size = 17, normalized size = 0.65 \begin {gather*} \frac {\log {\left (x \right )} - \log {\left (2 \right )}}{\log {\left (\frac {12 - 4 \log {\left (x \right )}}{\log {\left (x \right )}} \right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 55 vs.
\(2 (21) = 42\).
time = 0.38, size = 55, normalized size = 2.12 \begin {gather*} -\frac {\frac {{\left (\log \left (x\right ) - 3\right )} \log \left (2\right )}{\log \left (x\right )} - \log \left (2\right ) + 3}{\frac {{\left (\log \left (x\right ) - 3\right )} \log \left (-\frac {4 \, {\left (\log \left (x\right ) - 3\right )}}{\log \left (x\right )}\right )}{\log \left (x\right )} - \log \left (-\frac {4 \, {\left (\log \left (x\right ) - 3\right )}}{\log \left (x\right )}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.28, size = 20, normalized size = 0.77 \begin {gather*} \frac {\ln \left (\frac {x}{2}\right )}{\ln \left (-\frac {4\,\ln \left (x\right )-12}{\ln \left (x\right )}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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